Five years' Ph.d study coming to an end, many people have helped me with this project in one way or another, mathematically or not. Here I want to express my further gratitude to some of them.

First of all, I would like to thank one of my advisor Professor Qiu Weiyuan. During the three years in Fudan University, I was accompanied by his strict requirements, harsh criticism, frank conversation and constant encouragement. He led me into the world of mathematics research. He taught me how to read the articles, how to report the articles, how to think the problems and how to write articles. These to me are a valuable asset. As his student, I am feeling quietly warm and deep happiness all the time. The efforts and time he poured to me and his tireless teachings will encourage me all my life.

I am extremely grateful to my other advisor Professor Tan Lei. Because of her limitedless care and guidance, I was able to successfully complete this two-year study program in France. She takes good care of my basic needs. She patiently explained to me the mathematical concepts and theorems; she unreservedly share her rich life and scientific research experience with me. she kindly recommended various academic conferences to me. She gave very effective guidances on the topic of my Ph.D thesis, the research methods and the research directions. I will never forget her strong support and enthusiasm help on my life and work.

I want to thank Professor Yin Yongcheng. He taught me complex analysis knowledge, which lay a solid foundation for my entering the study of complex dynamic systems.

In the discussion sessions and daily communication, professor Yin demonstrated the unique charisma, humility passionate quality, rigorous academic style and profound academic knowledge gave me a deep and unforgettable memories, and will guide my lifestyle.

I want to thank Cui Guizhen for his patient guidance, generous assitance and invaluable advice, all of which have been of inestimable worth to the completion of my thesis.

I would like to offer my thanks to professor Ren Fuyao for his help. He has too much quality worthy of our young people to learn: optimistic, open-minded, healthy, positive, academic heart. He gave me a beautiful interpretation of the essence of mathematics, happy life.

Thank Professor Chen Jixiu for his precious advice in the seminars. Thank Professor Liu Lixin for his encouragements.

iii Many thanks go to Peter Haïssinsky. He has carefully read the total draft of the thesis and gave a lot of useful suggestion.

I also owe my sincere gratitude to my mathematical brothers and sisters: Su Weixu, Xiao yingqing, Wang Xiaoguang, Yang Jinghua, Yang Fei, Gao Yan, Xie Lan, Fang caiyun, Pan Hua, Zhu Guanzhou for their supports and encouragements.

I would also like to thank Li Ming, Shi Guanghua, Hou Yu, Lin Kui, Hu Long, Chen Xin, Fu Hua Zhang, Jin Yan, Chen Yuning, and many others friends for their essential helps.

I would like to appreciate the China Scholarship Council for support during the two years' studying in France.

Finally, my deepest gratitude goes to my parents for their understanding and support.

Chapter 0 Introduction 0.1 Quasisymmetric rigidity of F n,p

The quasisymmetric geometry of Sierpiński carpets is related to the study of Julia sets in complex dynamics and boundaries of Gromov hyperbolic groups.

Let S 2 be the unit sphere in R 3 . Let S = S 2 \ S i≥1 D i , where

D i , i = 1, 2, • • • , are open Jordan domains with D i T D j = ∅ for distinct i, j. S is called a (Sierpiński ) carpet if S
has empty interior and the spherical diameter diam(D i ) → 0 as i → ∞. For each i, ∂D i is called a peripheral circle of S. S is called a round carpet if each D i is a round disk. Topologically all carpets are the same [W58]. Much richer structure arises if we consider quasisymmetric geometry of metric carpets. The famous counjecture of Kapovich-Kleiner predicts that if G is a hyperbolic group with boundary ∂ ∞ G homeomorphic to a carpet, then G acts geometrically on a convex subset of H 3 with non-empty totally geodesic boundary. The Kapovich-Kleiner conjecture is equivalent to the conjecture that the carpet ∂ ∞ G, endowed with a visual metric, is quasisymetriclly equivalent to a round carpet on S 2 . The conjecture is true for carpets that can be quasisymmetrically embedding into S 2 .

Let us recall the concept of quasisymetric map between metric spaces defined by Tukia and Väisälä [TV80]. Let f : X → Y be a homeomorphism between two metric spaces

(X, d X ) and (Y, d Y ). f is quasisymmetric if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that d Y (f (x), f (y)) d Y (f (x), f (z)) ≤ η d X (x, y) d X (x, z)
, ∀ x, y, z ∈ X, x = z.

Let QS(X) be the group of quasisymmetric self-homeomorphisms of X. Let S p , p ≥ 3 odd, be the standard 1/p-Sierpiński carpets on the Euclidean plane. In [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF], Bonk and Merenkov proved that for each S p , p ≥ 3 odd, QS(S p ) is finite dihedral group. They further conjectured that, for any p ≥ 3 odd, QS(S p ) is a Euclidean isometry group. When p = 3, they has showed that the conjecture is true. They also established that a rigidity theorem that S p and S q are quasisymmetrically equivalent if and only if p = q.

The aim of Chapter 1 is to extend Bonk-Merenkov's results to a new class of Sierpiński carpets F n,p (see the definition below). In particular, we are able to prove that the group of quasisymmetric self-homeomorphisms of F n,p is a Euclidean isometry group.

Let 5 ≤ n, 1 ≤ p < n 2 -1 be integers. Let Q (0) n,p = [0, 1] × [0, 1] be the closed unit square in R 2 . We first subdivide Q (0) n,p into n 2 subsquares with equal side-length 1/n and remove the interior of four subsquares, each has side-length 1/n and is of distance √ 2p/n to one of the four corner points of Q (0) n,p . The resulting set Q (1) n,p consists of (n 2 -4) squares of side-length 1/n. Inductively,

Q (k+1) n,p , k ≥ 1, is obtained from Q (k)
n,p by subdividing each of the remaining squares in the subdivision of Q (k) n,p into n 2 subsquares of equal side-length 1/n k+1 and removing the interior of four subsquares as we have done above.

The Spierpiński carpet F n,p is the intersection of all sets Q (k) n,p , i.e.,

F n,p = +∞ \ k=0 Q (k) n,p .
See Figure 1. In Chapter 1, we improve the method in Bonk and Merenkov and prove the following theorem, Theorem 0.1. Let n ≥ 5, 1 ≤ p < n 2 -1 be integers. Then

• the carpet F n,p is not quasisymetrically equivalent to any standard Sierpiński carpet S m , m ≥ 3 odd;

• QS(F n,p ) = Isom(F n,p ), that is, every quasisymmetric self-homeomorphism of F n,p is a Euclidean isometry;

• F n,p and F n ,p are quasisymmetrically equivalent if and only if (n, p) = (n , p ).

Sierpiński carpets arising as Julia sets of rational maps

The first example of Sierpiński carpet as Julia set of a rational map, called carpet Julia set, was discovered by Tan Lei [START_REF] Milnor | Geometry and dynamics of quadratic rational maps, with an appendix[END_REF]. Later, carpet Julia sets appeared in many literatures including examples of McMullen maps, generated McMullen maps and quadratic rational maps, etc. [START_REF] Devaney | Sierpiński curve Julia sets for quadratic rational maps[END_REF] [DLU05] [START_REF] Qiu | Fatou components and Julia sets of singularly perturbed rational maps with positive parameter[END_REF].

There are two natural questions:

(Q 1 ) whether any two carpet Julia sets are quasisymmetrically equivalent? (Q 2 ) whether any carpet Julia set J f is quasisymmetrically equivalent to a round carpet?

Let X be a metric space. The conformal dimension of X is the infimum of the Hausdorff dimensions of all metric spaces which are quasisymmetrically equivalent to X. For the question (Q 1 ), Haïssinsky and Pilgrim [START_REF] Haïssinsky | Quasisymmetrically inequivalent hyperbolic Julia sets[END_REF] constructed a sequence of hyperbolic rational maps with carpet Julia sets such that their conformal dimensions tend to two. This means there are infinitely many quasisymmetrically inequivalent carpet Julia sets.

For the question (Q 2 ), Bonk gave a sufficient condition on which the carpet in C = C ∪ {∞} is quasisymmetrically equivalent to a round carpet. The relative distance ∆(A, B) of two subsets A and B in C is defined as ∆(A, B) := dist(A, B) min{diam(A), diam(B)} in the spherical metric. A set of Jordan curves C = {γ i } i∈N in C is called uniformly relatively separated if their pairwise relative distances are uniformly bounded away from zero, that is, there exists δ > 0 such that ∆(γ i , γ j ) ≥ δ for every distinct i, j. The set C are uniform quasicircles if there exists K ≥ 1 such that each γ i in C is a K-quasicircle.

Bonk proved that, if the peripheral circles of a carpet S are uniformly relatively separated and are uniform quasicircles, then S is quasisymmetrically equivalent to a round carpet. Recently, Bonk, Lyubich and Merenkov studied the carpet Julia set J f generated by a postcritically-finite rational map f [START_REF] Bonk | Quasisymmetries of Sierpiński carpet Julia sets[END_REF]. They showed that J f is quasisymmetrically equivalent to a round carpet. They also proved that the group QS(J f ) is a finite group, whose elements are restrictions of Möbius transformations on J f .

In Chapter 2, we study carpet Julia sets in the case of postcritically-infinite rational maps.

The ω-limit set ω(x) of a point x ∈ C under a rational map f is defined as the set of accumulation points of the orbit of x. Obviously f (ω(x)) ⊆ ω(x). A critical point c of f is called recurrent if c ∈ ω(c). A rational map f is called semi-hyperbolic if J f contains neither parabolic periodic points nor recurrent critical points.

We prove the following theorem in Chapter 2.

Theorem 0.2. Let f be a rational map whose Julia set J f is a Sierpiński carpet. Let C f be the set of all peripheral circles of this carpet. Then

(1) If elements in C f avoid the ω-limit sets of all critical points, then • J f is quasisymmetrically equivalent to a round carpet;

• QS(J f ) is discrete.

(2) If f is semi-hyperbolic, then • elements in C f are uniform quasicircles;

• C f are uniformly relatively separated if and only if elements in C f are disjoint with the w-limit sets of all critical points.

Criterion for rays landing together

Let f be a polynomials with degree d ≥ 2. If J f is connected, then the basin of infinity Ω f , which consists of points with the orbit attracted by ∞, is simply connected. Moreover, there exists a unique holomorphic parameterization

Ψ f : Ω f → C \ D such that Ψ f (∞) = ∞, Ψ f (∞) = 1 and Ψ f • f (z) = (Ψ f (z)) d .
Define R(θ) := Ψ -1 f {re 2πiθ : r > 1} to be external ray with the angle θ. We say that R(θ) lands at z ∈ J f if lim r→1 Ψ -1 f (re 2πiθ ) = z. By a theorem of Carathéodory, Ψ -1 f extends continuously to ∂D with f (∂D) = J f if and only if J f is locally connected. Throughout this thesis we only consider the case that J f is connected and locally connected.

Define α : R/Z → J f , θ → α(θ) where α(θ) is the landing point of ray R(θ). We have the following semi-conjugation,

f (α(θ)) = α(σ d (θ)),
where σ d : R/Z → R/Z is given by θ → dθ mod Z. Thus, in order to understand the topology of the Julia set and the dynamics of f on J f , it is necessary to figure out the semi-conjugation α.

Chapter 3 is devoted to give an answer to the following questions.

(Q 1 ) For any z in J f , is the fiber α -1 (z) finite?

(Q 2 ) Give a condition under which θ, θ are in the same fiber? For the first question, if the orbit of z is finite, the fiber α -1 (z) is finite [DH84]. If z is wandering, Kiwi gave an upper bound #α -1 (z) ≤ 2 d [Ki02]. We estimate the cardinal number in a more general case: a finite collection of wandering points with disjoint forward orbits. We give a sharp upper bound in Theorem 3.1, which is the same as that in Blokh and Levin's work [BL02], by using a totally different argument.

For the second question, following [BFH92] [Po93] [Ki05], we need the concept of critical portrait of f .

• For a critical point c in J f , let Θ(c) be the set of angles of external rays landing at c such that σ d maps Θ(c) onto exactly one angle and the external ray with this angle landing at f (c).

• For a strictly pre-periodic critical Fatou component U , let Θ(U ) be the collection of deg(f | U ) angles of external rays supporting U such that all of them are pre-image of some external ray supporting f (U ).

• For a cycle of critical Fatou component

U 0 , • • • , U p-1 with f i (U 0 ) = U i , i = 1, • • • , p and U p := U 0 , let U k 0 , • • • , U k l , 0 ≤ k 0 < • • • < k l ≤ p -1 be critical with degree n 0 , • • • , n l . For 0 ≤ i ≤ p, choose (z i , θ i ), z i ∈ ∂U i and R(θ i ) supporting U i at z i such that f i (z 0 ) = z i , f i (R(θ 0 )) = R(θ i
) and f p (z p ) = z p . Let Θ(U k j ) be the set of angles of external rays landing on ∂U k j which are inverse images of R(θ k j +1 ), for 1 ≤ j ≤ l.

Let A := {Θ(c 1 ),

• • • , Θ(c m ), Θ(U 1 ), • • • , Θ(U n )}, where {c 1 , • • • , c m } and {U 1 , • • • , U n }
are the set of critical points in J f and critical Fatou components, respectively. For any

Θ ∈ A, let Ò Θ := [ {Θ : ∃ a chain Θ 0 , • • • , Θ k = Θ in A such that Θ i \ Θ i+1 = ∅}.

The collection Ò

A := { Ò Θ 1 , • • • , Ò Θ N } is called critical portrait of f . A simple case is that f is a polynomial with J f locally connected and all cycles repelling. In this case the external rays with angles in Θ i are landing at a same critical point, and f maps these external rays to exact one external ray.

Let P := {I 1 , • • • , I d } be a partition of the unit circle, where each I i is a finite union of open intervals in R/Z\ S 1≤i≤N Ò Θ i with total length 1/d (See Section 3.5 for details). We say that θ, θ ∈ R/Z have the same sequence (itinerary) with respect to

P if σ k d (θ), σ k d (θ ) ∈ I i k for all k ≥ 0.
Biefield, Fisher and Hubbard showed that, for polynomials with all critical points strictly preperiodic, if θ, θ have the same sequence with respect to P, then α(θ) = α(θ ) [BFH92]. Poirier extended the above result to critical finite polynomials that admit periodic Fatou components [Po93]. In [Ki05], Kiwi considered polynomials with all cycles repelling and Julia set connected. He proved that if θ, θ have the same sequence with respect to P, then the impressions of R(θ) and R(θ ) intersect.

The following result is our main result in Chapter 3. Theorem 0.3. Let f be a polynomial with J f connected and locally connected. Let P be the partition induced by a critical portrait Ò A of f . If θ, θ have the same sequence with respect to P, then either R(θ), R(θ ) land at the same point or R(θ), R(θ ) land at the boundary of a Fatou component U , which is eventually iterated to a Siegel disk.

One of our motivation of the above study is to understand the core-entropy of polynomials, which was first introduced and explored by Thurston. The core-entropy of polynomial f is the topological entropy of f on its f -invariant set, the Hubbard tree. Let Acc(f ) be the set of all biaccessible angles θ, i.e., there exist at least two rays landing at α(θ). Then the core-entropy h(f ) is related to the Hausdorff dimension of Acc(f ) by the following formula:

h(f ) = log d • H.dimAcc(f ).
For more results on the core-entropy, we refer to [Do95]

[Ti13] [Ti14] [Ga13] [Li07] [Ju13].
As an application of Theorem 0.3, we prove the monotonicity of core-entropy for a family of quadratic polynomials {f c : z → z 2 + c, f c has no Siegel disks and J fc is locally connected } (see Theorem 3.3). This generalizes Tao Li's result on critical finite quadratic polynomials [Li07].

A landing theorem on non-recurrent polynomials

Let f be a polynomial with degree d ≥ 2. Let Ω f be basin of infinity consisting of points in C escaping to ∞. Denote the filled Julia set by K f := C \ Ω f . There exists a Green function G f that measures the escape rate of points to ∞, defined by

G f : C → [0, ∞), z → lim n→∞ log |f n (z)| d n .
Note that G f is positive and harmonic in Ω f . The derivative of G f vanishes at z if and only if z is a (pre-)critical point. Each locus G -1 f (r) = {z ∈ C, G f (z) = r} with r > 0 is called an equipotential curve around the filled Julia set K f .

There exists an unique normalized Böttcher map Ψ f which conjugates f with z → z d in a neighborhood of ∞. Ψ -1 f has an unique maximal radial extension to a subset of C\D. This radial extension terminates at a point w with |w| > 1 if and only if Ψ -1 f extends continuously to w and Ψ -1 f (w) is a (pre-)critical point of f . Then external radius R(t) with angle t is given by

R(t) := Ψ -1 f ((r t , ∞)e 2πit ),
where Ψ -1 f (r t e 2πit ) is a (pre-)critical point of f if t t > 1. If r t = 1, then R(t) is exactly the external ray defined in Section 0.3. Let P d be the set of monic centered polynomials of degree d. The shift locus S d is the subset of P d formed by polynomials with all critical points escaping to infinity. Let S d (r), r > 0, consist of polynomials f ∈ S d such that all critical points of f are in the same equipotential curve G -1 f (r) and let S d := S r>0 S d (r).

A collection Θ = {Θ 1 , • • • , Θ n } of finite subsets of R/Z is called an (abstract) critical portrait of degree d if the following conditions hold.

(1) For every j, |Θ j | ≥ 2 and |σ d (Θ j )| = 1, where σ d : θ → dθ mod Z and |A| denotes the cardinal number of the set A.

(2) Θ 1 , • • • , Θ n are pairwise unlinked.

(3)

P (|Θ j | -1) = d -1.
For another critical portrait Θ = {Θ 1 , • • • , Θ n }, we say Θ = Θ if there exists a permutation τ such that Θ i = Θ τ (i) for 1 ≤ i ≤ d. Let A d be the collection of all critical portraits of degree d. In [Ki05], Kiwi gave A d a compact-unlinked topology and proved that A d is compact and connected. Critical portraits of polynomials with connected Julia sets defined in Section 0.3 are obviously critical portraits defined here. Now we consider the map Π : S d → A d defined as following. Given any f ∈ S d , let {c j } i=1,••• ,n be set of critical points of f . For each c j , j = 1, • • • , n, there are exact deg f (c j ) external radius terminating at c j . Let Θ j be the set of angles of these external radius. Then Θ = {Θ 1 , • • • , Θ n } is a critical portrait in A d . We set Π(f ) := Θ.

In [START_REF] Goldberg | On the multiplier of a repelling fixed point[END_REF], Goldberg proved that Π is surjective. Kiwi showed that Π is continuous and, for any Θ ∈ A d , the preimage S Θ = Π -1 (Θ) is a one-dimensional real analytic manifold. Precisely, the map G : S Θ → (0, ∞) which sends f to G f (c i ) is bijective and analytic. Moreover, given r > 0, the restriction Π| S d (r) : S d (r) → A d is a homeomorphism [Ki05].

The connected locus C d is the set of monic centered polynomials with degree d such that all the critical orbits are bounded. We know that C d is a compact and connected subset of P d [BH88]. For instance, C 2 is the Mandelbrot set. To describe C d we look at it from outside S d .

The impression I C d (Θ) of a critical portrait Θ is a subset of C d , characterized by the property that f ∈ I C d (Θ) if and only if there exists a sequence {f n } in S d such that Π(f n ) = Θ and f n converges to f . Note that the impression here is slightly different from the definition in [Ki05], which is bigger and containing I C d (Θ) there. Kiwi proved that if all angles in Θ is strictly preperiodic, then the impression I C d (Θ) is a singleton [Ki05]. He conjectured that there exist critical portraits with aperiodic kneadings and non-trivial impressions.

In Chapter 4, we give an elementary proof of the following theorem based on the tools developed in [CT15].

Theorem 0.4. (1) The map

P : A d × (0, ∞) → S d , (Θ, r) → f Θ,r ,
such that f Θ,r ∈ S d (r) induces the critical portrait Θ, is well-defined, one-to-one and continuous.

(2) Let R Θ : (0, ∞) → S d , R Θ (t) = P (Θ, t). Then R Θ (t) forms a simple curve in S d . (R Θ is called a parameter ray with the angle Θ). Let f be a polynomial in C d with no recurrent critical points and all cycles repelling. Then R Θ (t) lands at f if and only if Θ is a critical portrait of f .

On the dynamics of a family of generated renormalization transformations

The statistical mechanical models on hierarchical lattices have attracted many interests recently since they exhibit a deep connection between their limiting sets of the zeros of the partition functions and the Julia sets of rational maps in complex dynamics [BL91,DSI83, Qi11, QL01, QYG10]. The well-known Yang-Lee theorem in statistical mechanics shows that the zeros of the partition function is dense in a line for many magnetic materials in a complex magnetic field plane. This means that the complex singularities of the free energy lie on this line, where the free energy is the logarithm of the partition function [START_REF] Yang | Statistical theory of equations of state and phase transitions. I. Theory of condensation[END_REF]. By the works of Fisher and others [START_REF] Fisher | The Nature of Critical Points[END_REF], it was generally believed that the zeros of the partition function condense to some simple curve.

Until 1983, Derrida et al. showed that the zeros of the partition function condense to the Julia set of the renormalization transformation of so-called standard hierarchical lattices [START_REF] Derrida | Fractal structure of zeros in hierarchinal models[END_REF]. They proved that the singularities of the free energy lie on the Julia set of the rational map

z → z 2 + λ -1 2z + λ -2 2 .
This means that the distribution of the singularities of the free energy is not as simple as one desired. For the ideas formulated in renormalization transformation in statistical mechanics, see [START_REF] Wilson | Renormalization group and critical phenomena I and II[END_REF].

Recently, Qiao considered the generalized diamond hierarchical Potts model and proved that the family of rational maps U mnλ (z) = (z + λ -1) m + (λ -1)(z -1) m (z + λ -1) m -(z -1) m n are actually the renormalization transformation of the generalized diamond hierarchical We prove the following theorem in Chapter 5.

Theorem 0.5. Let d ≥ 2 be integer and λ ∈ C * . Denote by J dλ the Julia set of U dλ .

Then

(1) J dλ is connected.

(2) If λ ∈ R, then J dλ is not a Sierpiński carpet.

(3) For sufficiently large |λ|, J dλ is a quasicircle and the Hausdorff dimension of J dλ is given by

dim H (J dλ ) = 1 + 1 4 log d |λ| -2 d+1 + O(λ -3 d+1 ).
(4) The non-escaping locus M d is connected.

Chapter 1

Quasisymmetric rigidity of Sierpiński carpets F n,p

Introduction

The quasisymmetric geometry of Sierpiński carpets is related to the study of Julia sets in complex dynamics and boundaries of Gromov hyperbolic groups. For background and research progress, we recommend the survey of M. Bonk [START_REF] Bonk | Quasiconformal geometry of fractals[END_REF].

Let S 2 be the unit sphere in R 3 . Let S = S 2 \ S i∈N D i be the complement in S 2 of countably many pair-wise disjoint open Jordan regions D i ⊂ S 2 . S is called a (Sierpiński) carpet if S has empty interior, diam (D i ) → 0 as i → ∞, and ∂D i ∩ ∂D j = ∅ for all i = j. The boundary of D i , denoted by C i , is called a peripheral circle of S. A round carpet is a carpet on S 2 such that all of its peripheral circles are geometric circles. Typical Examples of round carpets are limit sets of convex co-compact Kleinian groups.

Topologically all carpets are the same [W58]. Much richer structure arises if we consider quasisymmetric geometry of metric carpets. The famous conjecture of Kapovich-Kleiner [START_REF] Kapovich | Hyperbolic groups with low-dimensional boundary[END_REF] predicts that if G is a hyperbolic group with boundary ∂ ∞ G homeomorphic to a Sierpiński carpet, then G acts geometrically (the action is isometrical, properly discontinuous and co-compact) on a convex subset of H 3 with non-empty totally geodesic boundary. The Kapovich-Kleiner conjecture is equivalent to the conjecture that the carpet ∂ ∞ G (endowed with the "visual" metric) is quasisymetriclly equivalent to a round carpet on S 2 . The conjecture is true for carpets that can be quasisymmetrically embedding in

S 2 [Bo11].
The concept of quasisymmetric map between metric spaces was defined by Tukia and Väisälä [TV80]. Let f : X → Y be a homeomorphism between two metric spaces (X, d X ) and

(Y, d Y ). f is quasisymmetric if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that d Y (f (x), f (y)) d Y (f (x), f (z)) ≤ η( d X (x, y) d X (x, z) ), ∀ x, y, z ∈ X, x = z.
It follows from the definition that the quasisymmetric self-maps of X form a group QS(X).

A homeomorphism f : X → Y is called quasi-Möbius if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that for all 4-tuple (x 1 , x 2 , x 3 , x 4 ) of distinct points in X, we have

[f (x 1 ), f (x 2 ), f (x 3 ), f (x 4 )] ≤ η([x 1 , x 2 , x 3 , x 4 ]),
where

[x 1 , x 2 , x 3 , x 4 ] = d X (x 1 , x 3 )d X (x 2 , x 4 ) d X (x 1 , x 4 )d X (x 2 , x 3 )
is the metric cross-ratio.

It is not hard to check that a quasisymmetric map between metric spaces is quasi-Möbius. Conversely, any quasi-Möbius map between bounded metric spaces is quasisymmetric [TV80].

An important tool in the study of quasisymmetric maps is the conformal modulus of a given family of paths. The notion of conformal modulus (or extremal length) was first introduced by Ahlfors and Beurling [START_REF] Ahlfors | Conformal invariants: topics in geometric function theory[END_REF]. It has many applications in complex analysis and metric geometry [START_REF] Lehto | Quasiconformal Mappings in the Plane[END_REF][START_REF] Heinonen | Lectures on analysis on metric spaces[END_REF].

Motivation

In the work of Bonk and Merenkov [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF], it was proved that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpiński carpet S 3 is a Euclidean isometry. For the standard 1/p-Sierpiński carpets S p , p ≥ 3 odd, they showed that the groups QS(S p ) of quasisymmetric self-maps are finite dihedral. They also established that S p and S q are quasisymmetrically equivalent if only if p = q. The main tool in their proof is the carpet modulus, which is a certain discrete modulus of a path family and is preserved under quasisymmetric maps of carpets.

The following question is inspired by the above results of Bonk and Merenkov [BM11]:

Question 1.1. Determining sufficient condition on a carpet S on S 2 such that QS(S) is Isom(S), the isometry group of S.

Note that QS(S 3 ) = Isom(S 3 ) and QS(S p ) contains Isom(S p ) as a finite-index subgroup. Bonk and Merenkov [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] conjectured that QS(S p ) = Isom(S p ) for any p odd. The aim of this chapter is to extend Bonk-Merenkov's results to a new class of Sierpiński carpets F n,p (5 ≤ n, 1 ≤ p ≤ n 2 -1). We will show that QS(F n,p ) = Isom(F n,p ). This is a further generalization of the work of Bonk and Merenkov [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF].

Main results

Unless otherwise indicated, we will equip a carpet S = S 2 \ S i∈N D i with the spherical metric. Note that when a carpet is contained in a compact set K of C ⊂ C ∪ {∞} ∼ = S 2 , the Euclidean and the spherical metrics are bi-Lipschitz equivalent on K.

Let

5 ≤ n, 1 ≤ p < n 2 -1 be integers. Let Q (0) n,p = [0, 1] × [0, 1] be the closed unit square in R 2 . We first subdivide Q (0)
n,p into n 2 subsquares with equal side-length 1/n and remove the interior of four subsquares, each has side-length 1/n and is of distance √ 2p/n to one of the four corner points of Q (0) n,p . The resulting set Q (1) n,p consists of (n 2 -4) squares of side-length 1/n. Inductively,

Q (k+1) n,p , k ≥ 1, is obtained from Q (k)
n,p by subdividing each of the remaining squares in the subdivision of Q (k) n,p into n 2 subsquares of equal side-length 1/n k+1 and removing the interior of four subsquares as we have done above.

The Spierpiński carpet F n,p is the intersection of all the sets Q (k) n,p , i.e.,

F n,p = +∞ \ k=0 Q (k) n,p .
See Figure 1.1.

The following theorem will be proved in Section 1.4. It shows that, from the point of view of quasiconformal geometry, the carpets F n,p are different with the standard Sierpiński carpets S m , m ≥ 3 odd (note that the standard Sierpiński carpets S m is constructed from a similar process, by removing the interior of the middle square in each steps).

Theorem 1.1. Let 5 ≤ n, 1 ≤ p < n 2 -1 be integers. The carpet F n,p is not quasisymmetrically equivalent to the Standard Sierpiński carpet S m , m ≥ 3 odd. [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] that for m ≥ 3 odd the quasisymmetric group QS(S m ) is a finite dihedral group. Moreover, when m = 3, QS(S 3 ) is the Euclidean isometry group of S 3 . In Section 1.6, we will show that Theorem 1.2. Let f be a quasisymmetric self-map of F n,p . Then f is a Euclidean isometry.

It was proved by Bonk and Merenkov

Note that the Euclidean isometric group of F n,p (and S m ), consists of eight elements, is the group generated by the reflections in the diagonal {(x, y) ∈ R 2 : x = y} and the vertical line {(x, y) ∈ R 2 : x = 1 2 }. We will also prove that Theorem 1.3. Two Sierpiński carpets F n,p and F n ,p are quasisymmetrically equivalent if and only if (n, p) = (n , p ).

Idea of the proofs

The main tools to prove the above theorems are the carpet modulus and the weak tangent, both of which were investigated in [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF]. Our arguments follow the same outline as [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF]. One of the most important observations in [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] is that a quasisymmetric self-map f of S 3 should preserve the pair {M, O}, where M and O are the inner and outer peripheral circle of S 3 , respectively. By counting the orbits of points under the action of QS(S 3 ), Bonk and Merenkov [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] then showed that f maps distinguished points (points of S 3 on the corner or on the middle of peripheral circles) to distinguished points. One the other hand, f induces a "tangent map" Df between weak tangents of distinguished points, which is also quasisymmetry. The study of carpet modulus with respect to the normalized quasisymmetry group of weak tangent shows that f should map M to M and O to O.

We will first concentrate on carpet modulus of the families of curves connecting the boundary of the annulus domains bounded by pairs of distinct peripheral circles of F n,p . The extremal mass distribution of such a carpet modulus exists and is unique (Proposition 1.3). This, together with the auxiliary results in Section 1.3, allows us to show that (see Section 4) any quasisymmetric self-map f of F n,p should preserves the set

{O, M 1 , M 2 , M 3 , M 4 },
where O is the boundary of the unit square and M 1 , M 2 , M 3 , M 4 are the boundary of the first four squares removed from the unit square.

It is more difficult to see that f should maps O to O. To show this, we first study the weak tangents of the carpets (this is our main work on Section 1.5). In Section 1.6, we prove that f (O) = O by counting the orbit of a corner of O or M i under the group QS(F n,p ). The proofs of Theorem 1.2 and Theorem 1.3 are given at the end of this chapter. Theorem 1.2 is much stronger than the result of Bonk and Merenkov [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] for S m , m ≥ 5 odd.

Carpet modulus

In this section, we shall recall the definitions of conformal modulus and carpet modulus. The carpet modulus was introduced by Bonk-Merenkov [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] as a quasisymmetric invariant. There are several important properties of the carpet modulus that will be used in the rest of this chapter. In many cases, we will neglect the proof and refer to [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] instead.

Conformal modulus

A path γ in a metric space X is a continuous map γ : I → X of a finite interval I. Without cause of confusion, we shall identified the map with its image γ(I) and denote a path by γ. We say that γ is open if I = (a, b). The limits lim t→a γ(t) and lim t→b γ(t), if they exist, are called the end points of γ. If A, B ⊆ X, then we say that γ connects A and B if γ has endpoints such that one of them lies in A and the other lies in B.

If I = [a, b] is a closed interval, then the length of γ : I → X is defined by length(γ) := sup n X i=1 |γ(t i ) -γ(t i-1 )|
where the supremum is taken over all finite sequences a = t

0 ≤ t 1 ≤ t 2 ≤ • • • ≤ t n = b. If I is not closed, then we set length(γ) := sup J length(γ|J),
where J is taken over all closed subintervals of I and γ| J denotes the restriction of γ on J. We call γ rectifiable if its length is finite. Similarly, a path γ : I → X is locally rectifiable if its restriction to each closed subinterval is rectifiable. Any rectifiable path γ : I → X has a unique extension γ to the closure I of I.

Let Γ be a family of paths in S 2 . Let σ be the spherical measure and ds be the spherical line element on S 2 induced by the spherical metric (the Riemannian metric on S 2 of constant curvature 1). The conformal modulus of Γ is defined as

mod(Γ) := inf Z S 2 ρ 2 dσ ,
where the infimum is taken over all nonnegative Borel functions ρ :

S 2 → [0, ∞] satisfying Z γ ρds ≥ 1
for all locally rectifiable path γ ∈ Γ. Functions ρ satisfying (1.2.1) for all locally rectifiable path γ ∈ Γ are called admissible.

It is easy to show that (see [START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF])

mod(Γ 1 ) ≤ mod(Γ 2 ), (1.1) if Γ 1 ⊆ Γ 2 and mod( ∞ [ i=1 Γ i ) ≤ ∞ X i=1 mod(Γ i ). (1.2)
Moreover, if Γ 1 and Γ 2 are two families of paths such that each path

γ in Γ 1 contains a subpath γ ∈ Γ 2 , then mod(Γ 1 ) ≤ mod(Γ 2 ) (1.3)
If f : Ω → Ω is a continuous map between domains Ω and Ω in S 2 and Γ is a family of paths contained in Ω, then we denote by

f (Γ) = {f • γ | γ ∈ Ω}.
If f : Ω → Ω is a conformal map between regions Ω, Ω ⊆ S 2 and Γ is a family of paths in Ω, then mod(Γ)=mod(f (Γ)). This is the fundamental property of modulus: conformal maps do not change the conformal modulus of a family of paths.

In this chapter, we shall adopt the metric definition of quasiconformal maps ( [HK98], Definition 1.2) and allow them to be orientation-reversing. Suppose that f :

X → Y is a homeomorphism between two metric spaces X and Y . f is quasiconformal if there is a constant H ≥ 1, s.t. ∀x ∈ X, lim sup r→0 + max{d(f (x), f (y)) : d(x, y) ≤ r} min{d(f (x), f (y)) : d(x, y) ≥ r} ≤ H.
Quasiconformal maps distort the conformal modulus of path families in a controlled way. Let Ω and Ω be regions in S 2 and let Γ be a family of paths in Ω. Suppose that

f : Ω → Ω is quasiconformal map. Then 1 K mod(Γ) ≤ mod(f (Γ)) ≤ Kmod(Γ), (1.4)
where K ≥ 1 depends on the dilatation of f . From (1.4), a quasiconformal map preserves the modulus of a path family up to a fixed multiplicative constant. So if Γ 0 ⊆ Γ and mod(Γ 0 ) = 0, then mod(f (Γ 0 )) = 0.

Carpet modulus

If a certain property for paths in Γ holds for all paths outside an exceptional family Γ 0 ⊆ Γ with mod(Γ 0 ) = 0, then we say that it holds for almost every path in Γ.

Let S = S 2 \ S ∞ i=1 D i be a carpet with C i = ∂D i , and let Γ be a family of paths in S 2 . A mass distribution ρ is a function that assigns to each C i a non-negative number ρ(C i ).

The carpet modulus of Γ with respect to S is defined as

mod S (Γ) = inf ρ X i ρ(C i ) 2 ,
where the infimum is taken over all admissible mass distribution ρ, that is, mass distribution ρ satisfies

X γ T C i =∅ ρ(C i ) ≥ 1
for all most every path in Γ.

It is straightforward to check that the carpet modulus is momotone and countably subadditive, the same properties as conformal modulus in (1.1), (1.2) and (1.3). An crucial property of carpet modulus is its invariance under quasiconformal maps.

Lemma 1.1 ( [BM11]). Let D, f
D ⊂ S 2 be regions and f : D → D be a quasiconformal map. Let S ⊆ D be a carpet and Γ be a family of paths such that γ ⊂ D for each γ ∈ Γ.

Then mod f (S) (f (Γ)) = mod S (Γ).

Carpet modulus with respect to a group

We also need the notion of carpet modulus with respect to a group. Let S = S 2 \ S i∈N D i be a carpet and C i = ∂D i . Let G be a group of homeomorphisms of S. If g ∈ G and C ⊆ S is a peripheral circle of S, then g(C) is also a peripheral circle of S. Let O = {g(C) : g ∈ G} be the orbit of C under the action of G.

Let Γ be a familly of paths in S 2 . A admissible G-invariant mass distribution ρ :

{C i } → [0, +∞] is a mass distribution such that 1. ρ(g(C)) = ρ(C) for all g ∈ G and all peripheral circles C of S; 2. almost every path γ in Γ satisfies X γ T C i =∅ ρ(C i ) ≥ 1.
The carpet modulus mod S/G (Γ) with respect to the action of G on S is defined as

mod S/G (Γ) := inf ρ X O ρ(O) 2 ,
where the infimum is taken over all admissible G-invariant mass distributions. In the above definition, ρ(O) is defined by ρ(C) for any

C ∈ O. Since ρ is G-invariant, ρ(O)
is well-defined. Note that each orbit contributions with exactly one term to the sum [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF]). Let D be a region in S 2 and S be a carpet contained in D. Let G be a group of homeomorphisms on S. Suppose that Γ is a family of paths with γ ⊆ D for each γ ∈ Γ and f :

P O ρ(O) 2 . Lemma 1.2 ( [
D → f D a quasiconformal map onto another region f D ⊆ S 2 . We denote Ü S = f (S), e Γ = f (Γ) and Ü G = (f | S ) • G • (f | S ) -1 , then mod e S/ e
G ( e Γ) = mod S/G (Γ).

Lemma 1.3. Let S be a carpet in S 2 and Ψ : S 2 → S 2 be a quasiconformal map with Ψ(S) = S, ψ := Ψ| S . Assume that Γ is a Ψ-invariant path family in S 2 such that for every peripheral circle C of S that meets some path in Γ we have

ψ n (C) = C for all n ∈ Z. Then mod S/ ψ k (Γ) = kmod S/ ψ (Γ) for every k ∈ N.
This is ( [BM11], Lemma 3.3). In this Lemma, ψ denotes the cyclic group of homeomorphisms on S generated by ψ, and Γ is called Ψ-invariant if Ψ(Γ) = Γ. This lemma gives a precise relationship between the carpet modulus with respect to a cyclic group and its subgroups.

Existence of extremal mass distribution

Let S = S 2 \ {D i }, C i = ∂D i be a carpet and Γ be a family of paths on S 2 . An admissible mass distribution ρ for a carpet modulus mod S (Γ) is called extremal if mod S (Γ) is obtained by ρ:

mass(ρ) = X i ρ(C i ) 2 = mod S (Γ).
Similarly, an G-invariant mass distribution that obtains mod S/G (Γ) is also called extremal.

A criterion for the existence of an extremal mass distribution for carpet modulus (with respect to the group) is given by [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF]. Recall that the peripheral circles {C i } are uniform quasicircles if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that every C i is the image of an η-quasisymmetric map of the unit circle.

Proposition 1.1. Let S be a carpet in S 2 whose peripheral circles are uniform quasicircles, and let Γ be an arbitrary path family in S 2 with mod S (Γ) < +∞. Then the extremal mass distribution for mod S (Γ) exists and is unique. This is ( [BM11], Proposition 2.4). The uniqueness follows from elementary convexity argument.

Proposition 1.1. Let S be a carpet in S 2 whose peripheral circles are uniform quasicircles.

Let G be a group of homeomorphisms of S and Γ be a path family in S 2 with mod S/G (Γ) < +∞. Suppose that for each k ∈ N there exists a family of peripheral circles C k of S and a constant N k ∈ N with the following properties:

1. If O is any orbit of peripheral circles of S under the action of G, then #(O T C k ) ≤ N k for all k ∈ N.
2. If Γ k is the family of all paths in Γ that only meet peripheral circles in C k , then

Γ = S k Γ k .
Then extremal mass distribution for mod S/G (Γ) exists and is unique. This is ( [BM11], Proposition 3.2).

Auxiliary results

In this section, we collect a series of results obtained by M. Bonk and his coauthors [START_REF] Bonk | Rigidity of Schottky sets[END_REF][START_REF] Bonk | Uniformization of Sierpiński carpets in the plane[END_REF]. The theorems and propositions cited here are the cornerstone of our later proof (as well as they were for the proof in [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF]).

Quasiconformal extention of quasisymmetric map

Proposition 1.2. Let S be a carpet in S 2 whose peripheral circles are uniform quasicircles and let f a quasisymmetric map of S onto another carpet Ü S ⊆ S 2 . Then there exists a self-quasiconformal map F on S 2 whose restriction to S is f . This is ( [Bo11], Proposition 5.1).

Quasisymmetric uniformization and rigidity

The peripheral circles {C i } of S are called uniformly relatively separated if the pairwise distances are uniformly bounded away from zero. i.e., there exists δ > 0 such that

∆(C i , C j ) = dist(C i , C j ) min{diam(C i ), diam(C j )} ≥ δ
for any two distinct i and j. This property is preserved under quasisymmetric maps. See ( [START_REF] Bonk | Uniformization of Sierpiński carpets in the plane[END_REF], Corollary 4.6).

Theorem 1.1. Let S be a carpet in S 2 whose peripheral circles are uniformly relatively separated uniformly quasicircles, then there exists a quasisymmetric map of S onto a round carpet.

This is (

[Bo11], Corollary 1.2). Recall that a carpet S = S 2 \ S D i is called round if each D i is an open spherical disk.
Theorem 1.2. Let S be a round carpet in S 2 of measure zero. Then every quasisymmetric map of S onto any other round carpet is the restriction of a Möbius transformation.

This is ( [BKM], Theorem 1.2).

Here by definition a Möbius transformation is a fractional linear transformation on S 2 ∼ = Ĉ or the complex-conjugate of such a map. So we allow a Möbius transformation to be orientation-reversing.

Three-Circle Theorem

Let S ⊆ S 2 be a carpet. A homeomorphism embedding f : S → S 2 is called orientation-preserving if some homeomorphic extension F : S 2 → S 2 of f is orientationpreserving on S 2 (such an extension exists and the definition is independent of the choice of extension, see the proof of Lemma 5.3 in [START_REF] Bonk | Uniformization of Sierpiński carpets in the plane[END_REF]).

Corollary 1.1. Let S be a carpet in S 2 of measure zero whose peripheral circles are uniformly relatively separated uniform quasicircles and C i , i = 1, 2, 3 be three distinct peripheral circles of S. Let f and g be two orientation-preserving quasisymmetric selfmaps of S. Then we have the following rigidity results:

1. Assume that f (C i ) = g(C i ) for i = 1, 2, 3. Then f = g. 2. Assume that f (C i ) = g(C i ) for i = 1, 2 and f (p) = g(p) for a given point p ∈ S.
Then f = g.

3.

Assume that G is the group of all orientation-preserving quasisymmetric self-maps of S that fix C 1 , C 2 . Then G is a finite cyclic group.

4. Assume that G is the group of all orientation-preserving quasisymmetric self-maps of S that fix C 1 and fix a given point q ∈ C 1 , then G is an infinite cyclic group.

Proof. The proof we given here is contained in [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF]. Since its conclusion is important for the rest of our chapter, we include it here for completeness. By Theorem 1.1, there exists a quasisymmetric map h of S onto a round carpet Ü S. Using Proposition 1.2, we can extend h to a quasiconformal map on S 2 . Since quasiconformal maps preserve the class of sets of measure zero, Ü S has measure zero as well. We denote by G 0 and Ý G 0 the group of all orientation-preserving quasisymmetric self-maps of S and Ü S, respectively. By the quasisymmetric rigidity of round carpets (Theorem 1.2), Ý G 0 consists of the restriction of orientation-preserving Möbius transformations that fix Ü S. Now we look at the homomorphism h * induced by h:

h * : G 0 → Ý G 0 , ψ → h • ψ • h -1 .
We can check that h * is well-defined and is an isomorphic. Since h * (f ) and h * (g) are orientation-preserving Möbius transformation and h * (f )•(h * (g)) -1 fixes distinct spherical round circles h(C i ), i = 1, 2, 3, we know that h * (f ) • (h * (g)) -1 = id and (1) follows.

We can prove (2) from the fact that any orientation-preserving Möbius transformation fixing distinct spherical round circles and a given non-common center point p ∈ S 2 is the identity.

To prove (3), it suffices to show that Ü G = h * (G) is a finite cyclic. By post-composing fractional linear transformation to h, we can assume that h(C 1 ) and h(C 2 ) are distinct spherical round circles with the same center. Note that Ü G consists of orientationpreserving Möbius transformation, fixing h(C 1 ), h(C 2 ) and Ü S. Moreover, Ü G must be a discrete group as it maps peripheral round circles of S to peripheral round circles. Hence Ü G is a finite cyclic group, then (3) follows.

For (4), similarly, by post-composing fractional linear transformation to h, we can assume that h(C 1 ) = R S {∞}, h(q) = 0 and Ü S is contained in the upper half-plane. Then the maps in Ü G are of the form: z → λz with λ > 0, fixing Ü S. By the same reason as

(3), Ü G is a discrete group. So there exists a λ 0 ≥ 1 such that Ü G = {z → λ n 0 z|n ∈ N}. It follows that Ü
G, and hence also G, is the trivial group consisting only of the identity or an infinite cyclic group. Therefore, (4) follows. A C * -square Q is a Jordan region of the form

Square carpets

Q = {ρe iθ : a < ρ < b, α < θ < β} with 0 < log(b/a) = β -α < 2π.
We call the quantity

l C * (Q) = log(b/a) = β -α
its side length. Clearly, two opposite sides of Q parallel to the boundaries of A, while the other two perpendicular to the boundaries of A.

A square carpet T in a C * -cylinder A is a carpet that can be written as

T = A \ [ i Q i ,
where the sets Q i , i ∈ I, are C * -squares whose closures are pairwise disjoint and contained in the interior of A.

Theorem 1.3. Let S be a carpet of measure zero in S 2 whose peripheral circles are uniformly relatively separated uniform quasicircles, and C 1 and C 2 two distinct peripheral circles of S. Then there exists a quasisymmetric map f from S onto a square carpet T in a C * -cylinder A such that f (C 1 ) is the inner boundary component of A and f (C 2 ) is the outer one.

This is ( [Bo11]

, Theorem 1.6).

Let S be a carpet in S 2 and C 1 , C 2 be two distinct peripheral circles of S. Soppose that the Jordan regions D 1 and D 2 are the complementary components of S bounded by

C 1 and C 2 respectively. We let Γ(C 1 , C 2 ) be the family of all open paths in S 2 \ D 1 ∪ D 2 that connects D 1 and D 2 .
Proposition 1.3. Let S be a carpet of measure zero in S 2 whose peropheral circles are uniformly relatively separated uniform quasicircles, and C 1 and C 2 two distinct peripheral circles of S. Then

(1) mod S (Γ(C 1 , C 2 )) has finite and positive total mass.

(2) Let f be a quasisymmetric map of S onto a square carpet T in a C * -cylinder A = {z ∈ C; r ≤ |z| ≤ R} such that C 1 corresponds to the inner and C 2 to the outer boundary components of A. Then the extremal mass distribution is given as follows:

ρ(C 1 ) = ρ(C 2 ) = 0, ρ(C) = l C * (f (C)) log(R/r) with the peripheral circles C = C 1 , C 2 of S.
This is ( [Bo11], Corollary 12.2).

Let S be a carpet in a closed Jordan region D ⊂ Ĉ. S is called square carpet if ∂D is a peripheral circle of S and all other peripheral circles are squares with sides parallel to the coordinate axes.

Theorem 1.4. Let S and Ü S be square carpets of measure zero in rectangles

K = [0, a] × [0, 1] ⊆ R 2 and f K = [0, e a]×[0, 1] ⊆ R 2
, respectively, where a, e a > 0. If f is an orientationpreserving quasisymmetric homeomorphism form S onto Ü S that takes the corners of K to the corners of f K with f (0) = 0. Then a = e a, S = Ü S, and f is the identity on S.

This is ( [BM11]

, Theorem 1.4). Here the expression square carpet S in a rectangle K means that a carpet S ⊂ K so that ∂K is a peripheral circle of S and all other peripheral circles are squares with four sides parallel to the sides of K, respectively.

Distinguished peripheral circles

Let n ≥ 5, 1 ≤ p < n 2 -1 be integers. Let F n,p be the Sierpiński carpet as we defined in the introduction. We endow F n,p with the Euclidean metric in R 2 . Since F n,p is a subset of [0, 1] × [0, 1], the Euclidean metric (measure) is comparable with the spherical metric (measure).

If Q is a peripheral circle of F n,p , we denote by Q the Euclidean side length of Q. Denote by Q 0 the unit square [0, 1] × [0, 1].

Lemma 1.4. The carpet F n,p is of measure zero. The peripheral circles of F n,p are uniform quasicircles and uniformly relatively separated.

Proof. It follows from the construction that F n,p is a carpet of Hausdorff dimension log(n 2 -4)/ log n < 2.

So the measure of F n,p is equal to zero.

Since each peripheral circle of F n,p can be mapped to the boundary of Q 0 by a Euclidean similarity, the peripheral circles of F n,p are uniform quasicircles.

At last, the peripheral circles of F n,p are uniformly relatively separated in the Euclidean metric. Indeed, consider any two distinct peripheral circles C 1 , C 2 of F n,p . The Euclidean distance between C 1 and C 2 satisfies

dist(C 1 , C 2 ) ≥ min{ (C 1 ), (C 2 )} = 1 √ 2 min{diam(C 1 ), diam(C 2 )}.

Distinguished pairs of non-adjacent peripheral circles

We denote by O the boundary of the unit square Q 0 . In the first step of the inductive construction of F n,p , there are four squares

Q 1 , Q 2 , Q 3 , Q 4 of side-length 1
n , i.e., the lower left, lower right, upper right and upper left squares respectively, removed from Q 0 . We denote by

M i , i = 1, • • • , 4 the boundary of Q i , i = 1, • • • , 4, respectively.
In the following discussions, we call O the outer circle of F n,p and M i , i = 1, • • • , 4 the inner circles of F n,p . We say that two disjoint peripheral circles C, C are adjacent if there exists a copy F of F n,p (here F ⊂ F n,p can be considered as a carpet scaled from F n,p by some factor 1/n k ) such that C, C are inner circles of F . For example, two distinct inner circles M i and M j are adjacent. Two disjoint peripheral circles C, C which are not adjacent are called non-adjacent.

Lemma 1.5. Let {C, C } be any pair of non-adjacent distinct peripheral circles of F n,p . Then mod Fn,p (Γ(C, C )) ≤ mod Fn,p (Γ(O, M )).

Moreover, the equality holds if and only if {C, C } = {O, M } for some inner circle M = M i .

Proof. Assume that {C, C } = {O, M } for any inner circle M . By Lemma 1.4 and Proposition (1.3), mod Fp,q (Γ(C, C )) is a finite and positive number. Without loss of generality we may assume that (C) = 1/n m ≤ (C ). Note that there exists a copy F ⊂ F n,p , rescaled from F n,p by a factor 1/n m-1 , so that C corresponds to some inner circle, say, (1.5)

M 1 of F n,p .
On the other hand, since every path in Γ(C, C 0 ) meets exactly the same peripheral circles of F and F n,p , we have

mod Fn,p (Γ(C, C 0 )) = mod F (Γ(C, C 0 )).
Moreover, by the similarity of F n,p and F , mod F (Γ(C, C 0 )) = mod Fn,p (Γ(M, O)).

It follows that

mod Fn,p (Γ(C, C )) ≤ mod Fn,p (Γ(M 1 , O)).
We next show that the equality case in (1.7) cannot happen. We argue by contradiction. Assume that mod Fn,p (Γ(C, C )) = mod Fn,p (Γ(C, C 0 )).

Note that all carpet modulus considered above are finite by Proposition 1.3 and so there exist unique extremal mass distributions, say ρ and ρ , for mod Fn,p (Γ(C, C )) and mod Fn,p (Γ(O, M 1 )), respectively, by Proposition 4.9.

Let C be the set of all peripheral circles of F n,p . According to the description in Proposition 1.3, ρ and ρ are supported on C \ {C, C } and C \ {O, M 1 }, respectively.

By transplanting ρ to the carpet F using a suitable Euclidean similarity between F and F n,p , we get an admissible mass distribution e ρ for F supported only on the set of peripheral circles of F except C and C 0 . Note that the total mass of e ρ is the same as mass(ρ ).

We extend C → e ρ(C) by zero if C belonging to C does not intersect the interior region of C 0 . Then e ρ is an admissible mass distribution for mod Fn,p (Γ(C, C 0 )), thus for mod Fn,p (Γ(C, C )) as well. However, e ρ = ρ and mass( e ρ) = mod Fn,p (Γ(C, C )), we arrive at a contradiction by Proposition 4.9.

In summary, we get the following crucial inequality:

mod Fn,p (Γ(C, C )) < mod Fn,p (Γ(O, M 1 )) (1.6)
where {C, C } = {O, M i } i = 1, 2, 3, 4 and non-adjacent. So the lemma follows.

Proof. Assume that {C, C } = {O, M } for any inner circle M . By Lemma 1.4 and Proposition (1.3), mod Fp,q (Γ(C, C )) is a finite and positive number. Without loss of generality we may assume that (C) = 1/n m ≤ (C ). Note that there exists a copy F ⊂ F n,p , rescaled from F n,p by a factor 1/n m-1 , so that C corresponds to some inner circle, say, M 1 of F n,p . (1.7)

On the other hand, since every path in Γ(C, C 0 ) meets exactly the same peripheral circles of F and F n,p , we have

mod Fn,p (Γ(C, C 0 )) = mod F (Γ(C, C 0 )).
Moreover, by the similarity of F n,p and F , mod

F (Γ(C, C 0 )) = mod Fn,p (Γ(M, O)).
It follows that

mod Fn,p (Γ(C, C )) ≤ mod Fn,p (Γ(M 1 , O)).
We next show that the equality case in (1.7) cannot happen. We argue by contradiction. Assume that mod Fn,p (Γ(C, C )) = mod Fn,p (Γ(C, C 0 )).

Note that all carpet modulus considered above are finite by Proposition 1.3 and so there exist unique extremal mass distributions, say ρ and ρ , for mod Fn,p (Γ(C, C )) and mod Fn,p (Γ(O, M 1 )), respectively, by Proposition 4.9.

Let C be the set of all peripheral circles of F n,p . According to the description in Proposition 1.3, ρ and ρ are supported on C \ {C, C } and C \ {O, M 1 }, respectively.

By transplanting ρ to the carpet F using a suitable Euclidean similarity between F and F n,p , we get an admissible mass distribution e ρ for F supported only on the set of peripheral circles of F except C and C 0 . Note that the total mass of e ρ is the same as mass(ρ ).

We extend C → e ρ(C) by zero if C belonging to C does not intersect the interior region of C 0 . Then e ρ is an admissible mass distribution for mod Fn,p (Γ(C, C 0 )), thus for mod Fn,p (Γ(C, C )) as well. However, e ρ = ρ and mass( e ρ) = mod Fn,p (Γ(C, C )), we arrive at a contradiction by Proposition 4.9.

In summary, we get the following crucial inequality:

mod Fn,p (Γ(C, C )) < mod Fn,p (Γ(O, M 1 )) (1.8)
where {C, C } = {O, M i } i = 1, 2, 3, 4 and non-adjacent. So the lemma follows.

Corollary 1.2. Let f be a quasisymmetric self-map of F n,p . Then

f ({O, M 1 , M 2 , M 3 , M 4 }) = {O, M 1 , M 2 , M 3 , M 4 }.
Proof. We argue by contradiction. Assume that f maps {O, M 1 } to some pair of peripheral circles {C, C } {O, M 1 , M 2 , M 3 , M 4 } and f (O) = C. By Proposition 1.2, f extends to a quasiconformal homeomorphism on S 2 . In particular, Γ(C, C )=f (Γ(O, M 1 )). Then Lemma 1.1 implies mod Fn,p (Γ(O, M 1 )) = mod Fn,p (Γ(C, C )).

We distinguish the argument into two cases depending on the type of the squares C and C , i.e., whether they are adjacent or not.

Case (1): C, C are non-adjacent. This is only possible if {C, C } ⊆ {O, M 1 , M 2 , M 3 , M 4 } by Lemma 1.5. Then we get a contradiction.

Case (2): C, C are adjacent. Suppose C, C are inner circles of some copy F ⊂ F n,p . Consider f (M i ), i = 2, 3, 4. They must be inner circles of F as well. Otherwise, for example, suppose that f (M 2 ) is not an inner circle of F . Since C and f (M 2 ) are nonadjacent, we can apply Lemma 1.5 to show that mod Fn,p (Γ(C, f (M 2 ))) < mod Fn,p (Γ(O, M 1 )), which is contradicted with the fact that mod Fn,p (Γ(C, f (M 2 ))) = mod Fn,p (Γ(O, M 2 )) = mod Fn,p (Γ(O, M 1 )).

As a result, {f (O), f (M 1 ), f (M 2 ), f (M 3 ), f (M 4 ))} are pairwise adjacent and all of them are inner circles of F . However, F contains exactly four inner circles. So Case (2) can not happen.

By the same argument to pairs O and M i , i = 2, 3, 4, the corollary follows.

1.4.2 Quasisymmetric group QS(F n,p ) is finite

Let H denote the Euclidean isometry group which consists of eight elements: four of them rotate around the center by π/2, π, 3π/2, and 2π, respectively; the others are orientation-reserving and reflecting by lines x = 0, x = y, y = 0 and x+y = 0, respectively. It is obvious that H is contained in QS(F n,p ).

Corollary 1.3. Let 5 ≤ n, 1 ≤ p < n 2 -1 be integers. Then the group QS(F n,p ) of quasisymmetric self-maps of F n,p is finite.

Proof. According to Corollary 1.2, {O, M 1 , M 2 , M 3 , M 4 } are preserved under every quasisymmetric self-map of F n,p . The group G of all orientation-preserving quasisymmetric self-maps of F n,p is finite by the proof of Case (1) in Corollary (1.1). Since G is a subgroup of QS(F p,q ) with index two, QS(F p,q ) is finite.

Proof of Theorem 1.1

Recall that the standard carpet S m , m ≥ 3 odd, is obtained by subdivide [0, 1] × [0, 1] into m 2 subsquares of equal size, removing the interior of the middle square, and repeating these operations to every subsquares, inductively.

Proof of Theorem 1.1. Let M, O be the inner circle and outer circle of S m respectively. Lemma 5.1 of [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] states that mod Sm (Γ(O, M)) is strictly larger than the carpet modulus of any other path family Γ(C, C ) with respect to S m , where C and C are peripheral circles of S m . While for carpet F n,p , according to the symmetry, at least two pairs of peripheral circles the maximum of {mod Fn,p Γ(C 1 , C 2 ) : C 1 , C 2 ∈ C}. Since any quasisymmetric maps from F n,p to S m must preserve such a maximum property, there is no such quasisymmetric map.

Weak tangent spaces

The results in this section generalize the discussion in ( [BM11], Section 7). At first, we explain the definition of weak tangent of a carpet. Then we show that a quasisymmetric map between two carpets F n,p induces a quasisymmetric map between weak tangents.

Weak tangents

In general, the weak tangents of a metric space M at a point p ∈ M can be defined as the Gromov-Hausdorff limits of the pointed metric spaces

lim λ→∞ (λM, p)
where λM is the same set of points with M equipped with the original metric multiplied by λ. If the limit is unique up to multiplied by positive constants, then the weak tangents is usually called the tangent cone of M at p.

In the following, as in [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF], we will use a suitable definition of weak tangents for subsets of S 2 equipped with the spherical metric.

Suppose that a, b ∈ C, a = 0 and M ⊆ Ò C. We denote by

aM + b := {az + b : z ∈ M }.
Let A be a subset of Ò C with a distinguished point z 0 ∈ A, z 0 = ∞. We say that a closed set W A (z 0 ) ⊆ Ò C is a weak tangent of A if there exists a sequence (λ n ) with λ n → ∞ such that the sets A n := λ n (A -z 0 ) converge to W A (z 0 ) as n → ∞ in the sense of Hausdorff convergence on Ò C equipped with the spherical metric. In this case, we use the notation

W A (z 0 ) = lim n→∞ (A, z 0 , λ n ).
Since for every sequence (λ n ) with λ n → ∞, there is a subsequence (λ n k ) such that the sequence of the sets

A n k = λ n k (A -z 0 ) converges as k → ∞, A has weak tangents at each point z 0 ∈ A \ {∞}.
In general, weak tangents at a point are not unique. In particular, λW A (z 0 ) is also a weak tangent. Now we apply the notion to our carpets F n,p . In fact, the following arguments work for a general class of carpets, such as the standard Sierpiński carpet S m and carpets which satisfy some self-similarity property.

A weak tangent of a point

z 0 ∈ F n,p is a closed set W Fn,p (z 0 ) ⊆ Ò C such that W Fn,p (z 0 ) = lim j→∞ (F n,p , z 0 , n k j ),
where k j ≥ 1 and k j → ∞ as j → ∞.

At the point 0 the carpet F n,p has the unique weak tangent

W Fn,p (0) = lim j→∞ (F n,p , 0, n j ) = {∞} ∪ [ j∈N 0 n j F n,p .
(1.9)

This follows from the inclusions n j F n,p ⊆ n j+1 F n,p . Similarly, at each corner of O there exists a unique weak tangent of F n,p obtained by a suitable rotation of the set W Fn,p (0) around 0.

Let c = p/n + ip/n be the lower-left corner of M 1 . Then at c the carpet F n,p has unique weak tangent

W Fn,p (c) = lim j→∞ (F n,p , c, n j ) = {∞} ∪ [ j∈N 0 n j (iF n,p ∪ (-i)F n,p ∪ (-1)F n,p ).
Note that W Fn,p (c) can be obtained by pasting together three copies of W Fn,p . If z 0 is a corner of a peripheral circle C = O of F n,p , then F n,p has a unique weak tangent at z 0 obtained by a suitable rotation of the set W Fn,p (c) around 0.

Lemma 1.6. Let z 0 be a corner of a peripheral circle of F n,p . Then the weak tangent W F n,p (z 0 ) is a carpet of measure zero. If W Fn,p (z 0 ) is equipped with the spherical metric, then the family of peripheral circles of W Fn,p (z 0 ) are uniform quasicircles and uniformly relatively separated.

Proof. We can assume that z 0 equals 0. The proof works for other cases.

First note that (1.9) implies that W Fn,p (0) is a carpet of measure zero, since W Fn,p (0) is the union of countably many sets of measure zero.

Let Ω = {z ∈ C : Re(z) > 0, Im(z) > 0}. Then ∂Ω ia a peripheral circle of W Fn,p (0). It is easy to construct a bi-Lipschitz map between ∂Ω and the unit circle (both equipped with the spherical metric). Hence ∂Ω is a quasicircle. Note that all other peripheral circles of W Fn,p (0) are squares. As a result, the peripheral circles of W Fn,p (0) are uniformly quasicircles.

To show that the peripheral circles are uniformly relatively separated, we only need to check the following inequality:

dist(C 1 , C 2 ) ≥ min{ (C 1 ), (C 2 )} (1.10)
for any peripheral circles C 1 , C 2 = ∂Ω. Here dist(•, •) and (•) denote the Euclidean distance and Euclidean side length. The inequality implies that the peripheral circles are uniformly relatively separated with respect to the Euclidean metric. To see that they are uniformly relatively separated property with respect to the spherical metric, we can apply an argument of ( [BM11], Lemma 7.1).

Quasisymmetric maps between weak tangents

We are interested in quasisymmetric maps g : W → W between weak tangents W of F n,p and weak tangents W of F n,p . Note that 0, ∞ ∈ W, W . We call g normalized if Lemma 1.7. Let z 0 be a corner of a peripheral circle of F n 1 ,p 1 and let w 0 be a corner of a peripheral circle of F n 2 ,p 2 . Suppose that f : F n 1 ,p 1 → F n 2 ,p 2 be a quasisymmetric map with f (z 0 ) = w 0 . Then f induces a normalized quasisymmetric map g between the weak tangent W Fn 1 ,p 1 (z 0 ) and W Fn 2 ,p 2 (w 0 ).

Proof. By Proposition 1.2 we can extend f to a quasiconformal self-homeomorphism F of Ò C. There exists a relative neighborhood N 1 of z 0 in F n 1 ,p 1 and a relative neighborhood

N 2 of w 0 in F n 2 ,p 2 with F (N 1 ) = N 2 such that W Fn 1 ,p 1 (0) \ {∞} = [ j∈N 0 n j 1 (N 1 -z 0 )
and

W Fn 2 ,p 2 (0) \ {∞} = [ j∈N 0 n j 2 (N 2 -w 0 )
Pick a point u 0 ∈ N -z 0 , u 0 = 0. Then for each j ∈ N 0 we have

F (z 0 + n -j 1 u 0 )( = w 0 , ∞) in F n 2 ,p 2 .
We consider the following quasiconformal self-map

F j of Ò C with F j (n j 1 (N 1 -z 0 )) = n k(j) 2 (N 2 -w 0 ): F j : u → n k(j) 2 (F (z 0 + n -j 1 u) -w 0 ) for u ∈ Ò C,
where k(j) is the unique integer such that 1 ≤ |F j (u 0 )| < n 2 . Note that k(j) → ∞ as j → ∞ and F (∞) = w 0 . This implies that F j (∞) → ∞ as j → ∞. We also have F j (0) = 0. So the images of 0, ∞ and u 0 under F j have mutual spherical distance uniformly bounded from below independent of j. Moreover, F j is obtained from F by post-composing and pre-composing Möbius transformations. Hence the sequence (F j ) is uniformly quasiconformal, and it follows that we can find a subsequence of (F j ) that converges uniformly on Ò C to a quasiconformal map F ∞ . Without loss of generality, we assume that (F j ) converges uniformly to F ∞ .

Note that F ∞ (0) = 0 and F ∞ (∞) = ∞. To prove the statement of the lemma, it suffices to show that

F ∞ (W Fn 1 ,p 1 (z 0 ) ) = W Fn 2 ,p 2 (w 0 ), because then g := F ∞ |W Fn 1 ,p 1 (z 0 )
is an induced normalized quasisymmetric map between W Fn 1 ,p 1 (z 0 ) and W Fn 2 ,p 2 (w 0 ), as desired.

Let u be an arbitrary point in W Fn 1 ,p 1 (z 0 ). There exists a sequence (u j ) with u j ∈ n j 1 (N 1 -z 0 ) converging to u. We have F j (u j ) ∈ n j 2 (N 2 -w 0 ) and a subsequence of (F j (u j )) converging to some point v in W Fn 2 ,p 2 (w 0 ). By the definition of F ∞ , we have

F ∞ (u) = v. Hence F ∞ (W Fn 1 ,p 1 (z 0 )) ⊆ W Fn 2 ,p 2 (w 0 ).
For every point v in W Fn 2 ,p 2 (w 0 ), there exists a sequence (u j ) with u j ∈ n j 1 (N 1 -z 0 ) such that (F j (u j )) converges to v. Then we can choose a subsequence of (u j ) converging to some point u in W Fn 1 ,p 1 (z 0 ) and so F ∞ (u) = v.

It follows that F ∞ (W Fn 1 ,p 1 (z 0 )) = W Fn 2 ,p 2 (w 0 ) and we are done.

By Corollary 1.2, a quasisymmetric self-map f of F n,p maps {O, M 1 , M 2 , M 3 , M 4 } to {O, M 1 , M 2 , M 3 , M 4 }.
In the remaining part of this section, we will show that there is no quasisymmetric self-map f of F n,p with f (0) = c, where c is a corner of an inner circle. By Lemma 1.7, if such an f exists, then it would induce a normalized quasisymmetric map from W Fn,p (0) to W Fn,p (c). However, the following proposition shows that: Proposition 1.4. There is no normalized quasisymmetric map from W Fn,p (0) to W Fn,p (c).

To prove the proposition, we need two lemmas. Let G and Ü G be the group of normalized orientation-preserving quasisymmetric selfmaps of W Fn,p (0) and W Fn,p (c), respectively. By Corollary 1.1, G and Ü G are infinite cyclic groups. Note that the map µ(z)

:= nz is contained in G ∩ Ü G.
We assume that G =< φ > and µ = φ s for some s ∈ Z + . Since the peripheral circles of W Fn,p (0) are uniformly quasicircles and uniformly relatively separated, there exists a quasiconformal extension

Φ : Ò C → Ò C of φ.
Let H be the group generated by the reflection in the real and in the imaginary axes. We may assume that Φ is equivalent under the action of H (see Page 42, [START_REF] Bonk | Quasisymmetric rigidity of square Sierpiński carpets[END_REF] for the discussion).

Let

Ω = {z ∈ C : Re(z) > 0, Im(z) > 0}. Then C 0 := ∂Ω is a peripheral circle of W Fn,p (0). Since Φ(C 0 ) = C 0 and Φ is orientation-preserving, Φ(Ω) = Ω.
Let Γ be the family of all open paths in Ω that connects the positive real and positive imaginary axes. Since the paths in Ω are open, they don't intersect with C 0 . For any peripheral circle C of W Fn,p (0) that meets some path in Γ, note that φ k (C) = C for all k ∈ Z \ {0} (otherwise, φ would be of finite order, contradicted with the fact that φ is the generator of the infinite cyclic group G). So we can apply Lemma 1.3 to conclude that mod W Fn,p (0)/<µ> (Γ) = mod W Fn,p (0)/<φ s > (Γ) = smod W Fn,p (0)/G (Γ).

Note that without the action of the group G, the carpet modulus mod W Fn,p (0) (Γ) is equal to infinity.

Lemma 1.8. We have 0 < mod W Fn,p (0)/G (Γ) < ∞.

Proof. Let us first show that mod W Fn,p (0)/<µ> (Γ) < ∞ by constructing an admissible mass distribution of finite mass.

Let pr : C \ {0} → S 1 be the projection z → z |z| . If C = C 0 is a peripheral circle of W Fn,p (0), we let θ(C) be the arc length of pr(C). We set

ρ(C) := 8 > < > : 0, if C = C 0 ; 2 π θ(C), if C = C 0 . Note that ρ is < µ >-invariant.
Let Γ 0 be the family of paths γ ∈ Γ that are not locally rectifiable or for which γ∩W Fn,p (0) has positive length. Since W Fn,p (0) is a set of measure zero, we have mod(Γ 0 ) = 0, i.e., Γ 0 is an exceptional subfamily of Γ.

For any γ ∈ Γ \ Γ 0 , note that

X γ∩C =∅ ρ(C) = 2 π X γ∩C =∅ θ(C) ≥ 1.
As a result, ρ is admissible.

Let Q 0 = [0, 1] × [0, 1]. Note that every < µ >-orbit of a peripheral circle C = C 0 has a unique element contained in the set F = µ(Q 0 ) \ Q 0 . There is a constant K > 0 such that θ(C) ≤ K (C) for all peripheral circles C ⊂ F . It follows that 4 π 2 X C⊂F θ(C) 2 X C⊂F (C) 2 = Area(F ) = n 2 -1.
Hence ρ is a finite admissible mass distribution for mod W Fn,p (0)/<µ> (Γ).

To show that mod W Fn,p (0)/<µ> (Γ) > 0, we only need to show that the carpet satisfies the assumptions in Proposition 1.1. Then the extremal mass distribution for mod W Fn,p (0)/<µ> (Γ) exists and this is only possible if Γ itself is an exceptional family, that is, mod(Γ) = 0.

In fact, for k ∈ N we let C k be the set of all peripheral circles C of W Fn,p (0) with

C ⊂ F k = µ k (Q 0 ) \ µ -k (Q 0 ). Then 1. Every < µ >-orbit of a peripheral circle C = C 0 has exactly 2k elements in C k .
2. Let Γ k be the family of paths in Γ that only meet peripheral circles in C k . Then

Γ = S k Γ k .
As a result, the assumptions in Proposition 1.1 are satisfied.

Let Ü Ω = C \ Ω. The closure of Ü Ω contains W Fn,p (c) and C 0 = ∂Ω = ∂ Ü Ω is a peripheral circle of W Fn,p (c). Denote ψ = Φ| W Fn,p (c) . Then we have ψ ∈ Ü G. Let e Γ be

the family of all open paths in Ü

Ω that join the positive real and the positive imaginary axes.

Lemma 1.9. We have mod W Fn,p (c)/<ψ> ( e Γ) ≤ 1 3 mod W Fn,p (0)/G (Γ). Proof. Let ρ be an arbitrary admissible invariant mass distribution for mod W Fn,p (0)/G (Γ), with exceptional family Γ 0 ⊂ Γ. We set

e ρ( Ü C) := 8 > < > : 0, if Ü C = C 0 ; 1 3 ρ(α( Ü C))
if there is an α ∈ H such that α( Ü C) is a peripheral circle of W Fn,p (0) (such an α exits and is unique).

Since Φ is H-equivalent and ρ is G-invariant, e ρ is < ψ >-invariant. Let e Γ 0 be the family of paths in e Γ that have a subpath that can be mapped to a path in Γ 0 by an element of α ∈ H. Then mod( e Γ 0 ) = 0. Let γ ∈ e Γ. Note that γ has three disjoint open subpaths: one for each quarter-plane of Ü Ω and by suitable elements in H, the three subpaths are mapped to paths in Γ. Denote the images by γ 1 , γ 2 , γ 3 . If γ ∈ e Γ \ e Γ 0 , then γ i ∈ Γ \ Γ 0 , i = 1, 2, 3 and

X γ∩ e C =∅ e ρ( Ü C) ≥ 1 3 3 X i=1 X γ i ∩C =∅ ρ(C) ≥ 1.
Hence e ρ is admissible for mod W Fn,p (c)/<ψ> ( e Γ) and

mod W Fn,p (c)/<ψ> ( e Γ) ≤ mass W Fn,p (c)/<ψ> ( e ρ) ≤ 1 3 mass W Fn,p (0)/G (ρ).
Since ρ is an arbitrary mass distribution for 1 3 mod W Fn,p (0)/G (Γ), the statement follows.

Proof of Proposition 1.4. Suppose not, there exists a normalized quasisymmetric map f : W Fn,p (0) → W Fn,p (c). Precomposing f by the reflection in the diagonal line {x = y} if necessary, we may assume that f is orientation-preserving.

Then Ü G = f • G • f -1 and e φ = f • φ • f -1 is a generator for Ü G. Let F : Ò C → Ò
C be a quasiconfomral extension of f . Then e Γ = F (Γ). By quasisymmetric invariance of carpets modulus,

mod W Fn,p (c)/ e G ( e Γ) = mod W Fn,p (0)/G (Γ).
Assume that ψ = e φ m . Then similar to our discussion before Lemma 1.8, we have

mod W Fn,p (c)/<ψ> ( e Γ) = |m|mod W Fn,p (c)/ e G ( e Γ).
Hence by Lemma 1.9 we have

mod W Fn,p (0)/G (Γ) = mod W Fn,p (c)/ e G ( e Γ) = 1 |m| mod W Fn,p (c)/<ψ> ( e Γ) ≤ 1 3|m| mod W Fn,p (0)/G (Γ).
This is possible only if mod W Fn,p (0)/G (Γ) is equal to 0 or ∞. But this is contradicted with Lemma 1.8.

Quasisymmetric rigidity

Let D be the diagonal {(x, y) ∈ R 2 : x = y} and V be the vertical line {(x, y) ∈

R 2 : x = 1 2 }.
We denote the reflections in D and V by R D and R V , respectively. The Euclidean isometry group of F n,p is generated by R D and R V .

Let QS(F n,p ) be the group of quasisymmetric self-maps of F n,p . By Corollary 1.3,

QS(F n,p ) is a finite group. Proposition 1.5. Let f be a quasisymmetric self-map of F n,p . Then f ({O}) = {O} and f ({M 1 , M 2 , M 3 , M 4 }) = {M 1 , M 2 , M 3 , M 4 }.
Proof. From Corollary 1.2, we argue by contradiction and assume that there exists a quasisymmetric self-map f of F n,p and some i ∈ {1, 2, 3, 4} such that f ({O}) = {M i }. By pre-composing and post-composing suitable elements in the Euclidean isometry group, we can suppose that f is orientation-preserving and

f ({O}) = {M 1 }. Let G be the subgroup of QS(F n,p ), G = {g ∈ QS(F n,p ) | g(O) = O, g(M 1 ) = M 1 }.
G has a subgroup G with index two consisting of orientation-preserving elements. Then

G = G G G • R D .
We denote by

O G (z) = {g(z) : g ∈ G}
the orbit of z under the action of G for arbitrary z ∈ F n,p . Let c = (p/n, p/n) and c = ((p + 1)/n, (p + 1)/n) be the lower-left and upper-right corners of M 1 , respectively. Now we consider the map

Φ 0 : G -→ O G (0) g -→ g(0).
Note that Φ 0 is an isomorphism. In fact, for any g(0

) ∈ O G (0), if g is orientation- preserving, then Φ 0 (g) = g(0); otherwise, Φ 0 (g • R D ) = g(0). So Φ 0 is a surjection. On the other hand, if Φ 0 (g 1 ) = Φ 0 (g 2 )for any g 1 , g 2 ∈ G , then Case (2) of Corollary 1.1 gives g 1 = g 2 . So Φ 0 is a injection.
Similarly, we can also define the isomorphism

Φ c : G -→ O G (c) g -→ g(c).
These isomorphisms Φ 0 and Φ c imply that

#O G (0) = #G = #O G (c) (1.11)
On the other hand, f induces the following isomorphism

f * : G -→ G g -→ f • g • f -1 .
We denote by m = f (0). Then

O G (m) = {g(m) : g ∈ G} = {f • g • f -1 (m) : g ∈ G} = {f • g(0) : g ∈ G} = f (O G (0)). Hence #O G (m) = #G = #O G (0)
and so the orbits O G (m) and O G (c) have the same number of elements.

If G = {id}, we claim that G is a cyclic group of order 3. Indeed, for any g = id in

G , g(M 3 ) = M 3 , otherwise Case (1) of Corollary 1.1 implies g = id. By Corollary 1.2, either g(M 3 ) = M 4 , g(M 4 ) = (M 2 ) or g(M 3 ) = M 2 , g(M 2 ) = (M 4 ).
In both cases, g is of order 3, a.e., g 3 = id. Use Corollary 1.2 again we know that G is generated by g. So the claim follows. Hence, we have

#O G (m) = #G = 1 or 3. There must be some h ∈ G with h(m) = c or c . Otherwise, O G (m) does not contain c, c . For any point p ∈ O G (m), the point R D (p) ∈ O G (m) and R D (p) = p. Then #O G (m) is even, which is impossible.
By Lemma 1.7, h • f induces a normalizaed quasisymmetric map between the weak tangent W Fn,p (0) and W Fn,p (c) or W Fn,p (c ). This contradicts Proposition 1.4. So we have proved the proposition.

Proof of main theorems

Proof of Theorem 1.2. We adopt the notations as in previous. The proof of Proposition 1.5 implies that G is a cyclic group of order 3 or a trivial group. To prove the theorem, it suffices to show that the former case cannot happen. We argue by contradiction and assume that G is a cyclic group of order 3.

By Theorem 1.1, there exists a quasisymmetric map f from F n,p onto some round carpet S. After post-composing suitable fraction linear transformation, we can assume that the f (O) is the unit disc D and f (M 1 ) lies in D with center (0, 0). Then f induces the isomorphism Proof of the claim: In fact, by the proof of Proposition 1.5, we may assume that

f * : QS(F n,p ) -→ QS(S) g -→ f • g • f -1 . Combined with Theorem 1.2, f * (G ) is a cyclic group of order 3 consisting of Möbius transformations. Moreover, elements in f * (G ) preserves ∂D and the circle O 1 = f (M 1 ). Hence we have f * (G ) = {id, z → e 2πi/3 z, z → e 4πi/3 z}. Claim: O 2 = f (M 2 ), O 3 = f (M 3 ), O 4 = f (M 4 )
G =< g >, where g(M 3 ) = M 4 , g(M 4 ) = M 2 and g(M 2 ) = M 3 . Note that O 3 = f (M 3 ) = f • g(M 2 )) = f • g • f -1 (O 2 )
where f • g • f -1 is equal to the rotation z → e 2πi/3 z. Similarity, one can show that

O 4 = f • g • f -1 (O 3 ).
As a result, O 3 is obtained from O 2 by a rotation of angle 2π/3 and O 4 is obtained from O 2 by a rotation of angle 4π/3. The claim follows.

Let R be the rotation in the isometry group of

F n,p with R(M 1 ) = M 2 , R(M 2 ) = M 3 , R(M 3 ) = M 4 , and R(M 4 ) = M 1 . By Theorem 1.2, the composition h = f • R • f -1 : S → S is also a Möbius transformation which maps ∂D → ∂D, O 2 → O 3 , O 3 → O 4 . Such a Möbius transformation must be ϕ = z → e 2πi/3 z.
If not, let ϕ be other Möbius transformation satisfy the conditions. Then ϕ • ϕ -1 fixes three non-concentric circles

∂D, O 2 and O 3 and so ϕ • ϕ -1 = id. Hence ϕ = ϕ. But h(O 1 ) = O 2 , which is impossible.
So the theorem follows.

Proof of Theorem 1.3. Suppose there exists a quasisymmetric map f :

F n,p → F n ,p . Firstly, we claim that f (O) = O , f ({M 1 , M 2 , M 3 , M 4 }) = {M 1 , M 2 , M 3 , M 4 }.
Indeed, from Theorem 2, we know that every quasisymmetric self-map of F n,p and F n ,p is isometry and so preserves the peripheral circle O and O . For any

g in QS(F n,p ), f • g • f -1 is a quasisymmetric self-map of F n ,p and f • g • f -1 (f (O)) = f (O). So f (O) is fixed by any element in QS(F n ,p . Hence we have f (O) = O . If for some inner circles M i , say M 1 , of F n,p , f (M 1 ) is not an inner circle of F n ,p , then by Proposition 1.2, f extension to a quasiconformal self-map of S 2 . We have mod Fn,p (Γ(M 1 , O)) = mod F n ,p (Γ(f (M 1 ), O )) and mod F n ,p (Γ(M 1 , O )) = mod Fn,p Γ(f -1 (M 1 ), O). While Lemma 1.5 implies mod F n ,p (Γ(f (M 1 ), O)) < mod F n ,p (Γ(M 1 , O)) and mod Fn,p Γ(f -1 (M 1 ), O) ≤ mod Fn,p (Γ(M 1 , O)).
Hence mod Fn,p (Γ(M 1 , O)) < mod Fn,p (Γ(M 1 , O)) and we get a contradiction. Secondly, by pre-composing and post-composing with Euclidean isometries, we can assume that f is orientation-preserving and f (M 1 ) = M 1 . We claim that f ((0, 0)) = (0, 0) and f ((1, 1)) = (1, 1) or interchanges them and f (M 3 ) = M 3 . In fact, the orientationpreserving quasisymmetric map

f -1 • R D • f • R D : F n,p → F n,p fixes peripheral circles O and M 1 . Then, by Theorem 1.2, f -1 • R D • f • R D is a Euclidean
isometry and so it is the identity on F n,p . This implies f • R D = R D • f . Hence the claim follows.

We now distinguish two cases to analyze. Case (1) f ((0, 0)) = (0, 0) and f ((1, 1)) = (1, 1).

We denote the reflection in the line {(x, y) ∈ R 2 : x + y = 1} by R D . Then the map

f -1 • R D • f • R D is an orientation-preserving quasisymmetric map in QS(F n,p
), fixes peripheral circles O, M 1 , and the point (0, 0). Hence this map is the identity on F n,p and so

f • R D = R D • f . It follows that f fixes (1, 0
) and (0, 1) or interchanges them. Since f is orientation-preserving, the latter cannot happen. By Theorem 1.4 the map f must be the identity. Hence (n, p) = (n , p ).

Case (2) f ((0, 0)) = (1, 1) and f ((1, 1)) = (0, 0). The map g = R D • f • R D : F n,p → F n ,
p is an orientation-preserving quasisymmetry which fixes points (0, 0) and (1, 1) and peripheral circle O and maps

M 1 to M 3 . Similar to Case (1), g -1 • R D • g • R D is an orientation-preserving isometry map fixing (0, 0), (1, 1)
and O and so is the identity. Then g fixes (1, 0) and (0, 1) or interchanges them. The orientation-preserving of g implies the latter case is impossible. By Theorem 1.4 the map g is the identity, which contradicts with g(M 1 ) = M 3 . So case (2) can not happen. Our arguments in this chapter apply to a more general class of Sierpiński Carpets

Remark

F n,p,r , r ≥ 1, p ≥ 1, n ≥ 5, 1 ≤ p + r < n 2 . Let Q (0) n,p,r = [0, 1] × [0, 1]. Subdivide Q (0)
n,p,r into n 2 subsquares and remove the interior of four bigger subsquares with side-length r/n and is of distance √ 2p/n to one of the four corner points of Q (0) n,p,r . So the resulting set

Q (1)
n,p,r has (n 2 -4r 2 ) subsquares with side-length 1/n. Repeating the operation to the subsquares, we obtain Q (2) n,p,r . Inductively, we have

Q (k) n,p,r . Then the carpet F n,p,r = T k≥0 Q (k) n,p,r . See Figure 1.5. Note that F n,p = F n,p,1 .
Similarly, F n,p,r is not quasisymmetrically equivalent to S m , m ≥ 3 odd and QS(F n,p,r ) is the isometric group. Moreover, F n,p,r and F n ,r ,p are quasisymmetrically equivalent if and only if (n, p, r) = (n , p , r ). Since the proof of the above conclusions are of no essential difference from that of F n,p , we shall omit it.

Chapter 2 Quasisymmetric geometry of the carpet Julia sets

2.1 Introduction Let (X, d X ) and (Y, d Y ) be two metric spaces. If there exist a homeomorphism f : X → Y and a distortion control function η : [0, ∞) → [0, ∞) which is also a homeomorphism such that d Y (f (x), f (y)) d Y (f (x), f (z)) ≤ η d X (x, y) d X (x, z)
for every distinct points x, y, z ∈ X, then f is called a quasisymmetric map and (X, d X ), (Y, d Y ) are called quasisymetrically equivalent to each other. A basic question in quasiconformal geometry is to determine whether two given homeomorphic spaces are quasisymmetrically equivalent to each other. It is known that the question arises also in the classification of hyperbolic spaces and word hyperbolic groups in the sense of Gromov [BP, Kl]. See also [Bou] for examples of inequivalent spaces modelled on the universal Menger curve. In this chapter, we focus our attention on the Sierpiński carpets that arise as the Julia sets of rational maps.

According to [Wh], a set S ∈ C is called a Sierpiński carpet (carpet in short) if S has empty interior and can be expressed as

S = C \ S i∈N D i , where {D i } are pairwise disjoint Jordan disks with diam(D i ) → 0 as i → ∞.
The collection of the boundaries of the Jordan disk {∂D i } i∈N are called the peripheral circles of S. If each peripheral circle ∂D i is a round circle, then S is called a round carpet. All Sierpiński carpets are homeomorphic to each other, so the question about the quasisymmetric classification of the Sierpiński carpets arises naturally.

Actually, the study of the quasisymmetric equivalences between the Sierpiński carpets and round carpets was partially motivated by the Kapovich-Kleiner conjecture in the geometry group theory. This conjecture is equivalent to the following statement: if the boundary of infinity

∂ ∞ G of a Gromov hyperbolic group G is a Sierpiński carpet, then ∂ ∞ G is quasisymmetrically equivalent to a round carpet in C.
As the Julia set of a rational map, the first example of Sierpiński carpet was found by Tan [Mi1, Appendix F]. Later, the rational maps whose Julia sets are Sierpiński carpets appeared in many literatures. Such as the McMullen maps [DLU], the generated McMullen maps [XQY] and the quadratic rational maps [DFGJ] etc.

Let f be a rational map whose Julia set J f is a Sierpiński carpet. Two questions arise naturally: (Q1) Can one give another rational map g whose Julia set J g is a Sierpiński carpet, but J g is not quasisymmetrically equivalent to J f ? This question is equivalent to ask whether there exist quasisymmetrically inequivalent carpet Julia sets. (Q2) Can J f be quasisymmetrically equivalent to a round carpet?

Let X be a metric space. The conformal dimension of X is the infimum of the Hausdorff dimensions of all metric spaces which are quasisymmetrically equivalent to X. By definition, it is easy to see the conformal dimension is invariant under the quasisymmetric maps. For the first question stated above, Haïssinsky and Pilgrim constructed a sequence of hyperbolic rational maps with carpet Julia sets and showed that their conformal dimensions tend to two [START_REF] Haïssinsky | Quasisymmetrically inequivalent hyperbolic Julia sets[END_REF]Theorem 3]. This means that there are infinitely many quasisymmetrically inequivalent Sierpiński carpets as the Julia sets of rational maps.

The relative distance ∆(A, B) of two sets A and B in C is defined as

∆(A, B) := dist(A, B) min{diam(A), diam(B)} , (2.1)
where dist(A, B) := sup a∈A, b∈B |a -b| is the distance between A and B, and diam(A

) := sup a 1 ,a 2 ∈A |a 1 -a 2 | is the diameter of A. A set of Jordan curves C = {C i } i∈N is called
uniformly relatively separated if their pairwise relative distances are uniformly bounded away from zero. Specifically, there exists δ > 0 such that ∆(C i , C j ) ≥ δ for every two different i and j. The set C are uniform quasicircles if there exists K ≥ 1 such that each

C i in C is a K-quasicircle.
For the question (Q2), Bonk gave a sufficient condition on the carpets in C such that they can quasisymmetrically equivalent to some round carpets. He proved that a carpet S in C is quasisymmetrically equivalent to a round carpet if its peripheral circles are uniform quasicircles and is uniformly relatively separated [Bon, Corollary 1.2]. It is worth to mention that quasisymmetric maps preserve the uniform quasicircles and uniformly relatively separated properties. It is not hard to see that the peripheral circles of such S must be uniform quasicircles but are not necessarily uniformly relatively separated.

Recently, Bonk, Lyubich and Merenkov studied the postcritically-finite rational maps whose Julia sets are Sierpiński carpets. They proved that if the Julia set of a subhyperbolic rational map is a Sierpiński carpet, then it is quasisymmetrically equivalent to a round carpet [BLM, Theorem 1.10]. They also consider the quasisymmetric group between the carpet Julia sets of postcritically-finite rational maps and proved that any quasisymmetric map ξ defined from a carpet J f onto a carpet J g must be the restriction of a Möbius transformation, where f and g are postcritically-finite rational maps [BLM, Theorem 1.1]. As a corollary, they proved that the group QS

(J f ), consisting of quasisymmetric self-map of J f , is finite [BLM, Corollary 1.2].
In this chapter, we study carpet Julia sets in postcritically-infinite case.

2.1.1 Statement of the main results.

The ω-limit set w(x) of a point x ∈ C under a rational map f is defined as the set of accumulation points in the orbit of x. More precisely, ω(x) := {y ∈ C : there exists a sequence {k n } n∈N such that lim n→∞ f •kn (x) = y}. Obviously, ω(x) is f -forward invariant. We establish a sufficient condition on the carpet Julia sets such that they are quasisymmetrically equivalent to some round carpets.

Theorem 2.1. Let f be a rational map whose Julia set J f is a Sierpiński carpet. If the boundaries of the periodic Fatou components are disjoint with the ω-limit sets of the critical points, then the peripheral circles of J f are uniform quasicircles and uniformly relatively separated. In particular, J f is quasisymmetrically equivalent to a round carpet.

Recall that a rational map is sub-hyperbolic if every critical orbit is either finite or converges to an attracting periodic orbit. Note that the boundary of each Fatou component cannot contain any critical point if the Julia set is a Sierpiński carpet. By Theorem 2.1, we have following immediate corollary.

Corollary 2.1. Let f be a sub-hyperbolic rational map whose Julia set J f is a Sierpiński carpet. Then the peripheral circles of J f are uniform quasicircles and uniformly relatively separated. In particular, J f is quasisymmetrically equivalent to a round carpet.

A critical point c of f is called recurrent if c ∈ ω(c).
A rational map f is called semi-hyperbolic if and only if the Julia set J f contains neither parabolic periodic points nor recurrent critical points (see [Ma] and [Yin]). It was known that the Julia set of a semi-hyperbolic rational map is locally connected and has measure zero or equal to C. Theorem 2.2. Let f be a semi-hyperbolic rational map whose Julia set J f is a Sierpiński carpet. Then the peripheral circles of J f are uniform quasicircles. Moreover, they are uniformly relatively separated if and only if the ω-limit sets of the critical points are disjoint with the boundaries of periodic Fatou components.

If a rational map is not semi-hyperbolic, then the boundary of some Fatou component may not be a quasicircle although it is a Jordan curve. For example, one can construct a rational map f whose Julia set is a Sierpiński carpet but the Julia set J f contains a parabolic periodic point. The corresponding parabolic Fatou component contains exactly one petal and has infinitely many cusps on its boundary. Thus the boundary of this Fatou component cannot be a quasicircle. In this case, J f cannot quasisymmetrically equivalent to a round carpet. See Figure 2.1. As a corollary, we have the following theorem.

Theorem 2.3. Let f be a semi-hyperbolic rational map whose Julia set J f is a Sierpiński carpet. Then the quasisymmetric group QS(J f ) is discrete.

Outline of the proof and the organization of this chapter.

We are mainly interested on the condition when a carpet Julia set is quasisymmetrically equivalent to a round carpet. By Bonk's criterion, this motivates us to find the condition when the peripheral circles of a carpet Julia set are uniform quasicircles and when they are uniformly relatively separated.

In order to prove the peripheral circles of some carpet Julia sets are uniform quasicircles, we first discuss the periodic Fatou components and prove that they are quasicircles if their boundaries avoid the parabolic periodic points and the points in the ω-limit sets of the recurrent critical points (Lemma 2.9). Therefore, all peripheral circles are quasicircles by using Sullivan's eventually periodic theorem. In order to prove the uniformity, we discuss two cases. The first case, suppose that all the periodic Fatou components are disjoint with the ω-limit sets of the critical points. Then for each periodic Fatou component U , one can find a large Jordan disk V such that V \ U is an annulus and all components of the preimages of V \ U are annuli whose moduli have uniform lower bound. By using a distortion argument, one can prove that all peripheral circles are uniform quasicircles (Proposition 2.1). The second case, suppose that the rational map is semi-hyperbolic. Then the corresponding Julia set (and hence all the periodic Fatou components) contains neither parabolic periodic points nor recurrent critical points. One can also prove that all peripheral circles are uniform quasicircles by using Mañé's theorem and its variation (Theorem 2.5, Lemma 2.6 and Proposition 2.2).

In order to prove the peripheral circles of some carpet Julia sets are uniformly relatively separated, we first establish a lemma which asserts that the modulus can control the relative distance (Lemma 2.3). Then we prove the peripheral circles are uniformly relatively separated by showing that all moduli of the annuli between two different peripheral circles have a lower positive bound (Proposition 2.3).

This chapter is organized as follows: In §2.2, we prepare some distortion lemmas for the proofs of Theorems 2.1 and 2.2. Moreover, we prove that the modulus can control the relative distance. In §2.3, we first prove some propositions about the properties of uniform quasicircles and uniformly relatively separated. Then we prove Theorem 2.1 by using Bonk's criterion and prove Theorem 2.2 by combining Bonk's criterion and Mañé-Yin's characterization on semi-hyperbolic rational maps. In the last section, using the combinatorial method and renormalization theory, we construct a critically-infinite semi-hyperbolic rational map whose Julia set is quasisymmetrically equivalent to a round carpet.

Some distortion estimations

In this section, we give some distortion estimations and useful lemmas, which will be used in the next section. We use D := {z : |z| < 1} to denote the unit disk on the complex plane C.

Theorem 2.4 (Koebe's distortion theorem, [Pom, p. 9]). Let f : D → C be a univalent function. Then for every z ∈ D, one has

|f (0)| |z| (1 + |z|) 2 ≤ |f (z) -f (0)| ≤ |f (0)| |z| (1 -|z|) 2 ; and
(2.2)

|f (0)| 1 -|z| (1 + |z|) 3 ≤ |f (z)| ≤ |f (0)| 1 + |z| (1 -|z|) 3 .
(2.3)

Let A be an annulus with non-degenerated boundary components. Then there exists a conformal map sending A to a standard annulus {z ∈ C : 0 < r < |z| < 1}, where r > 0 is uniquely determined by A. As an invariant under conformal maps, the modulus of A is defined as mod(A) = 1 2π log(1/r). A set in C is called a Jordan disk if it is homeomorphic to the unit disk D and its boundary is a Jordan curve. Let A and B be two open sets in

C. We use the notation 'A B' if the closure A is contained in B. Lemma 2.1. Let U i V i = C be a pair of Jordan disks, where i = 1, 2. Suppose that mod(V 2 \ U 2 ) ≥ m > 0 and f : V 1 → V 2 a conformal map with f (U 1 ) = U 2 . Then there exists a constant C(m) ≥ 1 depending only on m such that for any x, y, z, w ∈ U 1 , one has 1 C(m) |x -y| |z -w| ≤ |f (x) -f (y)| |f (z) -f (w)| ≤ C(m) |x -y| |z -w| .
Proof. The proof is based on applying Koebe's distortion theorem. Without loss of generality, suppose that x = y and z = w are contained in the interior of U 1 . If not, we can enlarge U 1 appropriately. By Riemann's mapping theorem, there exists a conformal mapping g : (Ω, D) → (U 1 , V 1 ) which maps the unit disk D onto V 1 and a simply connected domain Ω onto U 1 . In particular, we require that g(0) = x.

We claim that there exists a positive constant r := r(m) < 1 depending only on m such that Ω ⊂ D r := {z : |z| < r}. Let ζ ∈ ∂Ω be the farthest point such that dist(0, ∂Ω) = |ζ|. Then D \ Ω is an annulus separating 0 and ζ from the unit circle. By Grötzsch's module theorem [LV, p. 54], we have

m ≤ mod(V 1 \ U 1 ) = mod(D \ Ω) ≤ µ(|ζ|),
where r → µ(r) is a continuous and strictly decreasing function defined on the interval (0, 1). This means that |ζ| ≤ µ -1 (m) and the claim follows if we set r = µ -1 (m).

Now we consider f • g : D → V 2 and g : D → V 1 . For every η ∈ Ω, by using (2.3) in Theorem 2.4, we have |f (x)| |g (0)| 1 -r (1 + r) 3 ≤ |(f • g) (η)| = |f (g(η))| |g (η)| ≤ |f (x)| |g (0)| 1 + r (1 -r) 3 . (2.4) Also, we have |g (0)| 1 -r (1 + r) 3 ≤ |g (η)| ≤ |g (0)| 1 + r (1 -r) 3 .
(2.5) Combine (2.4) and (2.5), it follows that for every ξ ∈ U 1 , we have

|f (x)| (1 -r) 4 (1 + r) 4 ≤ |f (ξ)| ≤ |f (x)| (1 + r) 4 (1 -r) 4 . (2.6)
Therefore, for x, y, z, w ∈ U 1 , by (2.6), we have

|f (x) -f (y)| ≤ (1 + r) 4 (1 -r) 4 |f (x)| • |x -y| and |f (z) -f (w)| ≥ (1 -r) 4 (1 + r) 4 |f (x)| • |z -w|. Set C(m) = (1 + r(m)) 8 /(1 -r(m)) 8 . The proof is complete.
Let U be a hyperbolic disk in C and E a connected and compact subset of U containing at least two points. For any z 1 , z 2 ∈ E, the turning of E about z 1 and z 2 is defined by

Λ(E; z 1 , z 2 ) = diam(E) |z 1 -z 2 | . It is easy to see that 1 ≤ Λ(E; z 1 , z 2 ) ≤ ∞ and Λ(E; z 1 , z 2 ) = ∞ if and only if z 1 = z 2 .
By definition (see for example, [LV, p. 100]), a Jordan curve C is called a quasicircle if there exists a positive constant K ≥ 1 such that for any different points x, y ∈ C, the turning of γ about x and y satisfies

Λ(γ; x, y) ≤ K,
where γ is one of the two components of C \ {x, y} with smaller diameter.

Lemma 2.2. Let U i V i = C be a pair of Jordan disks, where i = 1, 2. Suppose that mod(V 2 \ U 2 ) ≥ m > 0 and f : V 1 → V 2 a conformal map with f (U 1 ) = U 2 . If ∂U 2 is a K- quasicircle, then there is a constant C(K, m) ≥ 1 such that ∂U 1 is a C(K, m)-quasicircle.
Proof. By definition, if ∂U 2 is a K-quasicircle, then there exists a constant C(K) > 0 such that for any different points z 1 , z 2 ∈ ∂U 2 , the turning of γ about z 1 and z 2 satisfies

Λ(γ; z 1 , z 2 ) = diam(γ) |z 1 -z 2 | ≤ C(K), (2.7)
where γ is one of the component of ∂U 2 \ {z 1 , z 2 } with smaller diameter. Let x, y ∈ ∂U 1 be two different points which divide the quasicircle ∂U 1 into two closed subcurves α and β. Without loss of generality, let α ⊂ ∂U 1 be the subcurve with smaller diameter. Moreover, let z, w ∈ α such that diam(α) = |z -w|. By Lemma 2.1, we have

Λ(α; x, y) = |z -w| |x -y| ≤ C(m) |f (z) -f (w)| |f (x) -f (y)| , (2.8)
where C(m) is the constant appeared in Lemma 2.1. Note that f (x), f (y) divide the quasicircle ∂U 2 into two parts f (α) and f (β).

If diam(f (α)) ≤ diam(f (β)), then by (2.7) and (2.8), we have

Λ(α; x, y) ≤ C(m) diam(f (α)) |f (x) -f (y)| ≤ C(m)C(K).
(2.9)

If diam(f (α)) > diam(f (β)), let z , w ∈ β such that diam(β) = |z -w |.
By (2.7) and Lemma 2.1, we have

Λ(α; x, y) ≤ Λ(β; x, y) ≤ |z -w | |x -y| ≤ C(m) |f (z ) -f (w )| |f (x) -f (y)| ≤ C(m) diam(f (β)) |f (x) -f (y)| ≤ C(m)C(K).
(2.10) Combine (2.9) and (2.10), the Lemma follows.

Recall that the relative distance ∆(A, B) of two subsets A and B in C is defined in (2.1). Now we prove that relative distance of two disjoint Jordan curves can be controlled by the modulus of the annulus between them.

Lemma 2.3 (Modulus controls the relative distance). Let A ⊂ C be an annulus with two boundary components C 1 and C 2 . If the modulus of A satisfies mod(A) ≥ m > 0, then there exists a constant C(m) > 0 depending only on m such that the relative distance of

C 1 and C 2 satisfies ∆(C 1 , C 2 ) ≥ C(m) > 0.
Proof. Without loss of generality, we assume that A ⊂ C, C 1 , C 2 are not singletons and

0 < diam(C 1 ) ≤ diam(C 2 ) and dist(C 1 , C 2 ) = |x -y| (2.11) for x ∈ C 1 and y ∈ C 2 . There exists a point z = x in C 1 such that |x-z| = sup a∈C 1 |a-x|. Therefore, we have diam(C 1 ) ≤ 2|x -z|.
(2.12)

Consider the linear function h(t) = (t-x)/(x-z), which maps x, y, z to 0, (y-x)/(x-z) and -1. Then h(A) is an annulus separating the points 0 and -1 from h(y) and ∞, respectively. Let

R = |h(y)| = |(y -x)/(x -z)|.
By Teichmüller's Module Theorem (see for example, [LV, p. 56]), we have

m ≤ mod(A) = mod(h(A)) ≤ 2 µ s 1 1 + R ,
where r → µ(r) is a continuous and strictly decreasing map defined on the interval (0, 1). By (2.11) and (2.12), this means that the relative distance of C 1 and C 2 is

∆(C 1 , C 2 ) = dist(C 1 , C 2 ) diam(C 1 ) ≥ |x -y| 2|x -z| = R 2 ≥ 1 2 1 (µ -1 (m/2)) 2 -1 := C(m).
The proof is complete.

Lemma 2.4 ( [KL, Lemma 4.5]). Let U i V i = C be a pair of Jordan disks, where i = 1, 2. Suppose that g : V 1 → V 2 is a proper holomorphic map of degree d ≥ 1 and U 1 is a component of g -1 (U 2 ). Then mod(V 1 \ U 1 ) ≤ mod(V 2 \ U 2 ) ≤ d mod(V 1 \ U 1 ).
Let U be a hyperbolic disk in C and z ∈ U . The shape of U about z, denoted by Shape(U, z), is defined as

Shape(U, z) = max w∈∂U |w -z| min w∈∂U |w -z| = max w∈∂U |w -z| dist(z, ∂U ) .
It is obvious that Shape(U, z) = ∞ if and only if U is unbounded and Shape(U, z) = 1 if and only if U is a round disk centered at z. In other cases, 1 < Shape(U, z) < ∞.

Lemma 2.5 ( [QWY, Lemma 6.1]). (1) For all z ∈ U 1 , the shape satisfies

Let U i V i = C be a pair of Jordan disks with mod(V 2 \ U 2 ) ≥ m > 0, where i = 1, 2. Suppose that g : V 1 → V 2 is a proper holomorphic map of degree d ≥ 1 and U 1 is a component of g -1 (U 2 ).
Shape(U 1 , z) ≤ C 1 (d, m) Shape(U 2 , g(z)).
(2) For any connected and compact subset E of U 1 with the cardinal number E ≥ 2 and any z 1 , z 2 ∈ E, the turning satisfies

Λ(E; z 1 , z 2 ) ≤ C 2 (d, m) Λ(g(E); g(z 1 ), g(z 2 )).
Lemma 2.5 means that the shape and the turning of the interior boundary of an annulus can be controlled under a proper holomorphic map if the modulus of this annulus has a lower bound.

Proofs of the Main Theorems

If a rational map f whose Julia set J f is a Sierpiński carpet, then f cannot be a polynomial. In fact, the intersection of the closure of the bounded Fatou components (if any) and the basin of infinity of f is non-empty provided f is a polynomial since the Julia set J f is the boundary of the basin of infinity. If we want to prove Theorem 2.1, we need to prove that the peripheral circles of the carpets are uniform quasicircles and uniformly relatively separated by Bonk's criterion.

Mañé's Theorem and a lemma.

We first give a theorem due to Mañé, which will be used frequently later.

Theorem 2.5 ( [Ma, Theorem II]). Let f : C → C be a rational map with degree at least two. If a point x ∈ J f is not a parabolic periodic point and is not contained in the ω-limit set of a recurrent critical point, then for any > 0 there exists an open neighborhood U x of x such that:

(C1) For all n ≥ 0, every component of f -n (U x ) has diameter ≤ ; (C2) There exists d > 0 such that for all n ≥ 0 and every connected component V of

f -n (U x ), the degree of f •n : V → U x is ≤ d.
When we pull back a Jordan disk U by a rational map f , there maybe exist a component W of f -1 (U ) which is not simply connected. If the boundary ∂U avoids the critical values, then ∂W is the union of finitely many disjoint Jordan curves {C i }. Moreover, we have f (C i ) = ∂U for each i. Note that W is a connected set whose boundary consists of finitely many Jordan curves. We have C \ W = S i V i , where each V i is a Jordan disk bounded by the Jordan curve C i . Since the restriction of f on V i is a holomorphic branched covering and f

(∂V i ) = ∂U , we have f (V i ) = C or f (V i ) = C \ U . In other words, the image of each component of the complement of W under f is either C or C \ U . See Figure 2.2 for an example. U f V 2 V 3 V 1 W W C 3 C 2 C 1
= C 1 ∪ C 2 ∪ C 3 . The complement of W consists of 3 simply connected components V 1 , V 2 and V 3 . In particular, f (V 1 ) = f (V 2 ) = C \ U and f (V 3 ) = C.
Moreover, W contains 4 critical points of f and V 3 \ W (the white annulus) contains two.

In the rest of this chapter, we only consider the rational maps whose Julia sets are not the whole complex sphere. Therefore, after conjugating f by a suitable Möbius transformation, we always assume that ∞ lies in the Fatou set. This means that J f is a compact set in C. In the following, we equip J f the Euclidean metric if not special specified. We use D(a, r) := {z ∈ C : |z -a| < r} to denote the round disk in C with the center a ∈ C and radius r > 0.

Lemma 2.6. Let f be a rational map with degree at least two and J f ⊂ C. Suppose that x ∈ J f is not a parabolic periodic point and is not contained in the ω-limit set of a recurrent critical point. Then there exists an open neighborhood U x of x such that (C3) For all n ≥ 0, every connected component of f -n (U x ) is simply connected.

Proof. By the assumption that ∞ ∈ J f , the grand orbit of ∞ lies in the Fatou set of f . Let δ 0 > 0 be a small positive number such that

0 < δ 0 ≤ dist(f -1 (∞), J f )/2.
(2.13) By Theorem 2.5, there exists an open neighborhood U x of x such that every component of f -n (U x ) has diameter ≤ δ 0 for all n ≥ 0. Let U x := D(x, δ x ) be the largest round disk which is contained in U x . We claim that every component

W n of f -n (U x ) is simply connected. If not, let V n be a bounded component of C \ W n , where n ≥ 1. Then ∂V n ⊂ ∂W n and so diam(V n ) ≤ δ 0 . This means that V n cannot intersect f -1 (∞). Inductively, one can easily check that f •k (V n ) ∩ f -1 (∞) = ∅ for 0 ≤ k ≤ n -1. It follows that ∞ ∈ f •n (V n ), which is a contradiction since f •n (V n ) = C or f •n (V n ) = C \ U x .
Therefore, such V n does not exist. This means that W n is simply connected. The proof is complete.

Lemma 2.6 is useful in the following since we need to obtain the simply connected preimages of a simply connected domain.

Sufficiency for the property of uniform quasicircles.

In this subsection, we prepare some lemmas and give two sufficient conditions such that the boundaries of the Fatou components are uniform quasicircles. We first discuss the regularity of the boundaries of the periodic Fatou components and then spread the results to their all preimages.

Lemma 2.7. Let Γ be a Jordan curve in the plane C. Then there exists a constant δ Γ > 0 depending only on Γ such that, for any Jordan subarc γ ⊂ Γ with diam(γ) ≤ δ Γ , one has diam(γ) < diam(Γ \ γ).

Proof. Consider the function h : Γ × Γ → R which is defined by h(x, y) = diam(L (x, y)), where L (x, y) is one of the two components of Γ \ {x, y} with larger diameter. Obviously, the map h is continuous. Since Γ × Γ is compact, the function h has a minimum δ > 0. Then the lemma holds if we set δ Γ = δ /2. Lemma 2.8. Let f be a rational map with degree at least two and U a Fatou component which is a Jordan disk. Then f | ∂U is a local homeomorphism.

Proof. The image V = f (U ) is a Fatou component and hence a domain. Since f maps the boundary of U to that of V , it follows that V is a Jordan disk as well and f (∂U ) = ∂V . For an annulus A with the outer boundary ∂V and the inner boundary surrounding all the critical values in V , then A = (f | U ) -1 (A) is also an annulus in U with the outer boundary coinciding with ∂U by Riemann-Hurwitz's formula. Then f : A → A is an unbranched covering. Thus the restriction of f on ∂U is a local homeomorphism.

Lemma 2.9 (The boundaries of periodic Fatou components are quasicircles). Let f be a rational map with degree at least two. Suppose that U is a periodic Fatou component of f whose boundary ∂U is a Jordan curve and ∂U contains neither parabolic periodic points nor the points in ω(c) for any recurrent critical point c. Then ∂U is a quasicircle.

Proof. After iterating f by several times, we can assume that the periodic Fatou component U is fixed by f . Without loss of generality, we suppose that ∞ ∈ J f . Let = δ 0 > 0 be the number defined as in (2.13). For any x ∈ ∂U , by Theorem 2.5 and Lemma 2.6, there exists an open neighborhood U x := D(x, δ x ) of x satisfying (C1), (C2) and (C3). Since ∂U is compact and ∂U ⊆ S x∈∂U D(x, δ x /2), one can select a collection of finite number of elements

U = {D(x 1 , δ x 1 /2),• • • ,D(x N , δ x N /2)} such that ∂U is covered by U.
Let δ 1 > 0 be the Lebesgue number of U. Then every subset of ∂U with diameter ≤ δ 1 must be contained in at least one open disk D(x i , δ x i /2) for some 1 ≤ i ≤ N .

By Lemma 2.8, the restriction of f on ∂U is a local homeomorphism. This means that there exists a number δ 2 > 0 such that for any subset E ⊂ ∂U with diam(E) ≤ δ 2 , the restriction of f on E is a homeomorphism. Recall that δ ∂U > 0 is the number depending only on ∂U which is defined in Lemma 2.7. We define

δ := min ¨δ1 M , δ 2 , δ ∂U M « , (2.14)
where

M := 1 + sup{|f (z)| : dist(z, J f ) ≤ δ 0 } < +∞.
Let x, y be two different points in ∂U . We use γ := L(x, y) to denote one of the two components of ∂U \ {x, y} with the smaller diameter. Now we divide the argument into two cases.

Case 1: Suppose that diam(γ) ≥ δ.

Define E := {(ξ, η) ∈ ∂U × ∂U : diam(L(ξ, η)) ≥ δ}. Then E is compact and (ξ, ξ) ∈ E. The function h : ∂U × ∂U → R + defined by (ξ, η) → diam(L(ξ, η)) |ξ -η| is continuous on E.
Then h has a maximum K 1 on E since E is compact. In particular, the turning of γ about x and y satisfies

Λ(γ; x, y) = diam(γ) |x -y| ≤ K 1 . (2.15)
Case 2: Suppose that diam(γ) < δ. Denote γ n := f •n (γ) for n ≥ 0. Note that the forward orbit of γ will eventually cover ∂U . There is a smallest integer n ≥ 0 such that

diam(γ n ) < δ and diam(γ n+1 ) = diam(f (γ n )) ≥ δ. (2.16)
By the choice of δ in (2.14), we know that f •(n+1) | γ is a homeomorphism and so γ n+1 is a Jordan arc connecting f •(n+1) (x) and f •(n+1) (y). Note that there exist two points

z 1 , z 2 ∈ γ n , such that diam(γ n+1 ) = |f (z 1 ) -f (z 2 )| ≤ Z [z 1 ,z 2 ] |f (z)| |dz| ≤ M |z 1 -z 2 | ≤ M diam(γ n ) ≤ M δ ≤ min{δ 1 , δ ∂U },
(2.17)

where [z 1 , z 2 ] is the straight segment connecting z 1 and z 2 . By the definition of δ ∂U and Lemma 2.7, the Jordan arc γ n+1 is one of the two components of ∂U \ {f •(n+1) (x), f •(n+1) (y)} with smaller diameter. Since diam(γ n+1 ) ≥ δ by (2.16), as discussed in Case 1 above, we have

Λ(γ n+1 ; f •(n+1) (x), f •(n+1) (y)) ≤ K 1 .
(2.18)

By the definition of δ 1 , there exists a disk D(x i , δ

x i /2) such that γ n+1 ⊂ D(x i , δ x i /2) for some 1 ≤ i ≤ N since diam(γ n+1 ) ≤ δ 1 by (2.17). Let B n+1 (x i , δ x i /2) and B n+1 (x i , δ x i ), re- spectively, be the components of f -n-1 (D(x i , δ x i /2)) and f -n-1 (D(x i , δ x i )) both containing γ.
Note that both of them are simply connected by the choice of δ x i . Applying Lemma 2.5 to the case

(U 1 , V 1 ) = (B n+1 (x i , δ x i /2), B n+1 (x i , δ x i )), (U 2 , V 2 ) = (D(x i , δ x i /2), D(x i , δ x i )) and m = 1
2π log 2, together with (2.18), we have

Λ(γ; x, y) ≤ C 2 (d i )Λ(γ n+1 ; f •(n+1) (x), f •(n+1) (y)) ≤ C 2 (d i )K 1 , (2.19)
where C 2 (d i ) is a constant depending only on d i and d i > 0 is the number appeared in Theorem 2.5 which depends on x i . Let

K = K 1 (1 + max 1≤i≤N C 2 (d i )).
Then Λ(γ; x, y) ≤ K holds for any different x, y ∈ ∂U by (2.15) and (2.19). By the arbitrariness of x and y, this means that ∂U is a quasicircle. The proof is completed. Now we need to consider when the boundaries of all the Fatou components are uniform quasicircles. According to Sullivan [Sul], each Fatou component of a rational map is eventually periodic. It is natural to consider the pull back of the periodic Fatou components and then using some distortion lemmas to control the shape of pre-periodic Fatou components. To do this, it is necessary to construct a larger simply connected domain surrounding the periodic Fatou component such that all components of its preimages under the n-th iteration are still simply connected.

Proposition 2.1 (Uniform quasicircles I). Let f be a rational map such that the boundary of each Fatou component is a Jordan curve. Suppose that all the boundaries of periodic Fatou components are disjoint with the ω-limit sets of the critical points. Then the boundaries of all the Fatou components of f are uniform quasicircles.

Proof. If all periodic Fatou components of f are disjoint with the ω-limit sets of the critical points, then f has no parabolic periodic points. By Lemma 2.9 and Sullivan's eventually periodic theorem, all the boundaries of the Fatou components of f are quasicircles. We only need to prove that they are uniform quasicircles.

Let U be the collection of all the Fatou components such that each of them is either a critical Fatou component (contains at least one critical point) or a periodic Fatou component. We use

U := O + (U ) = {U 1 , • • • , U n }
to denote the union of the forward orbits of all the Fatou components in U . Note that the number of Fatou components in U is finite since U is. Therefore, there exists a constant

K 1 > 1 such that each ∂U i is a K 1 -quasicircle. For each 1 ≤ i ≤ n, let V i be a Jordan disk such that V i \ U i is
an annulus which is disjoint with the ω-limit sets of the critical points.

Let mod

(V i \ U i ) = m i > 0 for 1 ≤ i ≤ n. For each Fatou component U ∈ U, there exists a minimal number k ≥ 1 such that f •k (U ) = U i ∈ U for some i. Let V be the component of f -k (V i ) containing U . Then f •k : V → V i is conformal and V is a Jordan disk since V i contains no points in the critical orbits. By Lemma 2.2, the boundary ∂U is a C(K 1 , m i )-quasicircle, where C(K 1 , m i ) is a constant depending only on K 1 and m i . Let K = max 1≤i≤n C(K 1 , m i ). Then the boundary of each Fatou component of f is a K-quasicircle.
By the arbitrariness of U , this means that the Fatou components of f are uniform quasicircles. The proof is completed.

Recall that a rational map f is called semi-hyperbolic if and only if the Julia set J f contains neither parabolic periodic points nor recurrent critical points.

Proposition 2.2 (Uniform quasicircles II). Let f be a semi-hyperbolic rational map such that the boundary of each Fatou component is a Jordan curve. Then the boundaries of all the Fatou components of f are uniform quasicircles.

Proof. By Lemma 2.9 and Sullivan's eventually periodic theorem, it follows that all the boundaries of the Fatou components of f are quasicircles since f is semi-hyperbolic. We only need to prove that they are uniform quasicircles. According to [Yin, Theorem 1.2], the Julia set J f is locally connected. Then for any > 0, there are only finitely many Fatou components with diameter ≥ [Mi3, Lemma 19.5].

Without loss of generality, we suppose that ∞ ∈ J f . Let = δ 0 > 0 be the number defined as in (2.13). For any x ∈ J f , by Theorem 2.5 and Lemma 2.6, there exists an open neighborhood U x := D(x, δ x ) of x satisfying (C1), (C2) and (C3). Since J f is compact, there exists a collection of finite number of elements

U = {D(x 1 , δ x 1 /2),• • • ,D(x N , δ x N /2)} such that J f is covered by U.
We use δ > 0 to denote the Lebesgue number of U. Then every subset of J f with diameter ≤ δ must be contained in at least one open disk

D(x i , δ x i /2) for some 1 ≤ i ≤ N .
We divide the collection of all the Fatou components F of f into two classes as following. Let F 0 be the collection of all the Fatou components such that each U ∈ F 0 is one of the following cases: (1) U contains at least one critical point; (2) U is periodic; (3) diam(U ) ≥ δ. Let F 1 := O + (F 0 ) be the set of the union of the forward orbits of all the Fatou components in F 0 . Define F 1 := F 1 ∪ f -1 (F 1 ). By Sullivan's eventually periodic theorem, the number of Fatou components in F 1 is finite since F 0 is also. Therefore, there exists a constant

K 1 > 1 such that each Fatou component in F 1 is a K 1 -quasicircle.
For any Fatou component U ∈ F \ F 1 , we have diam(U ) < δ. There exists a minimal

n U ≥ 1 such that f •n U (U ) ∈ f -1 (F 1 ) \ F 1 ⊂ F 1 and diam(f •n U (U )) < δ. Moreover, the map f •n U : U → f •n U (U ) is conformal.
By the definition of δ, there exists some disk

D(x i , δ x i /2) in U such that f •n U (U ) ⊂ D(x i , δ x i /2). We use B U and B U , respectively, to denote the components of f -n U (D(x i , δ x i /2)) and f -n U (D(x i , δ x i )) both containing U . Let x, y ∈ ∂U be two different points such that ∂U \ {x, y} = γ 1 ∪ γ 2 . Then f •n U (γ 1 ) and f •n U (γ 2 ) are both Jordan arcs connecting f •n U (x) with f •n U (y). Applying Lemma 2.5 (2) to the case (U 1 , V 1 ) = (B U , B U ), (U 2 , V 2 ) = (D(x i , δ x i /2), D(x i , δ x i )), m = 1 2π log 2, g = f •n U and E = γ j , where j = 1, 2, we have Λ(γ j ; x, y) ≤ C 2 (d i ) Λ(f •n U (γ j ); f •n U (x), f •n U (y)),
where C 2 (d i ) is a constant depending only on d i and d i > 0 is the number appeared in Theorem 2.5 which depends on x i . Then

min j∈{1,2} {Λ(γ j ; x, y)} ≤ C 2 (d i ) min j∈{1,2} {Λ(f •n U (γ j ); f •n U (x), f •n U (y))} ≤ C 2 (d i )K 1 . Let K = max 1≤i≤N C 2 (d i )K 1 .
Then ∂U is a K-quasicircle by the arbitrariness of x and y. By the arbitrariness of U , we know that each Fatou component of f is a K-quasicircle and K is a constant depending only on f . The proof is completed.

Figure 2.1 shows a rational map having a parabolic periodic point whose Julia set is a Sierpiński carpet but the peripheral circles of J f are not uniform quasicircles. Note that in Propositions 2.1 and 2.2, we do not require the closure of Fatou components are disjoint to each other. They can touch each other at the points on their boundaries. It seems that the conditions in Proposition 2.1 is much stronger than in Proposition 2.2. However, it is not true. One can construct a rational map with recurrent critical points, whose w-limit sets are disjoint with boundaries of Fatou components, using similar method as stated in Section 2.4.

Sufficiency for the property of uniformly relatively separated.

By Lemma 2.3, if the lower bound of the annuli between the boundaries of the Fatou components can be controlled, then one can prove that the peripheral circles of the carpet Julia set are uniformly relatively separated.

Proposition 2.3 (Uniformly relatively separated). Let f be a rational map whose Julia set J f is a Sierpiński carpet. If the boundaries of all periodic Fatou components contain no points in ω(c) for any critical point c ∈ J f , then the boundaries of Fatou components are uniformly relatively separated.

Proof. After iterating f by some times, we can assume that all the periodic Fatou components X 1 , • • • , X n have period precise one. For 1 ≤ i ≤ n, let Y i be a simply connected domain containing X i such that Y 1 , • • • , Y n are mutually disjoint and each annulus

H i := Y i \ X i contains no points in the critical orbits. Define m = min 1≤i≤n mod(H i ) > 0.
For any two different Fatou components U 1 and U 2 , there exist two minimal numbers

n 1 , n 2 ≥ 0 such that f •n 1 (U 1 ) = X k 1 and f •n 2 (U 2 ) = X k 2 for 1 ≤ k 1 , k 2 ≤ n, where X k 1 , X k 2 are Fatou components with period one. Since H k i contains no critical values of f •n i for i ∈ {1, 2}
and so the restriction of f •n i on each components of f -n i (H k i ) is an unbranched covering. By Riemann-Hurwitz's formula, it follows that each component of their preimages is an annulus. Therefore, there exist two simply connected domains

V 1 , V 2 surrounding U 1 , U 2 such that V i \ U i is a component of f -n i (H n i ) and deg(f •n i : V i → Y n i ) = deg(f •n i : U i → X n i ). Note that f •j 1 (U i ) T f •j 2 (U i ) = ∅ for 0 ≤ j 1 < j 2 ≤ n i
and f has only finitely many critical points. So the degree of f •n i | U i is bounded by some number N ≥ 1 depending only on f . Denote by A the annulus bounded by ∂U 1 and ∂U 2 in C. We now divide the arguments into two cases.

Case 1: Suppose that n 1 = n 2 . Then V 1 and V 2 are two disjoint components of

f -n 1 (Y n 1 ∪ Y n 2 ). By Lemma 2.4, we have mod(A) ≥ mod(V 1 \ U 1 ) + mod(V 2 \ U 2 ) ≥ mod(H k 1 )/N + mod(H k 2 )/N ≥ 2m/N.
Case 2: Suppose that n 1 > n 2 . We claim that V 1 and U 2 are disjoint. Otherwise, the

annulus V 1 \ U 1 intersects U 2 and so f •n 2 (V 1 \ U 1 ) intersects the fixed Fatou component X k 2 . Then H k 1 = f •(n 1 -n 2 ) (f •n 2 (V 1 \ U 1 )) joints with X k 2 , which contradicts with the choice of H k 1 . Then we have mod(A) ≥ mod(V 1 \ U 1 ) ≥ m/N .
Above all, the annulus A has modulus not less than m/N . By Lemma 2.3, U 1 and U 2 are relatively separated with the relative distance ∆(∂U 1 , ∂U 2 ) depending only on m and N . By the arbitrariness of U 1 and U 2 , the peripheral circles of the carpet Julia set are uniformly relatively separated. The proof is completed.

Note that the condition in Proposition 2.3 does not exclude the existence of parabolic points on the Julia set. Actually, the peripheral circles of the parabolic rational map appeared in Figure 2.1 are uniformly relatively separated.

The property of non-uniformly relatively separated.

If the peripheral circles of a carpet Julia set are uniformly relatively separated, a natural question is whether it implies that all the boundaries of pre-periodic Fatou components avoid the accumulation points of the critical orbits in the Julia set. We give the answer in the following proposition.

Proposition 2.4 (Non-uniformly relatively separated). Let f be a semi-hyperbolic rational map whose Julia set is a Sierpiński carpet. Suppose that there exists a Fatou component U of f such that ∂U ∩ ω(c) = ∅ for some critical point c ∈ J f . Then the boundaries of Fatou components of f are not uniformly relatively separated.

Proof. Without loss of generality, we suppose that ∞ ∈ J f . Let = δ 0 > 0 be the number defined as in (2.13). Let x ∈ ∂U ∩ ω(c). By Theorem 2.5 and Lemma 2.6, there exists a number δ x > 0 such that the open neighborhood U x := D(x, δ x ) satisfies (C1), (C2) and (C3). Since ∂U ∩ ω(c) = ∅ and J f is a Sierpiński carpet, it follows that the forward orbit of c is infinite. Let c kn := f •kn (c) be the point in the forward orbit of c converging to

x. Set kn := |x -c kn |. We have kn → 0 as n → ∞. Given 0 < δ < δ x , there exists sufficiently large N such that D(c kn , δ/2) ⊆ D(c kn , δ) ⊆ D(x, δ x ) for any n ≥ N .

Evidently, the round disks D(c kn , kn ), D(c kn , δ/2) and D(c kn , δ) satisfy (C1), (C2) and (C3) in Theorem 2.5 and Lemma 2.6. Pulling these three disks back by f •(kn-1) and f •kn respectively, we denote by X kn-1 , Y kn-1 , Z kn-1 , respectively, X kn , Y kn , Z kn the simply connected components of their preimages containing the critical value c 1 and the critical point c respectively. Let U kn-1 be a component of f -(kn-1) (U ) such that ∂X kn-1 ∩∂U kn-1 = ∅. Then we can choose a point x kn-1 ∈ ∂X kn-1 ∩ ∂U kn-1 . Note that such x kn-1 may be not unique. See Figure 2.3. Since c 1 is a critical value, there exist at least two different Fatou components U kn and U kn , which are both the preimages of U kn-1 such that X kn ∩ U kn = ∅ and X kn ∩ U kn = ∅. Let x kn ∈ ∂X kn ∩ ∂U kn and x kn ∈ ∂X kn ∩ ∂U kn be the preimages of x kn-1 . We will show the relative distance between U kn and U kn converges to zero as kn converges to zero.

c kn x δ 1 δ δ/2 ∂U f •(kn-1) c 1 x kn-1 X kn-1 Y kn-1 Z kn-1 ∂U kn-1 f c x kn x kn X kn Y kn Z kn ∂U kn ∂U kn
Applying the Lemma 2.5 (1) to the case

(U 1 , V 1 ) = (Y kn , Z kn ), (U 2 , V 2 ) = (D(
∆(∂U kn , ∂U kn ) = dist(∂U kn , ∂U kn ) min{diam(∂U kn ), diam(∂U kn )} ≤ |x kn -x kn | dist(c, ∂Y kn ) -max w∈∂X kn |w -c| ≤ 2 max w∈∂X kn |w -c| dist(c, ∂Y kn ) -max w∈∂X kn |w -c| ≤ 2C 2 (d)dist(c, ∂X kn ) C -1 1 (d) max w∈∂Y kn |w -c| -C 2 (d)dist(c, ∂X kn ) = 2C 2 (d) max w∈∂Y kn |w-c| C 1 (d) dist(c,∂X kn ) -C 2 (d)
.

(2.22)

On the other hand, by Lemma 2.4, the modulus of Y kn \ X kn satisfies

1 2π log max w∈∂Y kn |w -c| dist(c, ∂X kn ) ≥ mod(Y kn \ X kn ) ≥ 1 d mod(D(c kn , δ/2) \ D(c kn , kn )) = 1 2πd log δ 2 kn .
(2.23)

Note that kn → 0 as n → 0, it follows that the relative distance ∆(∂U kn , ∂U kn ) of ∂U kn and ∂U kn tends to zero as n tends to ∞ by (2.22) and (2.23). This means that the peripheral circles of J f are not uniformly relatively separated. The proof is completed.

Proofs of the main results.

We now give the proofs of the main results in the introduction by combining some propositions.

Proof of Theorem 2.1. By Propositions 2.1 and 2.3, the peripheral circles of carpet J f are uniform quasicircles and uniformly relatively separated. According to Bonk [Bon, Corollary 1.2], J f is quasisymmetrically equivalent to a round carpet.

Proof of Theorem 2.2. The theorem follows immediately by Propositions 2.2, 2.3 and 2.4.

Proof of Theorem 2.3. By Theorem 2.1, let g : J f → S be a quasisymmetric map sending J f to a round carpet S. According to [START_REF] Bonk | Uniformization of Sierpiński carpets in the plane[END_REF]Theorem 1.1], one can extend g : J f → S to a quasiconformal map from C to itself. Since f is semi-hyperbolic, the corresponding Julia set J f has measure zero by [START_REF] Yin | On the Julia set of semi-hyperbolic rational maps[END_REF]Theorem 1.3]. It is well known that quasiconformal maps on the plane preserve the measure zero. So the round carpet S has measure zero as well. By the rigidity of Schottky sets (see [BKM, Theorem 1.1]), the quasisymmetric group QS(S) consists of the restriction of Möbius transformations.

Note that g induces a group isomorphism

g * : QS(J f ) → QS(S) with g * (h) = g • h • g -1 .
We are left to show that QS(J f ) is discrete, i.e., there exists δ > 0 such that

inf h∈QS(J f )\{id J f } max z∈J f |h(x) -x| ≥ δ.
If not, there exists a pairwise distinct sequence {h k } k≥1 ⊆ QS(J f ) converging to id J f . Let C 1 , C 2 and C 3 be three different peripheral circles of J f . Then the Hausdorff distance between C i and h k (C i ) tends to zero as k tends to ∞. Since all h k (C i ) are either disjoint or coincides for k ≥ 1 and i ∈ {1, 2, 3}, it follows that h k (C i ) = C i for sufficiently large k. This means, for sufficiently large k, the Möbius transformation g * (h k ) fixes three disjoint round disks bounded by g(C i ), 1 ≤ i ≤ 3.

By the rigidity of Möbius transformation, these g * (h k ) must be the identity id S . It follows that h k = id J f , for sufficiently large k. This contradicts the choice of {h k } k≥1 . The discreteness of QS(J f ) is proved.

An example of postcritically-infinite carpet Julia set

In this section, we will construct a carpet Julia set of a rational map such that it is quasisymmetrically equivalent to a round carpet. However, the rational map f is semihyperbolic and has an infinite critical orbit in J f .

Let q : R/Z → R/Z be the doubling map defined by q(t) = 2t mod Z and

l(t) := 8 < : 2t if 0 ≤ t < 1/2, 2 -2t if 1/2 ≤ t < 1.
(2.24) be the length of the component (R/Z) \ {t, 1 -t} containing 0. Let T (t) = min{2t, 2 -2t} be the tent map on the interval [0, 1]. One can easily check that

T • l(t) = l • q(t) (2.25)
for all t ∈ [0, 1]. Actually, the map l(t) is equal to T (t). We use these notations here by following Tiozzo's paper [START_REF] Tiozzo | Topological entropy of quadratic polynomials and dimension of sections of the Manderbrot set[END_REF]p. 24].

Lemma 2.10. Let 0 ≤ α ≤ 1 be a real number. Then α is rational if and only if α is (pre-)periodic under the iteration of the doubling map q.

Proof. Obviously, this lemma holds for α = 0 or 1. Hence we assume that 0 < α < 1. If α is (pre-)periodic, then there exist two different integers k 1 , k 2 ≥ 0 such that q •k 1 (α) ≡ q •k 2 (α) mod Z. This means that there exists an integer k 3 such that

2 k 1 α = 2 k 2 α + k 3 . Then α = k 3 /(2 k 1 -2 k 2 ) is a rational number.
Conversely, we only need to prove that, if α = m/n is a rational number with the simplest expression, where n is odd, then α is periodic under q. Consider the restriction of q on the set S := {0, 1/n, • • • , (n -1)/n}:

h := q| S : t n → 2 t mod n n .
We claim that h is injective. Indeed, if h(t

1 /n) = h(t 2 /n), then 2(t 1 -t 2 ) = k n holds for some integer k. Since n is odd, it follows that k is even and |t 1 -t 2 | = | k 2 | • n ≤ n -1.
This means that k = 0 and t 1 = t 2 . The finiteness of the cardinal number of S implies every element in S is pre-periodic under h. Then each element in S is periodic. Otherwise, there will be at least two elements which are mapped to a same element. This contradicts with that h is a injection. The proof is complete.

In the following, based on the combinatorial theory of quadratic polynomials and renormalization theory, we shall construct a semi-hyperbolic McMullen map whose Julia set is quasisymmetrically equivalent to a round carpet.

Theorem 2.6. There exists a suitable parameter λ > 0 such that the McMullen map

f λ (z) = z d + λ/z d (2.26)
is semi-hyperbolic and the corresponding Julia set is quasisymmetrically equivalent to a round Sierpiński carpet, where d ≥ 3.

Proof. We divide the construction into three main steps as following.

Step 1. For a given irrational number α ∈ (0, 1), one can write it as an infinite binary sequence α = 0.a 1 a 2 a 3 • • • by Lemma 2.10, where a i ∈ {0, 1}. Define a binary number

θ := 0.0 1 • • • 1 | {z } 100 0 • • • 0 | {z } b 1 1 • • • 1 | {z } b 2 0 • • • 0 | {z } b 3 • • • with b i = a i + 1 for i ≥ 1.
Then 1 ≤ b i ≤ 2 and we have:

• The number θ ∈ (0, 1/2) is irrational. If not, by Lemma 2.10, the number θ will be eventually periodic under the iteration of the doubling map q. Then there exist m ≥ 2 and p ≥ 2 such that

q •n (θ) = 0. 1 • • • 1 | {z } bm 0 • • • 0 | {z } b m+1 • • • 0 • • • 0 | {z } b m+p-1 , where n = 101+b 1 +• • •+b m-1 . This means that the sequence b m , b m+1 , • • • , b m+p-1 , b m+p , • • • is periodic with period p. Therefore, the sequence a m , a m+1 , • • • , a m+p-1 , a m+p , • • • is also periodic with period p since a i = b i -1 for each i. Then α = 0.a 1 • • • a m-1 a m a m+1 • • • a m+p-1
is a rational number by Lemma 2.10. This is a contradiction since α is irrational.

• Define a rational number with the binary form

θ = 0.0 1 • • • 1 | {z } 99 .
Then 0 < θ < θ < 1/2 and θ , θ are very close to 1/2. We have

l(θ ) = 0. 1 • • • 1 | {z } 99 0 1 • • • 1 | {z } 99 and l(θ) = 0. 1 • • • 1 | {z } 100 0 • • • 0 | {z } b 1 1 • • • 1 | {z } b 2 0 • • • 0 | {z } b 3 • • • .
For any n ≥ 2, one can easily check that

0 < l(q(θ )) < l(q •n (θ)), l(q •n (θ )) < l(θ ) < l(θ).
(2.27)

• Define a set

R := {t ∈ R/Z : T •n (l(t)) ≤ l(t) for all n ≥ 0}.
(2.28) By (2.25) and (2.27), we have θ , θ ∈ R.

Step 2. Construct a quadratic polynomial P c (z) = z 2 +c with the following properties:

(1) The critical orbit

O + Pc (0) = {P •n c (0) : n ≥ 0} is contained in the Julia set of P c and the cardinal number of O + Pc (0) is infinite. (2)
The critical point 0 is non-recurrent and the ω-limit set of 0 does not contain the β fixed point. Recall that a β fixed point of a polynomial is the landing point of dynamical external ray with angle zero.

The set R defined in (2.28) is exactly the set of all angles of parameter rays whose prime-end impression intersects the subset R ∩ M = [-2, 1/4] of the Mandelbrot set M (see [START_REF] Tiozzo | Topological entropy of quadratic polynomials and dimension of sections of the Manderbrot set[END_REF]Proposition 8.4]). By [Za, Theorem 3.3], there exists a real number c := c(θ) ∈ [-2, 1/4] in the boundary of the Mandelbrot set such that c is contained in the prime-end impression of the parameter rays R M (±θ) since θ ∈ R. Moreover, on the dynamical plane, the dynamical rays R c (±θ) land at the critical value c of P c (z) = z 2 + c.

In fact, such c is unique. Otherwise, suppose that there exists another c = c, such that c is contained in the prime-end impression of the parameter rays R M (±θ). By the density of hyperbolic parameters in R ∩ M (see [GS] and [Ly]), there is a real hyperbolic parameter e c between c and c with a pair of rational parameter rays landing at it. This means that c and c cannot lie in the same prime-end impression of R M (±θ) at the same time, which is a contradiction. Now we prove that the quadratic polynomial P c is the map what we want to find. Again by [START_REF] Tiozzo | Topological entropy of quadratic polynomials and dimension of sections of the Manderbrot set[END_REF]Proposition 8.4], the parameter rays R M (±θ ) land at a parabolic parameter c 0 ∈ R since θ ∈ R is a rational number. These two rays together with their landing point R M (θ ) ∪ R M (-θ ) ∪ {c 0 } bounds a wake W {-2} with the following property: The quadratic map P ξ (z) = z 2 + ξ has a repelling periodic point with exactly two dynamical rays R ξ (±θ ) landing at if and only if ξ ∈ W (see [Mi2, Theorem 1.2]). By the construction in Step 1, we have 0 < θ < θ < 1/2. Then R M (±θ) ∪ {c} ⊂ W and hence R c (±θ ) land at a repelling periodic point of P c on the real line. Also, the image R c (±q(θ )) of R c (±θ ) land at some point on the real line.

Denote by H the simply connected domain bounding by the four dynamical rays (d -1)} (Note that there exists a semiconjugacy between f λ and the rational map discussed in [Ste, Theorem 9]). The copy M is symmetric with respect to the positive real axis. Moreover, there exists a homeomorphism Φ : M → M such that, for every λ ∈ R + ∩ M = R + ∩ Λ d , there is a corresponding parameter Φ(λ) ∈ [-2, 1/4] and the Julia set J f λ contains an embedded set JP Φ(λ) , which is homeomorphic to the Julia set of the quadratic polynomial P Φ(λ) (z) = z 2 + Φ(λ). Moreover, the restriction of f λ in a neighborhood of JP Φ(λ) is quasiconformally conjugated to the restriction of P Φ(λ) in a neighborhood of J P Φ(λ) .

Chapter 3

Criterion for rays landing together

Introduction

Let f be a polynomial with degree d ≥ 2 in the complex plane C. The f illed Julia

set is K f := {z ∈ C : The orbit {f n (z)} n≥0 is bounded }
and the Julia set is the topological boundary of the filled Julia set

J f = ∂K f .
Both of them are nonempty and compact, and the filled Julia set is full, i.e., the complement C \ K f is connected. We call Ω f := C \ K f the basin of inf inity which consists of points with the orbit attracted by ∞. If J f is connected. Then Ω f is a simply connected and there exists an unique holomorphic parameterization

Ψ f : Ω f → C \ D such that Ψ f (∞) = ∞, Ψ f (∞) = 1 and Ψ f • f (z) = (Ψ f (z)) d . (3.1)
By the external ray R(θ) we mean the preimage of the radial line Ψ -1 f {re 2πiθ : r > 1}, where θ ∈ R/Z is the argument of the ray. We say that R(θ)

lands at z ∈ J f if lim r→1 Ψ f (re 2πiθ ) = z. By the theorem of Carathéodory Ψ -1 f extends continuous to ∂D with Ψ -1 f (∂D) = J f if and only if J f is locally connected.
Throughout this chapter we consider the case, J f is locally connected. Define α : R/Z → J f , θ → α(θ) where α(θ) is the landing point of ray R(θ). By (3.1), we have the following semi-conjugation,

f (α(θ)) = α(σ d (θ)), (3.2)
where σ d : R/Z → R/Z with θ → d θ mod Z. Thus, to study the topology of the Julia set and the dynamics of f on J f is necessarily to figure out the semi-conjugation α.

There are two questions arising naturally,

(1) For any z in J f , is the fiber α -1 (z) finite ? In other words, are there only finite rays landing at z ?

(2) Give a condition under which θ, θ are in the same fiber. That is, when two external rays R(θ), R(θ ) land at a same point ?

For the first question, if the orbit of z is finite, then the fiber α -1 (z) is finite [DH84]. If z is wandering, i.e., the orbit is infinite, J.Kiwi gave an upper bound

#α -1 (z) ≤ 2 d [Ki02].
A.Blokh and G.Levin consider the more general problem: counting the number of external rays landing at distinct wandering points with disjoint forward orbits. Blokh and Levin worked the abstract modeling invariant laminations and introduced a new tool called growing tree [BL02]. In this chapter, inspired by [Ki02], we reprove the inequality in a totally different way.

Theorem 3.1. Let z 1 , • • • , z m be wandering branched points such that their forward orbits avoid the critical points and are pairwise disjoint. Then

X 1≤i≤m (v(z i ) -2) ≤ d -2.
In the above theorem, a point z is called to be a branched point if the fiber α -1 (z) contains at least three angles and the valence v(z) is cardinal number of α -1 (z).

For the existence, W. Thurston proved that for quadratic polynomials there is no wandering branched points. He asked a deep question concerning their existence for higher degree in the preprint [START_REF] Thurston | The combinatorics of iterated rational maps[END_REF]. A.Blokh and L.Oversteegen answered the question by constructing an uncountable family of cubic polynomials, the Julia set of each one is a dendrite and containing wandering branched points [START_REF] Blokh | Wandering gaps for weakly hyperbolic polynomials[END_REF].

For the second question, following [BFH92], [Po93] and [Ki05] etc, we need a concept: critical portrait associated to a polynomial f .

• For critical point c in J f , Θ(c) is the set of arguments of external rays which land at c and are inverse images of one ray landing at critical value f (c). Obviously, #Θ(c) is deg f (c), the local degree of f at c.

• For strictly pre-periodic critical Fatou component U , Θ(U ) is a collection of deg(f | U )
arguments whose rays support U and are inverse images of one ray supporting f (U ).

• For Fatou component cycle U 0 , • • • , U p-1 with f i (U 0 ) = U i , U p := U 0 , let U k 0 , • • • , U k l with 0 ≤ k 0 < • • • < k l ≤ p -1 be critical with degree n 0 , • • • , n l . For 0 ≤ i ≤ p, choose (z i , θ i ), z i ∈ ∂U i and R(θ i ) supporting U i at z i , such that f i (z 0 ) = z i , f p (z p ) = z p and f i (R(θ 0 )) = R(θ i ). Then Θ(U k j )
is the set of arguments whose external rays land on ∂U k j and are preimages of R(θ k j +1 ), for 0 ≤ j ≤ l.

Let A := {Θ(c 1 ), • • • , Θ(c m ), Θ(U 1 ), • • • , Θ(U n )}. For any Θ ∈ A, set Ò Θ := [ {Θ : ∃ a chain Θ 0 = Θ, • • • , Θ k = Θ in A such that Θ i \ Θ i+1 = ∅}. The collection Ò A := { Ò Θ 1 , • • • , Ò
Θ N } is called critical portrait associated to f . In the unit circle, there is a partition

P := {I 1 , • • • , I d } of R/Z \ S 1≤i≤N Ò Θ i . Each I i is a finite union of open intervals with total length 1/d.
Given a partition, we say x, x have the same itinerary respect to the partition under a map g if and only if both g n (x) and g n (x ) lie in the same piece of the partition, for any n ≥ 0.

For polynomials with all critical points strictly preperiodic, B.Biefield, Y.Fisher and J.H.Hubbard showed that, if θ, θ have the same sequence respect to the partition P then α(θ) = α(θ ) [BFH92]. A.Porier extends this result to crtitical finite polynomials, admitting periodic Fatou component [Po93]. Both of their proofs rely on the orbifold metric in Julia set, on which f is expanding. In [Ki05], Kiwi considered the polynomials with all cycle repelling and Julia set connected. Based on the properties of maximal lamination, he proved that if θ, θ have the same sequence respect to P, then the impressions of R(θ) and R(θ ) intersect.

We prove the following theorem, which is the main result of this chapter.

Theorem 3.2 (Main Theorem). Let f be a polynomial with J f locally connected. Let P be the partition induced by critical portrait Ò A. If θ, θ have the same itinerary respect to P, then either R(θ),R(θ ) land at the same point or R(θ), R(θ ) land at the boundary of a Fatou component U , which is eventually iterated to a siegel disk.

Note that S.Zakeri in [START_REF] Zakeri | Biaccessibility in quadratic Julia sets[END_REF] proved that for Siegel quadratic polynomial f , i.e., f : z → z 2 + c has a fixed Siegel disk, no points has more that two rays landing at and if two rays landing at z then z must eventually hit the critical point 0.

The following Corollary holds immediately.

Corollary 3.1 (No wandering continua in J f ). Let f be a polynomial with J f locally connected. Then there is no wandering continua in J f .

We have to point out that A.Blokh and G.Levin also proved the above corollary [BL02]. And J.Kiwi proved that, for polynomials without irrational neutral periodic orbits f , J f is locally connected if and only if f has no wandering continua in J f . Kiwi's proof relies on constructing a puzzle piece around each pre-periodic or periodic point of a polynomial f with all cycles repelling [START_REF] Kiwi | Real laminations and the topological dynamics of complex polynomials[END_REF].

Motivation

One of our motivation is to study the core-entropy of polynomials. Suppose X is a compact metric space and g : X → X is continuous. The topological entropy of g is measuring the complexity of iteration from the growth rate of the number of distinguishable orbits. The core-entropy of polynomial f is the topological entropy of f on its f -invaritant set Hubbard tree, i.e., the convex hull of the critical orbits within the (filled) Julia set. Let Acc(f ) be the set of all biaccessible angles θ, i.e., there exist at least two rays landing at α(θ). Then the core-entropy h(f ) is related to the Hausdorff dimension of Acc(f ) in the following way,

h(f ) = log d • H.dim Acc(f ).
(3.3)

These quantities are according to W.Thurston who firstly introduced and explored the core-entropy of polynomials.

For quadratic polynomials, G.Tiozzo showed the continuity of core-entropy along principal veins of the Mandelbrot set M in [Ti13]. This result is generalized by W. Jung to all veins [Ju13]. Recently, G.Tiozzo proves that the function θ → h(f θ ) with f θ (z) = z 2 + c θ is continuous.

A.Douady proved the monotonicity of core-entropy along real vein

M T R [Do95]
. The monotonicity for all postcritically finite quadratic polynomials is proved in Tao Li's thesis [Li07]. As an application of theorem 3.2, we extend Tao Li's result to a quadratic family F := {f c = z 2 + c : f c has no Siegel disks and J fc is locally connected }.

Theorem 3.3 (Monotonicity of core-entropy). For any

f c , f c ∈ F, if f c ≺ f c , then Acc(f c ) ⊆ Acc(f c ) and so h(f c ) ≤ h(f c ).
For any f c , f c in F, we say f c ≺ f c if and only if I c ⊇ I c , where I c is the characteristic arc of f c . See section 3.7 for details.

Sketch of the proof and outline of the chapter

The proof of main theorem 3.2 is based on the analysis in the dynamical plane. There is a partition {Π i } 1≤i≤d of C, induced by critical portrait. It has nice properties: for any points x, y ∈ Π i T J f , the regulated arc [x, y] ⊆ Π i and F | [x,y] is one-to-one, where F is a topological polynomial which takes the same value as f in Ω f . Thus if x = y have the same itinerary respect to {Π i }, we obtain a sequence {F n [x, y]} of regulated arc. The sequence will eventually meet S 1≤i≤d ∂Π i T J f . However it is difficult to prove that the partition {Π i } 1≤i≤d separates f n (x), f n (y) for some n. To overcome this difficult, we use this sequence to construct a wandering arc in J f , which is a contradiction.

In section 3.2, we prove theorem 3.1. This key result is useful to show the fact of no wandering regulated arcs in Lemma 3.3. In section 3.3, we give the construction of regulated arcs and describe its properties. In section 3.4, we explain how to get a desired topological polynomial F by modifying f in Fatou set.

Section 3.5 analysis the properties of partition induced by critical portrait in the dynamic plane.

The main Theorem 3.2 is proved in section 3.6.

In the last section, we discuss characteristic arcs in details and give an application of the main theorem to the monotonicity of core entropy for a quadratic polynomial family.

Wandering Orbit Portrait

If not otherwise stated, we assume f to be a polynomial with degree d ≥ 2 and J f locally connected. Our objective is to prove the Proposition 3.1 in this section.

Portraits

Now we give some definitions by following

[Mi00] [GM93] [BFH92] [Ki02] etc.
For a point z in J f , the valence of z, written v(z), is the number of external rays landing at z. We define the annular size of a sector S, written l(S), by the length of the corresponding arc I(S) in R/Z. Number the n sectors of T by S 1 (T ), • • • , S n (T ) according to their length:

Then 1 ≤ v(z) ≤ ∞. If v(z) ≥ 3, z is called to be a branched point. z is called to be wandering if f m (z) = f n (z) for m = n ≥ 0. Let T := {θ 1 , . . . , θ n }, θ i ∈ R/Z, 3 ≤ n < ∞. T is called
l(S 1 (T )) ≤ l(S 2 (T ) • • • ≤ l(S n (T )).
By means of critical sector or critical value sector if a sector S contains critical points or critical values.

Lemma 3.1 (For portraits with distinct base points). Let T, T be two portraits with α(T ) = α(T ). Let S resp. S be the sector of T resp. T such that α(T ) resp. α(T ) is contained in S resp. S . Then all but S resp. S of the sectors of T resp. S are contained

Sector maps

Lemma 3.2 (Properties of sector maps). Let T = {θ 1 , • • • , θ v(T ) } be a portrait such that the base point α(T ) is not a critical point of f , here θ i are enumerated in cyclic order around the circle. Then Note that we distinguish the distinct definitions of σ d by acting on different categories.

Proof. Let z := α(T ). Since z is not critical. f is a locally orientation-preserving homeomorphism at z. Note that the v(T ) angles θ i in R/Z and rays R(θ i ) around z are identical in order. Moreover, the order of R(θ i ) can be measured within an arbitrarily small neighborhood of z. It follows that all rays with angels in T land together at f (α(T ) and σ d sends angles in T onto T bijectively and keeping the order. Thus (1) and (2) follows.

For

n 0 + 1 z ∈ Q n 0 z ∈ G d \ Q . (3.4)
By the Arguments Principle, every point z ∈ G d \ f (∂Q) has w(z) preimages, counting multiplicity, in Q. Now claim that every points z in ∂Q \ ∂G d , consisting of two segments of external rays, has n 0 + 1 preimages, counting multiplicity, in Q. Since such z can not be a critical value, choose sufficiently small enough neighborhood U z such that the restriction of f on every

component f -1 U z is homeomorphic. Since U z T Q has n 0 + 1 components in Q and Q is closed, z must have n 0 + 1 preimages in Q as well. Let v 1 , • • • , v n ∈ f (Q) be the critical value of f | Q . Let µ i be the total multiplicity of critical points in Q mapped to v i . Choose a cell subdivision ∆ of f (Q) such that the set of its 0-cells contains {f (z), v 1 , • • • , v n } and the set of 1-cells contains ∂Q . Let ∆ 1 := {complexes of ∆ contained in Q } and ∆ 2 := ∆ \ ∆ 1 . It follows that ∆ 1 is a cell subdivision of Q . Set x i ,
y i , z i to be the number of 0-cell, 1-cell and 2-cell of ∆ i . Computing the Euler characteristic, we have

X (Q ) = x 1 -y 1 + z 1 = +1 (3.5)
and

X (f (Q)) = (x 1 + x 2 ) -(y 1 + y 2 ) + (z 1 + z 2 ) = +1. (3.6)
After lifting every complexes in ∆ by f | Q , we obtain a cell subdivision ∆ 0 of Q. Then

X (Q) = [(n 0 + 1)x 1 + n 0 x 2 - X 1≤i≤n µ i ] -[(n 0 + 1)y 1 + n 0 y 2 ] + [(n 0 + 1)z 1 + n 0 z 2 ] = +1.
(3.7)

Combining (3.5), (3.6) and (3.7), we have

X 1≤i≤n µ i = n 0 .
Thus (3) is completed. For (4), we use the notations as above. If not, assume Q contains no critical values. Then every component of f | -1 Q (Q ) is simply connected and f on the closure of which is homeomorphic. Consider the component C with ∂Q \ γ ab ⊆ C. f | C cannot be one-to-one, a contradiction.

For (5), it follows directly by (3).

Dynamics of wandering portraits

Portrait T is called to be wandering if and only if the point α(T ) is wandering and not iterated to critical points of f . We denote by T n := σ •n d (T ).

Recall that S 1 (T ), • • • , S v(T ) (T ) are the v(T ) sectors of T enumerated by the order of their annular size. We have the following lemma. See also in [Ki02].

Lemma 3.3. Let T be a wandering portrait. Then

lim n→∞ l(S v(T )-2 (T n )) = 0.
Proof. If not, there exist a number a > 0 and an infinite sequence T n k such that 5a/6 < l(S v(T )-2 (T n k ) < 7a/6.

The sectors S v(T )-2 (T n k ) can not be pairwise disjoint. Because otherwise the total length of the infinite many intervals I(S v(T )-2 (T n k ) would be greater than 1.

Then there exist

n i = n j such that S v(T )-2 (T n i ) T S v(T )-2 (T n j ) = ∅. By Lemma 3.1, we can assume α(T n i ) ∈ S v(T )-2 (T n j ) and both sectors S v(T )-2 (T n i ) and S v(T )-1 (T n i ) are contained in S v(T )-2 (T n j ). Thus, l(S v(T )-2 (T n j )) > l(S v(T )-2 (T n i )) + l(S v(T )-1 (T n j )) > 5a/3, a contradiction.
By lemma 3.3, for any wandering portrait T , the annular size of sectors T n , except the two large ones, will converges to zero. Furthermore, a similar argument can show that lim inf l(S v(T )-1 (T n )) = 0. We will not use this fact. We are more interested in the moment when a "wide" critical sector is mapped to a "narrow" critical value sector.

For any sufficiently small > 0 and

1 ≤ k ≤ v(T ) -2, Set n ,k (T ) := min{n : l(S k (T n )) < }.
By lemma 3.3, l(S k (T n )) will eventually be smaller than as n → ∞. Thus n ,k (T ) is well defined. We have the following, Lemma 3.4. Let T be a wandering portrait. Then There exists δ > 0 such that for any

< δ, denote by n ,k := n ,k (T ), 1 ≤ k ≤ v(T ) -2, we have l(S k+1 (T n ,k
)) > and there exists at least one critical value sector S k 0 (T n ,k ) with 1 ≤ k 0 ≤ k.

Proof. By lemma 3.3, there exists an integer N ≥ 1 such that, for any n ≥ N ,

(S v(T )-2 (T n )) < 1 2v(T )d . Set δ := min 1≤i≤N { l(S 1 (T i )) }.
For any < δ, since n ,k is the first time that the k th sector has length strictly less than . We have

≤ l(S k (T n ,k -1 )) ≤ l(S v(T )-2 (T n ,k -1 )) < 1 2v(T )d . By Lemma 3.2 (5), f maps the v(T ) -2 sectors S 1 (T n ,k -1 ), • • • , S v(T )-2 (T n ,k -1
) onto sectors of T n ,k homeomorphic with their length multiplied by d. Then

l(σ d (S k (T n ,k -1 ))) ≥ d > and l(S k (T n ,k )) < .
This means σ d must map at least one of the two sectors S v(T )-1 (T n ,k -1 ) and S v(T ) (T n ,k -1 ) onto a "narrow" sector S k 0 (T n ,k ) with l(S k 0 (T n ,k )) < . By lemma 3.2 (4), S k 0 (T n ,k ) is a critical value sector. Actually, there are only one of the above two sectors mapped to such "narrow" sector. Because the total length of the v(T ) -1 images,

l(S k 0 (T n ,k )) + X 1≤i≤v(T )-2 l(σ d (S i (T n ,k -1 ))) < 1 2 .
It follows that the other sector is mapped to the widest sector S v(T ) (T ,k ) with length > 1 2 . Thus, we have

S k+1 (T n ,k ) = σ d (S k (T n ,k -1 )) ≥ d > and 1 ≤ k 0 ≤ k.
The proof is completed.

Proof of theorem 3.1

Proposition 3.1. Let T (1) , • • • , T (m) be wandering portraits such that α(T (i) ) have disjoint forward orbits. Then

X 1≤i≤m (v(T (i) ) -2) ≤ d -2. (3.8)
Proof. Let 0 > 0 be smaller than any δ T (i) , for 1 ≤ i ≤ m, as stated in the Lemma 3.4. Firstly, applying Lemma 3.4 to the case T = T (1) , k = 1 and = 0 , we obtain a critical value sector S 1 (T (1) n 0 ,1 ) and

:= l(S 1 (T (1) n 0 ,1 )) < 0 < l(S 2 (T (1) n 0 ,1 )). (3.9) Let n k,i := n ,k (T (i) ), for 1 ≤ i ≤ m, 1 ≤ k ≤ v(T (i) )
. By the definition of n k,i and orbits of α(T (i) ) disjoint in the condition, it is easy to see that

n k 1 ,i = n k 2 ,j = n 0 ,1 and α(T (i) n i,k 1 ) = α(T (j) n j,k 2 ) = α(T (1) n 0 ,1 ), (3.10) for 1 ≤ i, j ≤ m and (i, k 1 ) = (j, k 2 ), 1 ≤ k 1 ≤ v(T (i) ), 1 ≤ k 2 ≤ v(T (j)
). By Lemma 3.4 again, we obtain N := P 1≤i≤m (v(T (i) ) -2) critical value sectors, denoted by S τ (k,i) (T (i) n k,i ), and we have

l(S τ (k,i) (T (i) n k,i )) < < l(S k+1 (T (i) n k,i )), 1 ≤ τ (k, i) ≤ k. (3.11)
By (3.10) and Lemma 3.1, for any distinct two of the N + 1 critical value sectors S 1 (T (1) n 0 ,1 ) and S τ (k,i) (T (i) n k,i ), they are neither disjoint or one contains the other. We claim that the latter case can not happen. If not, suppose S τ (k 1 ,i 1 ) (T (i 1 )

n k 1 ,i 1 ) are contained in S τ (k 2 ,i 2 ) (T (i 2 ) n k 2 ,i 2
). By Lemma 3.1, we have

S k 1 +1 (T (i 1 ) n k 1 ,i 1 ) ⊂ S τ (k 2 ,i 2 ) (T (i 2 ) n k 2 ,i 2
) and l(S k 1 +1 (T (i 1 )

n k 1 ,i 1 )) < l(S τ (k 2 ,i 2 ) (T (i 2 ) n k 2 ,i 2 )).
This contradicts (3.11). If one of them is S 1 (T (1) n 0 ,1 ), similarly by (3.9), it is impossible. Thus the N + 1 critical values sectors are pairwise disjoint and each of them contains at least one critical value. Since it is known that, for degree d polynomials, there exist at most d -1 critical values. So N + 1 ≤ d -1. The proof is completed.

Proof of Theorem 3.1. The theorem follows immediately by Propositions 3.1.

Actually the result in this section can extended to polynomials with Julia set connected or not connected. We omit the details. See Appendix A in [Ki02].

Corollary 3.2. Let f be a polynomial with the Julia set J f locally connected. Then the number of grand orbits of wandering branched points is finite.

Regulated arcs

According to Fatou and Sullivan, every bounded Fatou components of polynomials must eventually be mapped to the immediate basin of attraction of an attracting periodic point, or to an attracting petal of a parabolic periodic point, or to a periodic Siegel disk [Mi06] [START_REF] Sullivan | Conformal Dynamical Systems[END_REF]. We refer to these cases simply as hyperbolic, parabolic and Siegel cases.

For any two points x, y ∈ K f there usually exist more than one arc γ in K f connecting

x and y. In the following, we will give the definition of internal ray and regulated arc in K f and show how to choose a canonical embedded arc between any two points in the filled Julia set. Under certain condition, such arc is unique (See Lemma 3.7).

Extended rays

Now consider the polynomial f with J f locally connected. We have, Lemma 3.5 (Bounded Fatou components are Jordan domains). For any bounded Fatou component U , ∂U is a Jordan curve.

Proof. Since J f is locally connected, then ∂U is locally connected. Consider the Riemann map: Φ U : D → U , it extends continuously to D by Carathéodory Theorem. Therefore, ∂U is the curve Φ U (S 1 ). If Φ U |S 1 is not injective. Then there exists t < t in S 1 with Φ U (t) = Φ U (t ). The two rays Φ U ([0, 1]e 2πit ) and Φ U ([0, 1]e 2πit ) will bound a domain U , which contains subset of the Julia set Φ U ({e 2πiη : t < η < t }. Since J f is the boundary of infinity attracting domain Ω f , some points in U will escape to infinity. This contradicts the Maximum Value Principle.

Given any bounded Fatou component U , pick a point c(U ) in U as center point and a Riemann map ϕ U : U → D with ϕ U (c(U )) = 0. Then extend it to a homeomorphism ϕ U : U → D by Carathédory Theorem.

An arc in U of the form ϕ -1 U {re iθ : 0 ≤ r ≤ 1} is called a internal ray of U with angle θ. All these internal rays meet at the center point c(U ). Each ray has a well defined landing point in the boundary of U . Conversely, for any point z in the boundary of U , there exists an unique internal ray of U landing at z. We denote this internal ray by

R U (z). For any θ ∈ R/Z, if α(θ) = z ∈ ∂U , define the extended ray Ò R U (θ) := R(θ) [ R U (z).

Components of J f \ {x} are arcwise connected

Recall that a topological space X is said to be arcwise connected provided that there is a topological embedding of [0, 1] into X (called arc ) joining any two given distinct points. If p ∈ X, then X is said to be locally arcwise connected resp. locally connected at p, provided that every neighborhood of p contains an arcwise connected neighborhood resp. connected neighborhood of p. The space X is said to be locally arcwise connected resp. locally connected, provided that X is locally arcwise connected resp. locally connected at every point. We have the following well-know result.

Lemma 3.6. If a compact metric space X is locally connected, then it is locally arcwise connected.

It follows directly by the Lemma 17.17 and Lemma 17.18 in [Mi06].

Corollary 3.3. If a compact metric space X is connected and locally connected, then it is arcwise connected. Moreover, every connected component of X \ {x} is arcwise connected for any x in X. X is locally connected, every z in C has a sufficiently small connected neighborhood W z avoiding x, thus W z ⊆ C. Since X is locally arcwise connected by Lemma 3.6, C is locally arcwise connected as well. Then one can show that C is arcwise connected in exactly the same way as above.

Hence all Julia sets and filled Julia sets discussed in this chapter are locally arcwise connected and arcwise connected.

Uniqueness of regulated arc

An arc γ in K f is called to be regulated if it joins two distinct points in J f and for any bounded Fatou component U , the intersection γ T U is an empty set or a point or exactly two internal rays.

Lemma 3.7 (Uniqueness of regulated arc). For any two distinct points x, y in J f , there exists only one regulated arc in K f joining x and y.

Proof. Let η(t) : [0, 1] → K f be the arc joining x and y with η(0) = x and η(1) = y. For any Fatou component U whose closure intersects the arc η, set x U = inf 0≤t≤1 {t : η(t) ∈ U }, i.e., the first time η meets U , and y U = sup 0≤t≤1 {t : η(t) ∈ U }, i.e., the last time η meets U . If x U = y U . Then we replace the segment η((x U , y U )) starting at η(x U ) ending at η(y U ) by the internal rays R U (η(x U ) and R U (η(y U )),

updating η = η[0, x U ] ∪ R U (η(x U )) ∪ R U (η(y U )) ∪ η[y U , 1].
After doing these processes for countable many Fatou components, we obtain a regulated arc η connecting x and y as required.

For the uniqueness, if η is the other one. Then C \ η ∪ η consists of several disjoint connected components. Let W be one of the bounded component in K f . Then W is a Jordan domain and ∂W ⊆ η ∪ η . Applying the Maximum value Principle, W belongs to the Fatou set. Let U be the Fatou component containing W . Thus W ⊆ U . Since

Quasi-buried regulated arc

A regulated arc γ is called quasi-buried if the intersection between γ and the closure of any bounded Fatou component is either empty or exactly one point. Obviously if K f = J f , every regulated arc is quasi-buried. But if K f = J f , does there exist quasi-buried arc? We conjecture that for some special locally connected J f such regulated arc exists.

Similarly as the quadratic case, for high degree polynomials, we still define β fixed point as the landing point of external ray R(0). It can be a branched point with at most d -1 external rays landing at.

Let E := S i≥0 {f -i (β)}, i.e., the preimages of β fixed points. Set E be the union of E and branched points in J f . If J f is a segment, then E = E. We know that E is dense in J f [Mi06] and thus E is dense in J f . Moreover, we have the following, Lemma 3.8 (Denseness of E in quasi-buried arcs). Let I := [x, y] be a quasi-buried regulated arc in K f . Then E is dense in I.

Proof. Let p be any point in I \ {x, y}. Since J f is locally arcwise connected by Lemma 3.6, we can choose sufficiently small arcwise connected neighborhood W p in J f such that Let ξ be the point at which γ zp meets I at the first time. Then ξ belongs to I \ {x, y} by (3.12). Let γ zξ be the subarc of γ zp joining z and ξ. It follows that the three arcs γ zξ , [x, ξ] and [y, ξ], meeting at ξ, form a "Y" shape.

We are left to show that ξ is a branched point. Due to the Theorem 6.6 in [START_REF] Mcmullen | Complex Dynamics and Renormalization[END_REF], we only have to proof that K f \ {ξ} has at least three connected components. Actually we have the following.

Claim that x, y and z lie in distinct connected components of K f \ {ξ}. Thus ξ is a branched point. The proof is completed.

The topological polynomial F

The regulated arcs in K f may not be preserved by the dynamic of f . In this section, we will construct a nice topological polynomial F by modifying f in each bounded Fatou set. F will coincide with f on the basin of infinity and the Julia set J f . The above difficulty can be most conveniently overcome by investigating F instead of f . Since we only interest in the Julia set and the combination of external angles. These changes make no essentially differences.

Branched covering map

Let X and Y be domains in C, g : X → Y be a continuous map. Then g is called a branched covering map if we can write it locally as the map z → z n for some n ∈ N after orientation-preserving homeomorphic changes of coordinates in domain and range. More precisely, we require that for each point q ∈ Y and any preimage p in g -1 (q) there exists n ∈ N, open neighborhoods U of p and V of q, open neighborhoods U and V of 0 ∈ C and orientation-preserving homeomorphisms φ : U → U and ψ : V → V with φ(p) = 0 and ψ(q) = 0 such that

(ψ • g • φ -1 )(z) = z n (3.
deg g (p) = deg(g)
for every q ∈ Y . A branched covering with no critical point is called unbranched covering.

A branched covering map g :

C → C is called topological polynomial if g -1 (∞) = ∞,
that is , ∞ is a fixed point with local degree deg(g).

From polynomial f to topological polynomial

F For polynomial f , a bounded Fatou component is called critical F atou component if it contains critical point of f . Its image is critical value F atou component. Given a bounded Fatou component U , f maps U to Fatou component U holomorphic. f | ∂U : ∂U → ∂U is an unbranched covering map with degree deg(f | U ). Recall that ϕ U : U → D c(U ) → 0 is a conformal parameterization . Set ϕ U U := ϕ U • f • ϕ -1 U | ∂D : ∂D → ∂D. Now we extend ϕ U U to be ϕ U U : D → D re 2πiθ → rϕ U U (e 2πiθ ).
One can check that ϕ U U is a branched covering. Define

F U := ϕ -1 U • ϕ U U • ϕ U : U → U by the following communicate diagram, (U, c(U )) F U ---→ (U , c(U )) ψ U ? ? ? y ? ? ? y ψ U (D, 0) ψ U U ---→ (D, 0).
By the construction, F U satisfies

• F U | ∂U = f | ∂U . • F U sends c(U ) to c(U ).
• F U is a branched covering with degree deg(f | U ) and the critical point can only be c(U ).

• F U sends internal rays to internal rays, more precisely,

F U (R U (z)) = R U (f (z)
). Now we define the topological polynomial F : C → C,

F (z) := 8 < : F U (z) If z in some bounded Fatou component U, f (z) Otherwise. (3.14)
Evidently, F takes the same value as f in the Julia set and the basin of infinity. Furthermore, we have the following.

Properties of the topological polynomial F

Lemma 3.9.

(1) F is continuous.

(2) F is a branched covering map.

(3) For any

x = y ∈ J f , [F (x), F (y)] ⊆ F ([x, y]). (4) F ( Ò R U (θ)) = Ò R U (σ d (θ))
, where U = F (U ), for any extended ray Ò R U (θ).

Proof.

(1) We only have to show that, for any z ∈ J f , F is continuous at z. Let {z k } be an arbitrary sequence such that z k → z as k → ∞. We continue the discussion into three cases,

• If {z k } ⊆ Ω f . Since F | Ω f = f and f is continuous, then F (z k ) → F (z) as k → ∞. • If {z k } are contained in C\Ω f . Let {U k } be a sequence of bounded Fatou components such that z k ∈ U k and U := {U k : k ≥ 1}. If # U < ∞, since F is continuous in any Fatou component, we f (z k ) → f (z) as k → ∞. If # U = ∞, since J f is
locally connected, the diameter of Fatou component F (U k ) converges to zero as k → ∞ (See for example Lemma 19.5 in [Mi06]). Thus,

|F (z k ) -F (z)| ≤ |F (z k ) -f (z k )| + |f (z k ) -F (z)| ≤ diam F (U k ) + |f (z k ) -f (z)| → 0 as k → ∞.
• In other cases, decompose {z k } into two subsequence {z k i }, contained in Ω f , and {z k i } in Fatou set. By the former arguments, both of the image of the two subsequence converge to

F (z) as k → ∞. So F (z k ) → 0 as k → ∞.
Thus F is continuous.

(2) Let Crit(F ) to be the union of critical points of f in J f and the center of critical Fatou components.

Firstly, claim that F :

C \ F -1 (F (Crit(F ))) → C \ F (Crit(F )
) is an unbranched covering. We only have to show that F is locally homeomorphic on C \ Crit(F ). For any z in some Fatou component U , It follows by the construction of F U . For any z ∈ J f \Crit(F ), choose a sufficiently small neighborhood W z such that

• f on W z is injective, • F | Wz T U is injective for any critical Fatou components, • f (U ) = f (U )
y z →z δ W -{F (z)} φ ---→ D -{0}
where ψ is a homeomorphism obtained by Lifting φ through F and z → z δ . Set ψ(z) = 0. Thus F satisfies (3.13) at z.

Therefore, F is a branched covering. The critical points set is Crit(F ).

(3) F ([x, y]), consisting of internal rays, is a curve connecting F (x) and F (y). There exists a regulated arc γ ⊆ F ([x, y]) joining F (x) and F (y). By Lemma 3.7, γ = [F (x), F (y)].

(4) Let z ∈ ∂U to be the landing point of R(θ). Then F (z) = α(σ d (θ)) ∈ ∂U . Since

F U maps internal ray R U (z) to internal ray R U (F (z)) and F (R(θ)) = R(σ d (θ)). Thus F ( Ò R U (θ)) = Ò R U (σ d (θ)).
The proof is completed.

Partitions induced by critical portraits

In this section our objective is to divide the plane into several simple connected domains by external rays and extended rays. These rays land at Crit(F ) and collide together after F . The restriction of F on each pieces is homeomorphic.

Supporting arguments resp. rays

Following [Po93], we give the definition of supporting arguments resp. supporting rays. Let U be a Fatou component and p ∈ ∂U with total k rays R(θ 1 ), • • • , R(θ k ) landing at. These rays, numbered in counterclockwise cyclic order, divide the plane into k sectors. Suppose U belong to the sector bounded by R(θ 1 ) and R(θ 2 ). The argument θ 1 resp. the ray R θ 1 is called the left supporting argument resp. left supporting ray of the Fatou component U . We can also define the right supporting arguments resp. right supporting rays in analogous way. If only one ray lands at p, then the two supporting rays coincide. Thus the lemma follows.

Definition of critical portraits

Firstly we define Θ(c), Θ(U ) resp. R(c), R(U ), for critical point c in J f and critical Fatou component U by the following way.

• For any critical point c ∈ J f , we set

Θ(c) := {θ 1 , • • • , θ deg F (c) } and R(c) := {R(θ 1 ), • • • , R(θ deg F (c) )}
such that the total deg F (c) external rays meet at c and F maps them onto exactly one external ray.

• For any strictly pre-periodic Fatou component U , we denote by

Θ(U ) := {θ 1 , • • • , θ deg F | U } and R(U ) := { Ò R U (θ 1 ), • • • , Ò R U (θ deg(F | U ) )}
such that the deg(F | U ) external rays R(θ i ) support U and collide onto one after F . Clearly, by Lemma 3.1, they are supporting U in the same direction.

• For any critical Fatou component cycle

U 0 , • • • , U p-1 with F i (U 0 ) = U i , U p := U 0
, it can only be attracting or parabolic [Mi06].

Let U k 0 , • • • , U k l , 0 ≤ k 0 < • • • < k l ≤ p -1, be critical with degree n 0 , • • • , n l respectively.
Firstly, For 1 ≤ i ≤ p, choose (z i , θ i ), z i ∈ ∂U i and R(θ i ) landing at z i , such that

F i (z 0 ) = z i , F p (z p ) = z p , F i (R(θ 0 )) = R(θ i ) and R(θ p ) supporting U p at z p . Since F p : ∂U 0 → ∂U 0 is δ := n 0 • • • n l to
U k i , 0 ≤ i ≤ l, Θ(U k i )
is the set of n i angles of external rays, which are supporting U k i and lie in the preimages of R(θ k i +1 ), and R(U

k i ) is the collection of n i extended rays of U k i with angles in Θ(U k i ).
After finishing the choice of Θ(U k i ) and R(U k i ) in critical Fatou cycle, we now state the following lemma by adopting the same notations as above,

Lemma 3.2. If z, z ∈ ∂U 0 have the same itinerary respect to R(U k 0 ), • • • , R(U k l ), then z = z .
Proof. Consider the covering F p : ∂U 0 → ∂U 0 . There are δ preimages of z p in ∂U 0 . These points cut ∂U 0 into open segments γ 0 , • • • , γ δ-1 , numbered in positive cyclic order which starts at z 0 . Denote by

[s 0 , • • • , s l ] := s 0 n 1 • • • n l + s 1 n 2 • • • n l + • • • + s l-1 n l + s l , where 0 ≤ s 0 ≤ n 0 -1, • • • , 0 ≤ s l ≤ n l -1. Let γ k i ,0 , • • • , γ k i ,n i -1 be the segments of ∂U k i \ S θ∈Θ(U k i ) α(θ), numbered in positive cyclic order which starts at z k i . Then F maps γ k i ,j onto ∂U k i +1 \ {z k i +1 } one to one.
By the construction above, we can see that

ξ ∈ γ [s 0 ,••• ,s l ] if and only if F k i (ξ) ∈ γ k i s i for 0 ≤ i ≤ l.
Hence by the condition, {F jp (z), F jp (z )}, for arbitrary j ≥ 0, are always contained in one segment of γ 0 , • • • , γ δ-1 . Now we show that it is impossible.

Let γ zz be the component of ∂U 0 \ {z, z } contained in some segment γ j . Since F p is expanding on ∂U 0 . There must exist a minimal positive s such that F sp (γ zz ) can not lie in one of γ 0 , • • • , γ δ-1 . Let F (s-1)p (γ zz ) ⊆ γ i 0 . Since F p | γ i 0 covers ∂U 0 \ {z p } by sticking the two endpoints into z p , which is the common boundary of γ j and γ (j+1)mod δ for some 0 ≤ j ≤ δ -1. Thus F sp (z) and F sp (z ) must be in distinct segments. The proof is completed.

It is easy to see that all the R(c) and R(U ) defined above are in star shape with a critical point in the center. ( Proof. By definition, (1) and (2) follow immediately.

Lemma 3.3 (Properties of R(c) and R(U )). (1) R(c) T R(c ) = ∅, for distinct critical points c, c in J f . (2) If R(c) T R(U ) = ∅,
) If R(U ) T R(U ) = ∅, 3
(3) Since for any two distinct Fatou component U, U , the intersection

U T U is at most one point. R(U ) T R(U ) = ∅ implies U T U := {p}. If Θ(U ) T Θ(U ) = ∅, then the latter case happens. Otherwise, we have R(U ) T R(U ) = {p}. In R/Z, let A := {Θ(c 1 ), • • • , Θ(c m ), Θ(U 1 ), • • • , Θ(U n )}. For any Θ ∈ A, let Ò Θ := [ {Θ : ∃ a chain Θ 0 := Θ, • • • , Θ k := Θ in A such that Θ i \ Θ i+1 = ∅}.

The collections Ò

A := { Ò Θ 1 , • • • , Ò Θ N } are called critical portrait of F .
One can check that the following conditions are satisfied.

(1)

P 1≤i≤N (# Ò Θ i -1) = d -1. (2) Ò Θ 1 , • • • , Ò
Θ N are pairwise unlinked, that is, for each i = j the sets Ò Θ i and Ò Θ j are contained in disjoint sub-intervals of R/Z.

(3) σ d sends Ò

Θ i onto exactly one argument.

Critical diagram associated to

z 1 b z 2 b z 3 D 1 D 2 D 3 W 1 W 2 W 3 W 4 W 5 W 6 W 7
Let L := {R(c 1 ), • • • , R(c m ), R(U 1 ), • • • , R(U n )}. For any R ∈ L, set c R := S {R : there exists a chain R 0 := R, • • • , R k := R in L such that, for R i and R i+1 , the latter case in Lemma 3.3(3) happens}. By Lemma 3.3, each c R corresponds to a Ò Θ, characterized by the property that R(θ) is in c R if and only if θ ∈ Ò Θ. Lemma 3.10 (Properties of Ò R ). (1) T := c R T K f is a tree. Namely, any z, z ∈ T T J f
can be joined by a regulated arc in T . Moreover, the branching points in the tree must be critical points in J f or c(U ) in critical Fatou component U .

(

) Suppose R(θ 1 ), • • • , R(θ l ) be all the external rays in c R, numbered in counter- clockwise order. Let L θ i θ i+1 := R(θ i ) S R(θ i+1 ) S [α(θ i ), α(θ i+1 )], 1 ≤ i ≤ l, θ l+1 := θ 1 . 2
Then L θ i θ i+1 cuts the plane into two domains Y , Y . Let Y be the one disjoint with

R(θ j ), 1 ≤ j ≤ l. Then for any x, y ∈ Y T J f , [x, y] ⊆ Y and F | [x,y]
T ∂Y is one-to-one.

(3) The image F (L θ i θ i+1 ) has only three types:

• Type I: one ray union the landing point,

• Type II: one extended ray union the landing point, • Type III: two internal rays and one external ray, which looks like "Y". (4) For another c R , if c R ∩ c R = ∅, then the intersection is a point.

Proof.

(1) By the construction of c R, it is clear that [z, z ] ⊆ T if z, z ∈ T T J f . The lemma 3.7 implies that regulated arcs cannot form a loop in K f . Thus T is a tree. Branched point z in Fatou component U is obviously a critical point c(U ). If z is in J f , there are at least three critical Fatou component

U i such that z ∈ R(U i ) ⊆ c R, i ∈ {1, 2, 3}. If z is not critical, R(U i
) share a common external ray which landing at z. Since one ray supports at most two Fatou components. It is impossible. Now we have to show that F | [γ(t 0 ),γ(t 1 )] is one-to-one. Note that [γ(t 0 ), γ(t 1 )] consists exactly several internal rays.

(2) Consider [α(θ i ), α(θ i+1 )]. It has only three possibilities (2.1) α(θ i ) = α(θ i+1 ), then [α(θ i ), α(θ i+1 )] is degenerated. (2.2) [α(θ i ), α(θ i+1 )] ⊆ U
In case (2.2), at least one of

R(θ i ), R(θ i+1 ) is supporting U , because R(U ) ⊆ Ò R. So [γ(t 0 ), γ(t 1 )]
T U is either a point or one internal ray. Thus F | [γ(t 0 ),γ(t 1 )] is one-to-one immediately.

In case

(2.3), let {p} = U T U . We have R(θ i ), R(θ i+1 ) supporting U, U respectively.
Otherwise, there exists a ray in R(U ) or R(U ) landing at p contained in Y , impossible. Thus the intersection between [γ(t 0 ), γ(t 1 )] and U resp. U is at most one internal ray. We are only left to consider the case [γ(t 0 ), γ(t (3) By the discussion in (2), it follows easily that F (L θ i θ i+1 ) is in Type I, Type II or Type III if and only if [α(θ i ), α(θ i+1 ] is in case (2.1), (2.2) and (2.3), respectively.

1 )] = [c(U ), c(U )]. Suppose F | [c(U ),c(U )] is not one-to-one. Then F ([p, c(U )]) = F ([p, c(U )]),
(4) It holds directly by the definition and Lemma 3.3.

c 0 c(U 1 ) c(U 2 ) c(U 3 ) c(U 4 ) Π 1 Π 2 Π 3 Π 3 Π 4 Π 5 Π 6 Π 6 Π 7
Let Ò L := { c R 1 , • • • , c R N }.
For simplification, the elements are numbered in such fine order that Ò R i consists of (extended) rays with their arguments in Ò Θ i . Let

P := C \ S 1≤i≤N c R i consists of finite unbounded pieces P 1 , • • • , P s . Consider the critical diagram D. Given W i , suppose it is bounded by S 1≤j≤k i (l θ j ∪ l θ j ) with θ j , θ j ∈ Ò
Θ j and l θ j ∪ l θ j ⊆ D j . Then, in the dynamic plane, L θ j θ j , 1 ≤ j ≤ k i , in Lemma 3.10 (2) are well defined. As in Lemma 3.10 (2), let Y j S Y j := C \ L θ j θ j where Y j be the component disjoint with S θ∈I i R(θ). Now we define the partition { Π i } 1≤i≤d of the dynamical plane by setting

Π i := C \ [ 1≤j≤k i Y j .
We have

• P = S 1≤i≤d Π i and Π i T Π j = ∅ if i = j.
• each Π i , maybe not a domain, consists of finite pieces P j and ∂Π i are the union of several (extended) rays.

• there is an one-to-one correspondence between { I i } 1≤i≤d and { Π i } 1≤i≤d by the property that θ ∈ I i if and only if R(θ) ⊆ Π i . See figure 3.5 and figure 3.6.

Based on the topological argument principle, we shall prove the following, Proposition 3.2. The restriction of F on each Π i is homeomorphic.

Proof. Recall G f : C → [0, ∞] is the Green's function which vanishes precisely on K f and G t := {z ∈ C : G(z) < t} a simply connected domain. Set Q t = G t T Π i , which is bounded by edges in two types,

• The segments of the equipotential cure G f (z) = t which lies in Π i . Each one corresponds to an arc in I i . We denote by Γ i the union of all these segments.

• The segments of L θ j θ j , 1 ≤ j ≤ k i satisfying the potential inequality G f (z) ≤ t. Each segments in Γ i is mapped to equipotential curve γ dt := {z ∈ C : G(z) = dt} locally homeomorphic. Since F pastes the segments of the latter two types together as in Lemma 3.10 (3). It follows that γ dt is covered by Γ i at least once. We know that

F | γt : γ t → γ dt is d to 1 and γ t is the union of Γ i , 1 ≤ i ≤ d, with their interiors disjoint. Thus F (Γ i ) covers γ d exactly once.
Let z 0 be any point of C which does not belong to the image F (∂Q t ). By the Topological Argument Principle, the number of solutions to the equation F (z) = z 0 with z ∈ Q t , counted with multiplicity, is equal to the winding number of F (∂Q t ) around z 0 . By the arguments above, it is not hard to check that this winding number is +1 for z 0 in G dt \ S 1≤j≤k i F (L θ j θ j ) and zeros for z 0 in C \ G dt . So F | Qt is one-to-one. By the arbitrariness of t, F on Π i is homeomorphic.

Regulated arcs in the partition

Lemma 3.4. For any distinct x, y ∈ Π i T J f , the regulated arc [x, y] is contained in Π i .

Moreover,

F : [x, y] → [F (x), F (y)] is homeomorphic. (3.15)
Proof. We adopt the notations as before. For 1 ≤ j ≤ k i , x, y ∈ Y j . Then the Lemma

3.10 (2) gives [x, y] ⊆ Y j . Thus [x, y] ⊆ T 1≤j≤k i Y j = Π i . Consider the set X := {z ∈ F ([x, y]) : there exist z 1 = z 2 ∈ [x, y] such that F (z 1 ) = F (z 2 ) = z}. Since F | Π i is one-to-one by Proposition 3.2, X ⊆ F ([x, y] T ∂Π i ).
We claim that X ⊆ F ([x, y]

T ∂Π i T J f ). If not, let z ∈ X T U for some bounded
Fatou component U . Then there exists two distinct z j ∈ U j such that F (z j ) = z. Firstly, If U 1 = U 2 , then U 1 must be critical. z 1 and z 2 are contained in two internal rays of R(U 1 ). It is impossible by Lemma 3.10 (2.2). If U 1 = U 2 , consider the branched covering F : U j → U . The image F (U j T Π i ) is either U or U \ R for some internal ray. In both of the cases we have

F (U 1 ∩ Π i ) \ F (U 2 ∩ Π i ) = ∅.
This contradicts the fact that F is one-to-one on Π i in Proposition 3.2. The claim follows.

Since [x, y]

T ∂Π i T J f is finite, then X is finite as well. This means F ([x, y]) has only finite many self-intersection points. If X = ∅, then we can easily obtain a loop in F ([x, y]), consisting of regulated arcs by Lemma 3.9 (3). Lemma 3.7 gives a contradiction. Thus we have X = ∅. Therefore, F : [x, y] → [F (x), F (y)] is homeomorphic.

Proof of the main theorem

In this section we aim to prove the main theorem, applying the tools prepared in the previous sections.

No wandering regulated arcs

Proposition 3.3. For any regulated arc

[x, y] in K f , there exist two integer m = n ≥ 0 such that F m [x, y] T F n [x, y] = ∅.
Proof. For any critical point, if [x, y] is mapped onto it twice, then of course we are done. So, by iterated [x, y] suitable times, we can assume f k | [x,y] is homeomorphic. We continue the analysis by distinguishing the regulated arc into two case.

• [x, y] is quasi-buried, i.e., #[x, y] T U ≤ 1, for any bounded Fatou component U .

• there exists a bounded Fatou component U such that #[x, y] T U ≥ 2.

In the first case, [x, y] ⊆ J f . Recall that E is the union of branched points and preimages of β fixed points in J f . By Lemma 3.8, E is dense in [x, y]. If some (pre-)periodic point lies in [x, y], we are done. Then E T [x, y] contains infinitely many wandering branched points. Since the number of grand orbits of wandering branched point is finite by Corollary 3.2. So there is at least a branched point z such that its grand orbit intersects [x, y] infinitely many times. Choose any two distinct z 1 , z 2 ∈ [x, y] in the grand orbit. Then we have

f m (z 1 ) = f n (z 2 ) for some m, n ≥ 0. Therefore f m [x, y] T f n [x, y] = ∅. Since f m | [x,y] and f n | [x,y] is injective. We must have m = n.
In the second case, let [x , y ] := [x, y] T U , consisting of two internal rays, particularly containing c(U ). By Sullivan's no wandering Fatou components, U will eventually be periodic. Then c(U ) ∈ [x , y ] is pre-periodic. So there exists m = n such that

f m [x , y ] T f n [x , y ] = ∅.
The proof is completed.

Quasi-buried case

Proposition 3.4. Let {Π i } 1≤i≤d be the partition of C induced by the critical portrait of f . Let [x, y] be quasi-regulated arc in K f . If x, y have the same itinerary respect to {Π i } 1≤i≤d , then x = y.

Proof. We argue by contradiction and suppose x = y. Denote by z n := F n (z) for any z ∈ C. By Lemma 3.4, for any m ≥ 0, n ≥ 1,

F n : [x m , y m ] → [x m+n , y m+n ] is homeomorphic. (3.16)
Firstly, we claim that there exist M = N ≥ 0 and ξ such that

• ξ ∈ [x M , y M ] T [x N , y N ],
• The orbit of ξ is disjoint with the finite set

X := S 1≤i≤d (∂Π i ∩ J f ).
Proof of Claim. Consider the set

Y := {z ∈ [x, y] : there exist m, n ≥ 0 and z = z ∈ [x, y] such that F m (z) = F n (z )}.
Since there is no wandering regulated arc by Proposition

3.3, Y is dense in [x, y]. For any z ∈ Y , there exist m = n ≥ 0 such that z m ∈ [x m , y m ] T [x n , y n ].
If the orbit {z i } i≥0 never hit X, we are done. If z n 0 ∈ X and the orbit {z n 0 +i } i≥0 is infinite, then there exists a large number N 0 such that the orbit {z N 0 +i } i≥0 avoids the finite points X.

Let M = N 0 + m, N = N 0 + n and ξ = z m+N 0 , we are done. Otherwise, we can suppose that all Y are eventually iterated to X 0 ⊆ X and points in X 0 are (pre-)periodic. Then there exist a periodic point w with period p and infinite many points in Y iterated to w. Thus we have (z , n ) and (z , n

), z = z ∈ Y , such that F n (z ) = F n (z ) = w and n = n mod p. Let n = n + kp, k > 0. Then F n (z ) = F n (z ) = w, which contradicts (3.16). The claim follows. For simplicity we write [x, y] = [x M , y M ]. Let ξ ∈ [x, y] T [x N , y N ], N ≥ 1, such that
its orbit never hits the boundary of the partition {Π i } 1≤i≤d . Let

H := [x, y] [ [x N , y N ] [ [x 2N , y 2N ] [ • • • . (3.17)
Then,

• For any ζ, η ∈ H, [ζ, η] ⊆ H. Indeed, suppose ζ ∈ [x n 1 N , y n 1 N ] and η ∈ [x n 2 N , y n 2 N ] with integers n 1 ≤ n 2 . Then the path γ ζη := [ζ, ξ n 1 N ] [ [ξ n 1 N , ξ (n 1 +1)N ] [ • • • [ [ξ n 2 N , η]
joins ζ and η. By the uniqueness of regulated arc in Lemma 3.7, It follows that

[ζ, η] ⊆ γ ζη . ξ ξ N ξ 2N ξ(ξ 2N ) ξ N ξ ξ 2N ξ N ξ 2N ξ ξ N ξ ξ N η ξ 2N (1) (2)
(3) (4)

(5)

Figure 3.7: Relations of [ξ, ξ N ] and [ξ N , ξ 2N ] Since [ζ, ξ n 1 N ] ⊆ [x n 1 N , y n 2 N ], [ξ kN , ξ (k+1)N ] ⊆ [x kN , y (k+1)N ] and [ζ n 2 N , η] ⊆ [x n 2 N , y n 2 N ], then γ ζη ⊆ H. Thus [ζ, η] ⊆ H. • For any n ≥ 0, if F n (ξ) ∈ Π i(n) , then F n (H) ⊆ Π i(n) . Indeed, since ξ is never mapped into S 1≤i≤d ∂Π i , such Π i(n) exists.
We claim that x kN , ξ kN , ξ (k+1)N , y kN have the same itinerary respect to {Π i }. Since ξ, ξ N ∈ [x, y], then ξ kN , ξ (k+1)N ∈ [x kN , y kN ]. By Lemma 3.4 and (3.15), [x kN +j , y kN +j ] must be contained in some Π j(n) for any j. In particularly, we have F j (x kN ), F j (ξ kN ), F j (ξ (k+1)N ), F j (y kN ) ∈ Π j(n) . By the arbitrariness of j, the claim follows. Therefore we obtain a sequence ξ, ξ N , ξ 2N , • • • , ξ kN , x kN , y kN , which have the same itinerary. Thus if

F n (ξ) ∈ Π i , F n [x kN , y kN ] ⊆ Π i . By the arbitrariness of k, it follows that F n (H) ⊆ Π i .
• For any n ≥ 0, F n | H is homeomorphism and F N (H) ⊆ H. The latter follows immediately by definition. For the former, if not, there exists a minimal number n 0 ≥ 0 such that we have ζ = η ∈ F n 0 (H) with F (ζ) = F (η). By the above conclusions, we see that (

[ζ, η] ⊆ F n 0 (H) and is contained in some Π i(n 0 ) . Since [ζ, η] is quasi-buried, there exists [ζ (i) , η (i) ] ⊆ [x, y] with ζ (i) , η (i) ∈ Π i(n 0 ) such that ζ (i) → ζ, η (i) → η as i → ∞. Then F | [ζ (i) ,η (i) ] is one-to-one by Lemma 3.4. Thus F [ζ, η] is a loop. It is impossible by Lemma 3.7.
1) [ξ, ξ N ] T [ξ N , ξ 2N ] = {ξ N }. (2) [ξ, ξ N ]=[ξ N , ξ 2N ]. (3) [ξ N , ξ 2N ] ⊂ [ξ, ξ N ]. (4) [ξ, ξ N ] ⊂ [ξ N , ξ 2N ]. (5) [ξ, ξ N ] T [ξ N , ξ 2N ] = [η, ξ N ] for some η ∈ (ξ, ξ N ).
We will show that all of them are impossible and so the proof is completed.

For case (1), we have

[ξ, ξ 2N ] = [ξ, ξ N ] S [ξ N , ξ 2N ] ⊆ H. Then F | [ξ,ξ 2N ] is homeomorphic. Note that F N [ξ, ξ N ] = [ξ N , ξ 2N ]. It follows that [ξ 2N , ξ 3N ] T [ξ N , ξ 2N ] = {ξ 2N }. We also have [ξ 2N , ξ 3N ] T [ξ, ξ N ] = ∅. Otherwise, the three arcs [ξ, ξ N ] S [ξ N , ξ 2N ] S [ξ 2N , ξ 3N ] would form a loop. By induction, it follows that [ξ nN , ξ (n+1)N ] T [ξ, ξ nN ] = {ξ nN } for n ≥ 0. Then (ξ, ξ N ) is a wandering regulated arc of F N . By Proposition 3.3, it is impossible. Case (2) can not happen. Indeed, otherwise F N : [ξ, ξ N ] → [ξ, ξ N ] is homeomorphic. Choose any subarc I in [ξ, ξ N ] such that F N (I) T I = ∅. Then I is a wandering regulated arc of F N . For case (3), choose an arbitrary subarc I in (ξ, ξ 2N ). Then F N (I) ⊆ (ξ N , ξ 2N ). Since F N : [ξ, ξ N ] → [ξ N , ξ 2N ] is homeomorphic and [ξ N , ξ 2N ] ⊂ [ξ, ξ N ], I is a wandering regulated arc of F N , a contradiction.
For case (4), by the intermediate value theorem, there is a fixed point ν ∈ (ξ, ξ N ) of

F N . Then [ν, ξ] ⊂ [ν, ξ 2N ] and the map F 2N : [ν, ξ] → [ν, ξ 2N ] is homeomorphic. Let ξ -2N ∈ [ν, ξ] such that F 2N (ξ -2N ) = ξ. Then [ξ -2N , ξ] T [ξ, ξ 2N ] = {ξ}. Similar to case (1), it is impossible. For case (5), let η -N ∈ [ξ, ξ N ] with F N (η -N ) = η.
We distinguish three possibilities to analyze.

(5.1)

η -N ∈ (ξ, η). Then η N ∈ (η, ξ 2N ). Therefore [η -N , η] T [η, η N ] = {η}. By case (1)
again, it is impossible.

(5.2) η -N = η. Then η is a fixed point of F N . We claim that there exist ν ∈ (η, ξ) and n 0 ≥ 3 such that

F n 0 N [η, ν] ⊆ [η, ξ]. Indeed, since F 3N [η, ξ] = [η, ξ 3N ] and F N | H is injective, hence [η, ξ 3N ] T ([η, ξ N ] ∪ [η, ξ 2N ]) = {η}. If [η, ξ 3N ] T [η, ξ] = {η}, the claim follows. Otherwise, continue the process to [η, ξ 3N ] • • • , until [η, ξ kN T [η, ξ] = {η}. Oth-
erwise, we obtain an infinity sequence {(η, ξ kN ]} k≥0 which are pairwise disjoint. This contradict Proposition 3.3. Hence the claim follows.

Choose

ν ∈ (η, ν) such that ν n 0 N = ν . If ν n 0 N ∈ (η, ν ), similarly in case (3), it is impossible. If ν n 0 N ∈ (ν, ξ), let ν -n 0 N ∈ (η, ν ) be the preimage of F n 0 N | [η,ν ] , then similar in case (1), (ν -n 0 N , ν ) is a wandering regulated arc of F n 0 N . It is impossible. (5.3) η -N ∈ (ξ N , η). Applying intermediate value theorem to F : [η -N , η] → [η, η N ], we obtain a fixed point ν ∈ (η -N , η). Since [ν, ξ N ] T [ν, ξ 2N ] = {ν}. So this is the case (5.2),
impossible. The proof is completed.

Characteristic arc I c

In order to introduce a partial order on F, we need the following definition of characteristic arc I c . (2) If f c is in case (C1) and I c = R/Z, then (2.1) L ηcξc separates critical point 0 and critical value c.

Recall L ηcξc := R(η c ) S R(ξ c ) S {z}. Therefore, |I c | < 1 2 . (2.2) R(η c ) and R(ξ c ) resp. R(η c ) and R(ξ c ) land together at z resp. z with {z , z } := f -1 (z).
(2.3) Let S c be the sectors bounded by R(η c ) and R(ξ c ) avoiding the critical point and H c the domain bounded by L η c ξ c and L η c ξ c , then H c T S c = ∅ and f : H c → S c is a branched covering of degree two.

(3) For any

f c , f c ∈ F, if f c ≺ f c , then I c S I c ⊆ I c S I c .
Proof.

(1) Since both R(η c ), R(ξ c ) land at critical value c. Then, at the critical point 0, there exist preimages, rays R(η c ), R(η c ), R(ξ c ), R(ξ c ). If I c is a single point, we have

S c S c S c R(η c ) R(ξ c ) R(η c ) R(ξ c ) R(η c ) R(ξ c ) S c L ηcξc L η c ξ c L η c ξ c H c
(S c S S c ) T S c = ∅. Since f | S c resp. f | S c is conformal. Thus l(f (S c )) = 2l(S c ) = 2l(S c ). Note that l(C \ S c ) > l(S c ) + l(S c ). It follows that f (S c ) = f (S c ) = S c .
(2) Let p be the period of the critical value Fatou component U and z

0 := z, z 1 := f (z), • • • , z p := f p (z) with z p = z 0 . Since this orbit is disjoint with critical point, we can set L z i the preimage of f -(p-i) L ηcξc at each z i . Obviously, L z 0 = L zp ,
because both of them support Fatou component U . Let S z i be one of the components C \ L z i containing 0 and S z i the other.

For (2.1), suppose L z 0 does not separate 0 and c, then S z 0 contains both of them. For i = p -1, By Lemma 3.2 (4), the sector map σ 2 must send the critical sector S z p-1 to critical value sector S zp , and thus σ 2 (S z p-1 ) = S zp . We have l(S z p-1 ) = 1 2 l(S z 0 ). Claim L z p-1 cannot separate 0 and c. Otherwise, using Lemma 3.1 and properties of supporting rays, we have S z p-1 ⊃ S z 0 , thus l(S z p-1 ) > l(S z 0 ), impossible. For i = p -2, • • • , 0, the same argument as above implies l(S z i ) = 1 2 l(S z i+1 ) and S z i contains both 0 and c. Thus l(S z 0 ) = 1 2 p l(S z 0 ), a contradiction. For (2.2), since z is not a critical value. We have two z = z preimages of z. We discuss by contradiction and assume R(η c ), R(ξ c ) resp. R(η c ), R(ξ c ) land at z resp. z . Then consider the sector S c :=

S θ∈I c R(θ) resp. S c := S θ∈I c R(θ). We have σ 2 (S c ) = σ 2 (S c ) = S z 0 . Since l(S c ) = l(S c ) = l(I c ) = l(I c ) < 1 2 , by Lemma 3.2, f | S c , f | S c are conformal.
Therefore, the image S z 0 cannot contain critical value c. This contradicts (2.1).

For (2.3), note that both of L η c ξ c and L η c ξ c support the critical Fatou component and are symmetry respect the original. Then the fact

|I c | > |I c | = |I c | implies H c T S c = ∅.
For (3), one can easily check it by definition.

Dynamic of biaccessible angles

Given f c ∈ F, an angle θ in R/Z is called to be biaccessible, if there exists θ = θ such that R(θ) and R(θ ) landing together. Evidently, if θ is biaccessible, then the preimages σ -1 2 (θ) are biaccessible. Inversely, if θ is biaccessible and α(θ) is not the critical point, then σ 2 (θ) is also biaccessible. Denote by Acc(f c ) the set of all biaccessible angles of f c . Then if

I c = R/Z, Acc(f c ) = ∅ by lemma 3.2.
Lemma 3.12. Let I c = R/Z and not a single angle. Let θ be a biaccessible angle of f c and the orbit of the landing point ζ 0 := α(θ) avoid critical point 0. Then there exists a N ≥ 0 such that the orbit of

ζ N := f N (ζ 0 ) is disjoint with H c , where H c is defined in Lemma 3.11 (1.2)(2.
3). Therefore, there exists ϑ = θ N := σ N 2 (θ) such that, for any ν ∈ (η c , ξ c ), ϑ and θ N have the same itinerary respect to R/Z \ σ -1 2 (ν).

Proof. Let θ = θ with α(θ ) = α(θ) = ζ 0 . Since ζ 0 will never meet the critical point. For n ≥ 0, L θnθ n = f n (L θθ ) bounds two sectors S ζn and S ζn , where we assume S ζn is the one containing 0. Firstly, there exists a N ≥ 0 such that L θ N θ N separates 0 and c. If not, for each n ≥ 0,

σ 2 must send S ζn to S ζ n+1 and S ζn to S ζ n+1 , therefore, l(S ζ n+1 ) = 2l(S ζn ) by Lemma 3.2 (2), (3) and (4)
. It follows that l(S ζn ) → ∞ as n → ∞, impossible. By Lemma 3.11 (1.2)(2.3), points in H c will be mapped to S c . Thus we only have to show that ζ n / ∈ S c , n ≥ N . Claim l(S ζn ) ≥ l(S c ). If not, let n 0 > N be first integer such that l(S ζn 0 ) < l(S c ). Thus S ζn 0 must be a critical value sector. This means S ζn 0 ⊇ S c or S ζn 0 ⊇ C \ S c , both of which imply l(S ζn 0 ) ≥ l(S c ), a contradiction. Therefore,

ζ n / ∈ S c , n ≥ N .

Monotonicity of core-entropy

Proof of Theorem 3.3.

If I c = R/Z, Acc(f c ) = ∅. If #I c = 1, then I c = I c , hence I c = I c and I c = I c . By Theorem 3.2, Acc(f c ) = Acc(f c ).
In other cases, we have either

I c = I c or I c I c . If I c I c . We can assume η c ∈ (η c , ξ c ).
For any θ ∈ Acc(f c ), if the orbit of α(θ) is disjoint with critical point 0, by Lemma 3.12, there exist N and θ = θ N such that θ N and ϑ have the same itinerary respect to partition R/Z \ σ -1 2 (η c ). By theorem 3.2, in the dynamic plane of f c , external rays with arguments θ N , ϑ land together. Thus θ ∈ Acc(f c ). If α(θ) is iterated to 0, then critical point is not periodic. Evidently, the above N and ϑ exist as well.

If

I c = I c . For any θ ∈ Acc(f c ), if α(η c
) is not periodic, by the same argument as above, such ϑ and N exist. If α(η c ) is periodic. If the orbit of α(θ) avoids α(η c ), then such

ϑ and N exist. If α(θ) is mapped to α(η c ). Then θ is iterated into {η c , ξ c } ⊆ Acc(f c ). Thus θ ∈ Acc(f c ). The proof is completed.
Chapter 4

A landing theorem on non-recurrent polynomials

Introduction

In this section we give some notations and recall some results of polynomial dynamics. We refer to [Mi06], [START_REF] Goldberg | On the multiplier of a repelling fixed point[END_REF] and [Ki05] for details.

Let f be a monic polynomial with degree d ≥ 2. Let Ω f be basin of infinity consisting the set of all points in C escaping to ∞ and the filled Julia set K f := C \ Ω f . There exists a green function G f measures the escape rate of points to ∞, defined by

G f : C → [0, ∞) z → lim n→∞ log |f n (z)| d n .
It is a continuous function which vanishes on the filled Julia set and satisfies

G f (f (z)) = dG f (z).
Moreover, G f is positive and harmonic in Ω f . In Ω f , the derivative of G f vanishes at z if and only if z is a pre-critical point. Each locus

G -1 f (r) = {z ∈ C, G f (z) = r} with r > 0 is called an equipotential curve around the filled Julia set K f .
Near ∞, there exists an unique normalized Böttcher map Ψ f which conjugates f with

z → z d in a neighborhood of ∞. Ψ -1
f has an unique maximal radial extension to a subset of C \ D. This radial extension terminates at a point w with |w| > 1 if and only if Ψ -1 f extends continuously to w and

Ψ -1 f (w) is a (pre)critical point of f . Then external radius R f (t) with argument t is R f (t) := Ψ -1 f ((r t , ∞)e 2πit ),
where Ψ -1 f (r t e 2πit ) is (pre)critical point of f . If all critical points has bounded orbits, then r t = 1 and so Ω f is simply connected.

We are working in the parameter space

P d ∼ = C d-1 of monic centered polynomials, that is polynomials z → z d + a d-2 z d-2 + • • • + a 0 .
The shift locus S d is the subset of P d formed by polynomials with all critical points escaping to infinity. Then K f is cantor set for each f in S d . Let S d (r), r > 0, consist of polynomials in S d such that all the critical points are in the same equipotential curve G -1 (r) and let S d := S r>0 S d (r).

A collection Θ = {Θ 1 , • • • , Θ n } of finite subsets of R/Z is called a critical portrait of degree d if the following conditions hold, (1) for every j, |Θ i | ≥ 2 and |σ d (Θ j )| = 1, where σ d : θ → d • θ mod 1. (2) Θ 1 , • • • , Θ n are pairwise unlinked. (3) P (|Θ j | -1) = d -1.
For another critical portrait Θ = {Θ 1 , • • • , Θ n }, we say Θ = Θ iff there exist a permutation τ such that Θ i = Θ τ (i) for 1 ≤ i ≤ d. Let A d be the collection of all critical portrait of degree d. In [Ki05], Kiwi gave A d a compact-unlinked topology and proved that A d is compact and connected.

Now we consider the map

Π : S d → A d f → Θ.
Indeed, since there are deg f (c j ) external radius with argument θ k terminating at c j . Denote these arguments θ k by Θ j . Then Π(f

) := {Θ 1 , • • • , Θ n } is the critical portrait induced by f .
In [START_REF] Goldberg | On the multiplier of a repelling fixed point[END_REF], L.R.Goldberg proved that Π is surjective. Indeed, for each Θ ∈ A, she constructed a degree d topological polynomial g which maps X g (r) onto X g (dr) conformally. All the critical points of g are in ∂X g (r). Moreover, g induces the prescribed critical portrait Θ. Then g pullbacks the standard complex structure on X g (r) to the space

R = [ n≥0 g -n (X r ).
Thus g : R → R is a complex analytic map. Since R is a planar Riemann surface, it can be conformally embedded in C by Koebe's general uniformization Theorem 9.1 in [START_REF] Springer | Introduction to Riemann Surfaces[END_REF]. The complement C \ R is a holomorphically removable Cantor set [START_REF] Sario | Classification Theory of Riemann Surfaces[END_REF]. Thus g extends to a holomorphic map f : C → C which is a degree d polynomial with the required critical portrait Θ.

In [Ki05], Kiwi showed that Π is continuous and, for any Θ, the preimage S Θ = Π -1 (Θ) is a 1-real dimensional analytic manifold. Precisely, the map G : S Θ → (0, ∞) which sends f to G f (c i ) is bijective and analytic. Moreover, given r > 0, the restriction

Π| S d (r) : S d (r) → A d is a homeomorphism.
The connected locus C d is the set of monic centered polynomials with degree d such that all the critical orbits are bounded. We know that C d is a compact and connected subset of P d [BH88]. For instance, C 2 is the Mandelbrot set. To describe C d we look at it from outside S d .

The impression

I C d (Θ) of critical portrait Θ is a subset of C d , characterized by the property that f ∈ I C d (Θ) if and only if there exists a sequence {f n } in S d such that Π(f n ) = Θ and f n converges to f .
Note that the impression here is slightly different from the definition in [Ki05], which is bigger and containing I C d (Θ). J.Kiwi proved that if all arguments in Θ is strictly preperiodic, then the impression I C d (Θ) is a singleton [Ki05]. He conjectured that there exist critical portraits with aperiodic kneading and non-trivial impressions.

Main results. In this chapter, we shall give an elementary proof of the following two theorems based on the tools in [CT15].

Theorem 4.1. The map

P : A d × (0, ∞) → S d (Θ, r) → f Θ,r
where f Θ,r ∈ S d (r) induces critical portrait Θ, is well-defined, one-to-one and continuous.

The well-defined and one-to-one properties are proved by quasiconformal surgery. In parameter space we will call the simple curve

R Θ (t) := P (Θ, •) : (0, ∞) → S d
parameter ray in S d with argument Θ. For quadratic polynomials, R Θ is exactly parameter ray outside of Mandelbrot set. We will say that R Θ lands if and only if the impression

I C d (Θ) is a singleton.
In dynamical plane, let f be a polynomial in C d with J f locally connected and all cycles repelling. A critical portrait

Θ = {Θ 1 , • • • , Θ n } is called a critical portrait of f if each external rays with arguments in Θ i lands at critical point c i , for 1 ≤ i ≤ n.
We have the following landing theorem.

Theorem 4.2. Let f be a polynomial in C d with non recurrent critical points and all cycles repelling. Then the parameter ray R Θ (t) lands at f if and only if Θ is a critical portrait of f .

The only if part comes from combinatorial continuity [Ki05] Theorem 1.

Preliminaries

In this section, we collect some known result on spherical metric, the distortions of modulus, as well as shapes and turning, by holomorphic maps. The convergence of rational maps on C is also discussed. These preliminaries will be used in the rest of this chapter.

Spherical metric

We will denote by B e (z, r), dist e (x, y), diam e W Area e S the Euclidean balls, distances, diameters and Euclidean area. While B(z, r), dist(x, y) and diam W are measured in spherical metric.

Recall that the spherical line element and spherical area element on C are

ds = 2|dz| 1 + |z| 2 and dσ = 4dxdy (1 + |z| 2 ) 2 .

So we have

• 2 5 |dz| ≤ ds ≤ 2|dz| on B e (0, 2), • the holomorphic map α : z → 1/z preserves the spherical distance and spherical area,

• let 0 = dist (B e (0, 1), ∂B e (0, 2)), then any subset S of C with diam S < 0 is either contained in B e (0, 2) or C \ B e (0, 1).

• inf x∈C,0<r< 0 /2 ¦ mod B(x, 2r) \ B(x, r) © ≥ m 0 > 0.

Mañé Lemma

Lemma 4.1 ( [Ma93],Theorem II). Let f : C → C be a rational map with degree at least two. If a point x ∈ J f is not a parabolic periodic point and is not contained in the ω-limit set of a recurrent critical point, then for any > 0 there exist δ = δ(x, ) < and integer η = η(x, ) such that for any r ≤ δ and any n ≥ 0,

(1) every component of f -n B(x, r) has spherical diameter less than , (2) for every component W of f -n B(x, r), degree of f n : W → B(x, r) is less than η, (3) every component of f -n B(x, r) is a topological disk.

Distortions of modulus, shape and turning

Lemma 4.2 ( [KL09],Lemma 4.5). Let U i ⊆ V i in C be a pair of Jordan disks, where

i = 1, 2. Suppose that g : V 1 → V 2 is a proper holomorphic map of degree d and U 1 is a component of g -1 (U 2 ). Then mod V 1 \ U 1 ≤ mod V 2 \ U 2 ≤ d mod V 1 \ U 1 .
Let U be a domain in C and z ∈ U . The Shape e resp. Shape of U about z is defined as Shape e (U, z) = max w∈∂U dist e (w, z) min w∈∂U dist e (w, z) resp. Shape(U, z) = max w∈∂U dist(w, z) min w∈∂U dist(w, z)

Obviously, B(z, r) ⊆ U ⊆ B(z, kr) for some r, where k := Shape(U, z). Thus U is a round disk centered at z if and only if Shape(U, z) = 1.

Let E be a compact set in C and z 1 , z 2 ∈ E, the turning is defined as

Λ e (E, z 1 , z 2 ) = diam e E dist e (z 1 , z 2 ) resp. Λ(E, z 1 , z 2 ) = diamE dist(z 1 , z 2 )
.

We have the following lemma,

Lemma 4.3 ( [QWY12], Lemma 6.1 ). Let U i ⊆ V i in C be a pair of Jordan disks with mod(V 2 \ U 2 ) ≥ m > 0, where i = 1, 2. Suppose that g : V 1 → V 2 is a proper holomorphic map of degree ≤ d and U 1 is a component of g -1 (U 2 ).
Then there are two positive constants C 1 (d, m) and C 2 (d, m) depending only on d and m, such that (1) for all z ∈ U 1 , the shape satisfies

Shape e (U 1 , z) ≤ C 1 (d, m)Shape e (U 2 , g(z)), (4.1)
(2) for any connected and compact subset E of U 1 with the cardinal number #E ≥ 2 and any z 1 , z 2 ∈ E, the turning satisfies From now on when we apply Lemma 4.3, we always assume U i , V i and E satisfying this additional condition.

Λ e (E; z 1 , z 2 ) ≤ C 2 (d, m)Λ e (g(E); g(z 1 ), g(z 2 )).

Convergence of rational map sequences

Throughout this chapter, if not otherwise stated, the convergence of maps on C is measured in spherical metric.

If a sequence of rational maps {f n } uniformly converges on C, then it converges to a rational map g and deg (f n ) = deg (g) as n is large enough. Moreover, the coefficients of f n converges to that of g as well. For more results, we have the following lemma. See also in [CT15]. Proof. By composing Möbius transformations, we may assume that ∞ ∈ U and f n (∞) → 1. Thus as n is large enough, the function f n has the form

f n (z) = k n (z -a 1,n ) • • • (z -a d,n ) (z -b 1,n ) • • • (z -b d,n ) , and
k n → 1 as n → ∞. (4.3) Since f -1 n (0) = {a 1,n , • • • , a d,n }, f -1 n (∞) = {b 1,n , • • • , b d,n
} and f n (∞) → 1, both {a i,n } and {b j,n } are bounded in C. Passing to a subsequence {f n k }, we have

(a 1,n k , • • • , a d,n k ; b 1,n k , • • • , b d,n k ) → (a 1 , • • • , a d ; b 1 , • • • , b d ) as n k → ∞.
(4.4)

Without loss of generality, we assume that a i = a j , b i = b j for i = j and

a d 0 +1 = b d 0 +1 , • • • , a d = b d , a i = b j for 0 ≤ i, j ≤ d 0 (4.5) for some 0 ≤ d 0 ≤ d. Let g 1 (z) = (z -a 1 ) • • • (z -a d 0 ) (z -b 1 ) • • • (z -b d 0 ) or g 1 (z) = 1 if d 0 = 0.
We claim that f n k converges locally uniformly to

g 1 on C \ {a d 0 +1 , • • • , a d }. Proof. Consider the metric d(•, •) on C d(z, z ) := 2|z -z | È 1 + |z| 2 • È 1 + |z | 2 and d(∞, z) := lim ξ→∞ d(ξ, z)
for z, z ∈ C. We know that it is equivalent to the spherical metric on C.

For any z ∈ C, the distance d(f n k (z), g 1 (z)) equals

2 k n Q 1≤i≤d (z -a i,n k ) Q 1≤i≤d 0 (z -b i ) - Q 1≤i≤d 0 (z -a i ) Q 1≤i≤d (z -b i,n k ) É k n Q 1≤i≤d (z -a i,n k ) 2 + Q 1≤i≤d (z -b i,n k ) 2 • É Q 1≤i≤d 0 (z -a i ) 2 + Q 1≤i≤d 0 (z -b i ) 2 =: ∆ z,n k ∆ z,n k
By assumption the claim follows at the point ∞.

Let

δ := 1 3 • min x =x ∈{a 1 ,••• ,a d ,b 1 ,••• ,b d 0 ,∞} dist (x, x ). For any x ∈ C \ {a d 0 +1 , • • • , a d }, B(x, δ) ⊆ C. Moreover, by (4.3) (4.4) (4.5), sup z∈B(x,δ) ∆ z,n k → 0 as k → ∞
and there exists δ x > 0 such that inf z∈B(x,δ),k≥0

∆ z,n k ≥ δ x .
This is because, as k large enough, at most one sequences of {a i,n k } and {b i,n k } stays in the disk B(x, 2δ). Therefore, {f n k } is uniformly convergent on B(x, δ). The claim is proved.

Since {f n } converges uniformly to g on U , we have {f n } converges uniformly to g(= g 1 ) on any compact subset in C \ {a

Modulus distortion dominates the spherical distortion

In this section, for our purpose we introduce a new quantity, namely the maximal distortion of modulus in [CT15], to control the distance between univalent map and Möbius transformation. Although there are many existent measurements, such as the norm Schwarzian derivative, or the maximal distortion of cross-ratios or extremal lengths etc.

Maximal distortion of modulus

Let V be an open set in C and φ : V → C a univalent map. For any two disjoint full continua E 1 , E 2 in V , we denote by

A(E 1 , E 2 ) := C \ (E 1 S E 2 ) which is annulus induced by E 1 , E 2 .
Define the maximal distortion of modulus D(φ, V ), as follows

D(φ, V ) := sup E 1 ,E 2 ⊆V |mod A(E 1 , E 2 ) -mod A(φ(E 1 ), φ(E 2 ))|,
where

E 1 , E 2 are disjoint full continua in V with mod A(E 1 , E 2 ) < ∞.
This quantity may not be finite, the first example being z 2 acting on the right half plane (see the theorem below for a proof). Also, although it is clear that D(φ, V ) = 0 if φ is the restriction of a Möbius transformation on V , but it is not at all obvious that the converse is also true.

In order to understand the following theorem, let us look at a guiding example: consider V a neighborhood of {0, ∞} and define a univalent map φ on V as

φ(z) = 8 > < > : λ 0 z, near 0 z/λ ∞ , near ∞ with λ 0 , λ ∞ = 0.
For sufficiently small, let (1) For any Möbius transformations α, β, we have

E 1 = {z ∈ C : |z| ≤ } and E 2 = {z ∈ C : |z| ≥ 1 }. We have A(φ(E 1 ), φ(E 2 )) = {|λ 0 | < |z| < 1 |λ∞| }. So |mod A(E 1 , E 2 ) -mod A(φ(E 1 ), φ(E 2 ))| = 1 2π log |λ 0 λ ∞ || ≤ D(φ, V ).

Properties of maximal distortion of modulus

D(α • φ • β, β -1 (V )) = D(φ, V ).
(2) If V contains 0, ∞ and φ fixes 0, ∞, then, setting φ (∞) := lim z→∞ z φ(z) ,

1 2π |log |φ (0)φ (∞)|| ≤ D(φ, V ).
(3) If V contains 0, 1, ∞ and φ fixes 0, 1 and ∞, then, for any z ∈ V \ {∞},

1 5π |log |φ (z)|| ≤ D(φ, V ).
(4) D(φ, V ) = 0 if and only if φ is the restriction of a Möbius transformation on V .

(5) If an extension of φ has a critical point on the boundary of V , then D(φ, V ) = +∞. (6) If V contains 0, D(1, r 0 ) and ∞. Then there exists constant 0 < C(r 0 ) < ∞ such that, for any univalent map φ fixing 0, 1 and ∞, we have

sup z∈V {dist (φ(z), z)} ≤ C(r 0 ) • D(φ, V ).
Proof. (1) It follows evidently. Because Möbius transformations preserve the modulus of annulus.

(2) Let M 0 ( ), m 0 ( ) be the supremum and infimum of |φ(z)| on the circle {|z| = }, and M ∞ ( ), m ∞ ( ) be the supremum and infimum of |ψ(z)| on the circle {|z| = 1/ }.

Set

E 1 := {|z| ≤ }, E 2 := {z ∈ C : |z| ≥ 1 } and A φ := A(φ(E 1 ), φ(E 2 )). Then {M 0 ( ) < |z| < m ∞ ( )} ⊆ A φ ⊆ {m 0 ( ) < |z| < M ∞ ( )} and 1 2π log 2 m ∞ ( ) M 0 ( ) ≤ mod A φ -mod A ≤ 1 2π log 2 M ∞ ( ) m 0 ( ) . Since |φ (0)| = lim →0 M 0 ( ) = lim →0 m 0 ( ) and |φ (∞)| = lim →0 1 • M ∞ ( ) = lim →0 1 • m ∞ ( ) ,
we get

1 2π |log |φ (0)φ (∞)|| = lim →0 |mod A φ -mod A| ≤ D(φ, V ).
(3) For arbitrary distinct z, w ∈ V \ {∞}, there exist Möbius transformations

β a (ξ) = w • ξ + z • a ξ + a and α b (ξ) = b • ξ -φ(z) ξ -φ(w)

Controlling modulus distortions by areas

In this section, firstly we recall some classic results on modulus of annulus without proof. Then we give a lemma, which provides a way to control the maximal distortion of modulus. Height(ρ, A) is the infimum of Length (ρ, γ) over all γ ∈ Γ Height . Width (ρ, A) is the infimum of Length (ρ, γ) over all γ ∈ Γ Width . The modulus of annulus A is defined to be the modulus of the family of curves Γ Width , that is, mod A := inf Area (ρ, A), (4.7)

Modulus of annulus

where the infimum is taken over all nonnegative Borel functions ρ with Width (ρ, A) ≥ 1.

There are some well-known facts about modulus of annulus.

• It is preserved by conformal maps.

• It equals the extremal length of the family of curves Γ Height , that is,

mod A = sup Height (ρ, A) 2 Area (ρ, A) (4.8)
where ρ is taken over all nonnegative Borel function ρ with 0 < Area(ρ, A) < ∞.

• If mod A < ∞, there exists an unique conformal map χ A : A → A(1, e 2πmod A ) up to a post composition of a rotation.

• ρ 0 reaches the infimum in (4.7) if and only if

ρ 0 (z) = |( 1 2π • log • χ A ) (z)|,
which is the pullback of Euclidean metric in C. ρ 0 is called extremal metric on A.

• Extremal metric ρ 0 realizes the supremum in (4.8), thus mod A = Height (ρ 0 , A) = Area (ρ 0 , A) and Width (ρ 0 , A) = 1. (4.9)

• If A in C is annulus with mod A > 5log2 2π . Then there exists an annulus A(z 0 ; r 1 , r 2 ) contained essentially in A such that mod A(z 0 ; r 1 , r 2 ) ≥ mod A -5log2 2π .

A conformal metric ρ, i.e., nonnegative Borel measurable function, on A is called to be length increasing if Width (ρ, A) ≥ 1 and Height (ρ, A) ≥ Height (ρ 0 , A).

(4.10)

Area difference controls modulus distortion

Lemma 4.5. Let Z be a compact set in C and φ univalent on

C\Z. Let E 1 , E 2 be disjoint full continua in C \ Z. Then |mod A(E 1 , E 2 ) -mod A(φ(E 1 ), φ(E 2 ))| ≤ Area(ρ, A) -Area(ρ 0 , A)
for arbitrary length increasing conformal metric ρ on A(E 1 , E 2 ) with ρ| Z = 0.

Proof. We write A := A(E 1 , E 2 ) and A φ := A(φ(E 1 ), φ(E 2 )). Let ρ 0 be an extremal metric on A.

123

For arbitrary length increasing conformal metric ρ on A with ρ| Z = 0, set

e ρ(z) = 8 > < > : 0, on A φ \ φ(A \ Z) ρ(φ -1 (z)) |φ (z)| , on φ(A \ Z).
Then it is not difficult to check that

• Area (ρ, A) = Area ( e ρ, A φ ), • Width ( e ρ, A φ ) ≥ Width (ρ, A), • Height ( e ρ, A φ ) ≥ Height (ρ, A).
Combining with (4.7), (4.8), (4.9) and (4.10), we have

mod A φ -mod A ≤ Area ( e ρ, A φ ) -mod A = Area (ρ, A) -Area (ρ 0 , A) and mod A -mod A φ = Area (ρ 0 , A) -mod A φ ≤ Area (ρ 0 , A) - Height ( e ρ, A φ ) 2 Area ( e ρ, A φ ) ≤ Area (ρ 0 , A) - Height (ρ, A) 2 Area (ρ, A) ≤ Area (ρ 0 , A) - Height (ρ 0 , A) 2 Area (ρ, A) ≤ Area (ρ, A) -Area (ρ 0 , A).
The last inequality holds by the fact Height (ρ 0 , A) = Area (ρ 0 , A) and the Cauchy Inequality. Thus the lemma is proved.

Univalent maps off a finite nested disc system

The result in this section generalizes the discussion in ( [CT15], Section 8.2). At first, we give the definition of nested disk systems, the m-nested and λ-scattered properties. Then we pay great effort to prove Theorem 4.4.

Nested disk system

Let Y be a finite set in C. A collection of open topological disks {D x } x∈Y is called nested disk system if

• each D x contains x, • if D x T D y = ∅ then either D x D y or D x D y .
Let {D x } x∈Y be a nested disk system. Let {D x } x∈Y and {D x } x∈Y be topological disks. Then {D x , D x , D x } x∈Y is said to be m-nested if

• x ∈ D x ⊆ D x ⊆ D x , • for any D y = D x with D y T D x = ∅, we have D y ⊆ D x , • inf x∈Y {mod D x \ D x } ≥ m > 0.
Let W be an open set in C such that S x∈Y D x ⊆ W . Let V x be the union of all D y ( = D x ) contained in D x and W x the component of W containing x. Then the nested disk system {D x } x∈Y is said to be λ-scattered in W with λ ∈ (0, 1) if and only if for each

x ∈ Y and any univalent map h :

W x → C * , Area(ρ * , h(V x )) ≤ λ • Area(ρ * , h(D x )),
where ρ * := 1 2π|z| is a planar metric on C * .

The boundedness of Area p (E, W )

The following lemma will become natural by the end of the proof of Theorem 4.4. We estimate,

Area (ρ * , φ(E)) = ZZ φ(E) ( 1 2π|ξ|
)

2 dxdy = ZZ E |φ (z)| 2 4π 2 |φ(z)| 2 dxdy ≤ 4 π 2 d 2 Area e (E) < ∞ (4.11)
By the arbitrariness of φ, the proof is completed.

D = D 1 D 2 • • • D n .
(4.12)

Secondly, for 2 ≤ k ≤ n, set

I k := {x ∈ I k : D x \ D y = ∅ for any y ∈ I 1 [ • • • [ I k-1 } ⊆ I k and I k := {x ∈ I k : D x \ D y = ∅ for some y ∈ I 1 [ • • • [ I k-1 } ⊆ I k .
For k = 1, set I 1 := I 1 and I 1 := ∅. Obviously, I k is the union of the two disjoint sets I k and I k .

Thirdly, for any disjoint full continua E 1 , E 2 in C \ W , we write A := A(E 1 , E 2 ). Suppose mod A < ∞. Let χ A : A → A(1, e 2πmod A ) be conformal. Then

ρ 0 (z) = |( 1 2π log • χ A ) (z)|
is an extremal metric on A and satisfies (4.9). Define ρ k (z) on A for 1 ≤ k ≤ n inductively by

ρ k (z) = 8 > > > > > < > > > > > : 0, z ∈ S x∈I k D x ρ 0 (z)(1 -e -πm ) -k , z ∈ S x∈I k D x \ D x ρ k-1 (z), z ∈ S x∈I k D x ρ k-1 (z), z ∈ A \ D k .
When reaching the deepest nest, the metric stabilizes to ρ n . This is the metric we were looking for.

Claim 1. For 1 ≤ k ≤ n, ρ k (z) ≤ ρ 0 (z)(1 -e -πm ) -k on A.
Proof. It is obviously true for k = 1. By induction, for k ≥ 2, we have

ρ k (z) ≤ max {ρ 0 (z)(1 -e -πm ) -k , ρ k-1 (z)} ≤ max {ρ 0 (z)(1 -e -πm ) -k , ρ 0 (z)(1 -e -πm ) -k+1 } ≤ ρ 0 (z)(1 -e πm ) -k
for any z in A. Thus the Claim follows.

Claim 2. For 2 ≤ k ≤ n, ρ k (z) = ρ k-1 (z) = 0 on S x∈I k D x .
Proof. For any z ∈ S x∈I k D x , by (4.12) there exists an unique sequence

{x i } 1≤i≤k in Y with x i ∈ I i such that z ∈ D x k • • • D x 2 D x 1 . (4.13) Since x k ∈ I k , then there exists x l 1 with 1 ≤ l 1 ≤ k -1 such that D x k T D x l 1 = ∅. By (4.13), for l 1 -1 ≤ i ≤ k, we have D x i T D x l 1 = ∅ and thus x i ∈ I i . Therefore, ρ k (z) = ρ k-1 (z) = • • • = ρ l 1 (z). (4.14)
The m-nested property gives

z ∈ D x k D x l 1 D x l 1 . (4.15)
If x l 1 is contained I l 1 . We have ρ l 1 (z) = 0 by (4.15) and the definition of ρ l 1 . Otherwise we have x l 1 ∈ I l 1 . Then we continue the discussion and obtain l 1 > l 2 > l 3 > • • • satisfying (4.14) and (4.15). After at most k -1 steps, we must have some x l j in I l j such that x ∈ D x l j D x l j . This is because I 1 = ∅. So ρ k (z) = ρ l j (z) = 0. The Claim is proved.

Claim 3. ρ n (z) = 0 on D .

Proof. Recall that D = S x∈Y D x . For any x ∈ Y and any z ∈ D x . Suppose x ∈ I k . By (4.12), there exist a maximal k ≤ l ≤ n and sequence {x i } k≤i≤l with x i ∈ I i and x k := x such that

z ∈ D x l D x l-1 • • • D x k . It follows that D x i T D x k = ∅ and so x i ∈ I i for k ≤ i ≤ l.
By Claim 2, we have

0 = ρ k (z) = • • • = ρ l (z) = • • • = ρ n (z).
The Claim is proved. So there exists an annulus A(z x ; r x , R x ) essentially contained in

H x such that mod A(z x ; r x , R x ) = 1 2π log R x r x ≥ m - 5log2 2π ≥ m 2 . (4.17) Let A x := h -1 x (A(z x ; r x , R x )). Let t 0 := inf {t : γ(t) ∈ A x } and t 1 := sup {t : γ(t) ∈ A x }. Set ξ := γ(t 0 ), η := γ(t 1 ), γ 1 := γ| [0,t 0 ] , γ 2 := γ| [t 0 ,t 1 ] and γ 3 := γ| [t 1 ,1] . Then γ 1 , γ 3 ⊆ D x \ D x
and γ 2 crosses A x at least two times. For i ∈ {1, 3},

Length (ρ k , γ i ) ≥ Length (ρ k-1 , γ i ).
(4.18) Together with (4.17), we estimate

γ 1 γ 2 γ 3 γ 2 γ A x A(x 0 ; r x , R x ) D x D x h x (γ) H x h x (ξ) h x (η) h x
Length (ρ k , γ 2 ) = Z γ 2 ρ 0 (z)(1 -e -πm ) -k |dz| = (1 -e -πm ) -k Z hx(γ 2 ) |dz| ≥ (1 -e -πm ) -k (2R x -2r x ) ≥ (1 -e -πm ) -k+1 2R x ≥ (1 -e -πm ) -k+1 Z γ |dz| = Z γ 2 ρ k-1 (z)|dz| = Length (ρ k-1 , γ 2 ), (4.19)
where γ ⊆ h x (D x ) is a straight line segment connecting h x (ξ), h x (η) and γ 2 := h -1 x (γ ). Set e γ := γ 1 S γ 2 S γ 3 . By (4.18) and (4.19), the Claim follows.

Claim 5. ρ n is a length increasing conformal metric on A with ρ n | D = 0.

Proof. By Claim 3 ρ n | D = 0. For arbitrary γ ∈ Γ Height , applying Claim 4, we have

Length (ρ n , γ) = Length (ρ n , γ \ A \ D n ) + X x∈In Length (ρ n , γ \ D x ) ≥ Length (ρ n-1 , γ \ A \ D n ) + X x∈In Length (ρ n-1 , e γ x ) = Length (ρ n-1 , e γ) ≥ • • • ≥ Length (ρ 0 , γ ).
where γ ∈ Γ Height . Hence it follows that Height (ρ n , A) ≥ Height (ρ 0 , A) ≥ 1. By the same arguments, we have Width (ρ n , A) ≥ Width (ρ 0 , A). The Claim is proved.

Since {D x } x∈Y is λ-scattered. By (4.12), for any 1 ≤ k ≤ n -1,

Area (ρ 0 , D k+1 ) ≤ λArea (ρ 0 , D k ) ≤ • • • ≤ λ k Area (ρ 0 , D 1 ) (4.20)
Now applying Claim 5, Lemma 4.5, Claim 1 and (4.20), we have

0 ≤ Area (ρ n , A) -Area (ρ 0 , A) = ZZ Dn (ρ 2 n -ρ 2 0 )dxdy + X 1≤k≤n-1 ZZ D k \D k+1 (ρ 2 n -ρ 2 0 )dxdy = ZZ Dn (ρ 2 n -ρ 2 0 )dxdy + X 1≤k≤n-1 ZZ D k \D k+1 (ρ 2 k -ρ 2 0 )dxdy ≤ ZZ Dn ρ 2 0 (1 -e -πm ) 2n -ρ 2 0 dxdy + X 1≤k≤n-1 ZZ D k \D k+1 ρ 2 0 (1 -e -πm ) 2k -ρ 2 0 dxdy ≤ X 1≤k≤n 1 (1 -e -πm ) 2k -1 Area (ρ 0 , D k ) ≤ X k≥1 1 (1 -e -πm ) 2k -1 λ k-1 Area (ρ 0 , D 1 ).
If m is sufficient large such that (1 -e -πm ) 2 > λ. Then the last term of the above inequalities is

e -πm (2 -e -πm ) (1 -λ) [(1 -e -πm ) 2 -λ] Area (ρ 0 , D 1 ) := C(m.λ) • Area (ρ 0 , D 1 ).
By the arbitrariness of E 1 , E 2 in C \ W , Lemma 4.6 and Lemma 4.5, we have proved the theorem.

Application to rational maps

The aim of this section is to construct a sequence of m k -nested λ-scattered nested disk systems from pullback disk systems with m k → ∞ as k → ∞. We require the nested disk systems are uniformly λ-scattered. Moreover, each one contains a pullback disk system. See Proposition 4.1 for details.

One of our challenge is to deal with the situation that one critical point c is contained in the w-limit set of another critical point c . Thus for arbitrary small disk B(c, r), after pulling it back, the preimages will eventually meet c . This makes it impossible to obtain a desired nested disc system from pullback disk system directly. To overcome this problem, we select nice disks B(c i , r i ) and discard all of the "bad" components in preimages, namely the component containing points of lower level. The remaining components still contains all the preimages. See (4.32).

Throughout this section, we shall adopt the assumption in the following subsection.

Proof. Firstly, we claim that for any x ∈ X 0 , diam B -n (x, δ 0 /2) → 0 as n → ∞.

Proof. Since S k≥0 f -k B(x, δ 0 ) contains at most finite critical points. Thus the degree of covering map f n : B -n (x, δ 0 ) → B(x, δ 0 ) is uniformly upper bounded by a integer η x . By Lemma 4.3, Shape(B -n (x, δ 0 /2)) ≤ C 1 (η x , m 0 ).

We know that all disks {B -n (x, δ 0 /2)} n≥0 are pairwise disjoint (see Subsection 4.6.1 (4)).

Then the claim follows.

For any x ∈ X 0 , choose η x and δ x ≤ δ 0 /2 satisfying (1)-( 4) by the claim. For any

x ∈ Orb(X 0 ), let δ x , η x be in Lemma 4. For (3), if not, there exists a sequence of disks

{B -n k (x k , δ)} k≥0 , x k ∈ K, n k → ∞ as k → ∞,
such that their diameters are greater than some positive number. Since shapes of these disks are bounded by C 1 (η, m 0 ). There exist a ball B(x ∞ , r 0 ) with x ∞ ∈ J f contained in infinitely many disks of {B -n k (x k , δ)}. This contradicts the fact that J f ⊆ f N (B(x ∞ , r 0 )) for sufficiently larger N . 4.6.3 A sequence of arbitrary small nice disks N x around x Lemma 4.8 (Key lemma). Let x be a point in X 0 such that ω(x) is disjoint with the subset Y x of X 0 . Then there exist N x = {(E n,x , O n,x , U n,x )} n≥1 , δ x > 0, and C x ≥ 1, such that

(1) O n,x , U n,x are open disks with x ∈ O n,x ⊆ U n,x , (2) x / ∈ E n,x ⊆ O n,x and E n,x is contained in the Fatou set, (3) Shape (E n,x , ξ) ≤ C x , for some ξ ∈ E n,x , and diam U n,x ≤ C x • diam E n,x , (4) diam U n,x → 0 as n → ∞, (5) If B -k (y, δ x ) T ∂U n,x = ∅, then B -k (y, δ x ) T O n,x = ∅ for any y ∈ Y x , k ≥ 1 and components B -k (y, δ x ) of f -k B(y, δ x ), (6) E n,x T f -k B(y, δ x ) = ∅ for any y ∈ Y x and k ≥ 1.
Proof. If x ∈ X 0 . Then {f n (x)} either locally uniformly converges to a periodic cycle or eventually conjugates to a irrational rotation. Thus there exists a neighborhood B(x, δ)

in Fatou set with δ ≤ δ 0 such that Orb(f B(x, δ))

T Y x = ∅. Let δ x := 1 2 • dist (Orb(f B(x, δ)), Y x ).
Then B(x, δ) T f -k B(y, δ x ) = ∅ for any y ∈ Y x and k ≥ 1. Indeed, if not, suppose

ζ ∈ B(x, δ) T f -k B(y, δ x ). We have f k (ζ) ∈ Orb(f B(x, δ)) T B(y, δ x ), a contradiction. Set O n,x := U n,x := B(x, δ/2 n ), E n,x := B(ξ n , δ/(3 • 2 n )) ⊆ O n,x with x / ∈ E n,x
and C x := 3. Then N x , δ x , C x satisfy the conditions (1)-( 6).

If x ∈ X 0 . Denote by x n := f n (x) and

0 := 1 2 • dist (Orb(f (x)), Y x ).
Applying Lemma 4.7 by setting = 0 , we obtain δ and η 1 . Let B -n (x n , δ) be the component of f -n B(x n , δ) containing x. Then we have

1 := inf n≥1,1≤k≤n {dist (f k B -n (x n , δ), Y x )} > 0. (4.21)
Every ball B(z, δ/5) with z ∈ Orb(x) intersects the Fatou set. Thus there exists a sufficiently small ball E z ⊆ B(z, δ/5) such that E z is contained in the Fatou set and Orb(E z )

T Y x = ∅. Let {B(z, δ/5)} z∈Σ be a finite open covering of Orb(x). Using Lemma 4.7 again by setting

= min { 1 , min x∈Σ dist (Orb(E z ), Y x ), δ/2}, (4.22)
we obtain δ x and η 2 . Let η := max {η 1 , η 2 }. Now we set U n,x := B -n (x n , δ) and O n,x := B -n (x n , δ/2), both of which contain x. Since there must exist ξ ∈ Σ such that

E ξ ⊆ B(ξ, δ/5) ⊆ B(x n , δ/2), we can set E n,x to be a component of f -n E ξ in O n,x .
We are left to check that N x := {(E n,x , O n,x , U n,x )} n≥1 is as required. For (3), by Lemma 4.7 (6),

diam U n,x diam E n,x ≤ C 2 (η, m 0 ) 2δ diam E ξ ≤ C 2 (η, m 0 ) 2δ min z∈Σ {diam E z } =: C x We claim that for any y ∈ Y x and k ≥ 1, if U n,x T f -k B(y, δ x ) = ∅, then k ≥ n + 1.
Indeed, if not, we have dist (f k U n,x , {y}) < δ x ≤ 1 , this contradicts (4.21). Thus, for (5), we have

f n B -k (y, δ x ) = B -k (y, δ x ) with k := k -n ≥ 1 and B -k (y, δ x ) T ∂B(x n , δ) = ∅. Since diam B -k (y, δ x ) < δ/2 by (4.22). We have B -k (y, δ x ) T B(x n , δ/2) = ∅. It follows that B -k (y, δ x ) T O n,x = ∅.
For (6), since Orb(E n,x ) T B(y, δ x ) = ∅ for any y ∈ Y x by (4.22) and the claim. It follows evidently.

Pullback disk systems

For

n ≥ 1, let X n := f -n (X 0 ) \ S 0≤k≤n-1 X k and X := S n≥0 X n . Let n(x) ≥ 0 be the integer such that x ∈ X n(x) .
Let {U x } x∈X 0 be a collection of pairwise disjoint open disks with x ∈ U x such that, for any y ∈ X with f n(y) (y) = x, the component U y of f -n(y) (U x ), which contains y, is an open disk. Then we say that {U x } x∈X is a pullback disk system of f induced by {U x } x∈X 0 .

Let {U x } x∈X be a pullback disk systems. For each x ∈ X with y = f n(x) (x), define

χ x := χ y • f n(x) : U x → D x → 0, and U x (r) the component of χ -1 x (D r ) containing x, where χ y : U y → D with y → 0 is a conformal map. It is easy to check that (1) f (U x (r)) = U f (x) (r) for n(x) ≥ 1, r ∈ (0, 1), (2) either U x T U y = ∅ or U x = U y for distinct x, y with n(x) = n(y) ≥ 0, (3) if U x (r) T S = ∅ resp. ∂U x (r) T S = ∅, then U y (r) T f n (S) = ∅ resp. ∂U y (r) T f n(x) (S) = ∅
, where y = f n(x) (x), for any x ∈ X and set S ⊆ C.

The construction of

{(U x , O x , E x , U x (r x ), U x (r x ))} x∈X 0 in W .
Let W be an open set on C such that X ⊆ W . Let m be a given positive number. Firstly we define a partial order on X 0 . For arbitrary x, x ∈ X 0 x ≺ x ⇐⇒ either x = x or x ∈ ω(x ).

One can check that this order satisfies transitivity property, because of the fact that if

x ∈ ω(x ) then ω(x) ⊆ ω(x ).
Let L 1 be the set of all maximal elements in X 0 . Inductively, L k , k ≥ 2, is the set of all maximal elements in X 0 \ S 1≤i≤k-1 L i . Then

X 0 = L 1 [ L 2 [ • • • [ L N . It is not difficult to check that x / ∈ ω(x ) for any x ∈ L k , x ∈ L k with 1 ≤ k ≤ k ≤ N . Applying Lemma 4.7 by setting = 1 2 • min {δ 0 , dist (∂W, X)},
we obtain δ 0 ≤ , η. For each x in X 0 , let Y x be the collection of points y in X 0 such that y / ∈ ω(x). Let N x = {(E k,x , O k,x , U k,x )} k≥0 , δ x > 0 and C x satisfying the conditions (1)-(6) in Lemma 4.8. Set

δ 0 := min x∈X 0 {δ x } > 0, C 0 := max x∈X 0 {C x } ≥ 1.
(4.23)

Let A x be the forward orbits of points y ∈ X 0 with x / ∈ ω(y). Set can choose

(E x , O x , U x ) ∈ N x such that diam U -n x ≤ min {δ k-1 , δ k-1 , δ k-1 }. (4.25)
where U -n

x takes over all components of f -n U x , n ≥ 0. Let χ x : U x → D be a conformal map with χ x (x) = 0. We denote by U x (r) := χ -1

x D r . Since x / ∈ E x , we can choose r x sufficiently small such that U

x (r x ) ⊆ O x and U x (r x ) T E x = ∅, moreover, mod O x \ U x (r x ) ≥ η • m (4.26) For any components U -n y of f -n U y , y ∈ L 1 S • • • S L k , n ≥ 1, since it can not contain
x by (4.24), there exists r x < r x independent of y, n such that if U -n

y T ∂U x (r x ) = ∅ then U -n y ⊆ U x (r x ) by the Shrinking Lemma 4.7 (3). Set δ k := min x∈L 1 ∪•••∪L k {dist (E x , J f ), dist (E x , x)}, δ k := min x∈L 1 ∪•••∪L k {dist (O x , ∂U x )}, (4.27)
and

δ k := min x∈L 1 ∪•••∪L k {dist (x, ∂U x (r x )), dist (U x (r x ), ∂U x (r x )}. (4.28)
4.6.6 Properties of the desired pullback disk system.

Let U := {U x } x∈X be the pullback disk system in W induced by {U x } x∈X 0 as constructed above. Let X be the collection of elements x in X such that there exists some y ∈ X with y ∈ U x and n(y) ≤ n(x) and X := X \ X .

Consider the sets U := {U x } x∈X and U := {U x } x∈X .

(1) For any U x in U , it does not contain points x in X of lower level, that is, x /

∈ U x if n(x ) ≤ n(x). (2) U is a forward invariant set, that is, f (U x ) = U f (x) ∈ U if U x ∈ U and n(x) ≥ 1. While U is a backward invariant set, that is, U x ∈ U if U f (x) ∈ U .
(3) For U x ∈ U , critical points of the covering mapping

f x := f n(x) : U x → U y
can only be the point x, where y := f x (x). Thus we can define

O x := f -1
x (O y ), r x := r y , r x := r y (4.29)

and E x any one of the component f -1

x (E y ). (4) For any x ∈ X , set

Z x (r) := {y ∈ X , U y \ ∂U x (r) = ∅ and n(y) ≥ n(x)}, r ∈ (0, 1].
Then the followings hold by (4.25), (4.27), (4.28), subsection 4.6.4(1)(3), and the choice of U

x (r x ), [ y∈Zx(r x ) U y ⊆ U x (r x ) ⊆ O x ⊆ U x \ [ y∈Zx(1) U y .
(4.30) and

U y \ E x = ∅ if y ∈ X \ {x} and n(y) ≥ n(x).
(4.31)

(5) For any x ∈ X , there exists ξ ∈ X with n(ξ) ≤ n(x) such that U x ⊆ U ξ (r ξ ). Indeed, there exists k such that f k U x ∈ U and f k+1 U x ∈ U . Then f k U x contains a point y in X 0 . By (4.24) and (4.28), we have

f n(x) (x) ∈ ω(y) and f k U x ⊆ U y (r y ). So U x ⊆ U y 1 (r y 1 ) with f n(y 1 ) (y 1 ) = y and n(y 1 ) < n(x). If y 1 / ∈ X , continue the same process to U y 1 , U y 2 , • • • . Since n(x) > n(y 1 ) > n(y 2 ) > • • • ≥ 0. After finite steps, we have ξ := y k ∈ X . Thus it follows that, for any n ≥ 0, [ x∈X ,n(x)≤n U x ⊆ [ x∈X ,n(x)≤n U x (r x ).
(4.32)

4.6.7 From pullback disk system to nested disk systems.

Let U be the pullback disk system of X as mentioned above. Given n ≥ 0, for

n(x) = n with x ∈ X , set D x := U x . Obviously O x ⊆ D x ⊆ U x . Inductively, for k = n -1, n -2, • • • , 0 and n(x) = k with x ∈ X , let b k (x) := {y ∈ X , k + 1 ≤ n(y) ≤ n and D y \ ∂U x = ∅}.
By (4.26) (4.30) and (4.31), there exists a Jordan domain D x , which is the component of

U x \ S y∈b k (x) U y containing x, such that the following holds x ∈ U x (r x ) ⊆ U x (r x ) ⊆ O x ⊆ D x ⊆ U x . (4.33) Moreover, mod D x \ U x (r x ) ≥ m and E x ⊆ D x \ U x (r x ). (4.34)
By the construction, it is not difficult to check that {D x } n(x)≤n,x∈X is a nested disk system. In summary, we have the following proposition. By (4.31), (4.33) and (4.34), we have

E x ⊆ D x and E x \ V x = ∅.
Consider the map f n(x) : U x → U y , where y := f n(x) (x). The Lemma 4.7(5)(6), Lemma 4.8(3) and equations (4.23) (4.33) give Let ξ h be in W such that |h(ξ h )| = max ξ∈W dist e (h(ξ), 0). Then for any ξ ∈ W , we have

Shape (E x , ξ) ≤ C 1 • Shape (E y , f n(x) (ξ)) ≤ C 1 C 0 and diam D x ≤ diam U x ≤ C 2 C y • diam E x ≤ C 2 C 0 • diam E x .

Thus we have

|h(ξ h )| |h(ξ)| ≤ |h(ξ)| + diam e h(W ) |h(ξ)| ≤ 1 + diam e h(W ) dist e (∂h(W ), ∂h(W )) ≤ 1 + C 4 (4.38)
Now we estimate, using equations (4.36), (4.37) and (4.38),

Area (ρ * , h(E x )) = ZZ h(Ex) 1 4π 2 |ξ| 2 dxdy = ZZ Ex |h (ξ)| 2 4π 2 |h(ξ)| 2 dxdy ≥ 1 C 2 3 ZZ Ex |h (ξ h )| 2 4π 2 |h(ξ h )| 2 dxdy ≥ 1 CC 2 3 ZZ Dx |h (ξ h )| 2 4π 2 |h(ξ h )| 2 dxdy ≥ 1 CC 4 3 (1 + C 4 ) ZZ Dx |h (ξ)| 2 4π 2 |h(ξ)| 2 dxdy =: (1 -λ) Area (ρ * , h(D x )) Therefore, Area (ρ * , h(V x )) ≤ Area (ρ * , h(D x )) -Area (ρ * , h(E x )) ≤ λ • Area (ρ * , h(D x )).
To prove (4.35), let {B x } x∈X be the pullback disk system by choosing δ := min x∈X 0 {r x }.

Then it follows by (4.29) and (4.32).

Univalent maps off a pullback disk system

Theorem 4.5 (Controlling distortion of univalent map). Let V be a Jordan domain on C such that X ⊆ C \ V . Let z 1 , z 2 , z 3 be three distinct points in V . Then for any > 0, there exists δ > 0 such that sup

z∈V {dist (φ(z), z)} ≤ , for any univalent map φ : C \ S n(x)≤n B x → C fixing z 1 , z 2 , z 3 , where {B x } x∈X is the pullback disk system in C \ V induced by {B(x, δ)} x∈X 0 . Proof. Let W be Jordan domain on C such that W ⊆ W := C \ V and X ⊆ W . By Proposition (4.1), there exist m-nested system {(D x , D x , D x )} x∈Yn , λ-scattered {D x } x∈Yn in W and pullback disk system {B x } x∈X induced by disks {B(x, δ)} x∈X 0 .
Let α be the Möbius transformation sending 0, 1, ∞ to z 1 , z 2 , z 3 respectively. Suppose

D(1, r 0 ) ⊆ α -1 (V ). Denote by ψ := α -1 • φ • α. Applying Theorem 4.
∈ α -1 (V ), dist (ψ(z), z) ≤ C(r 0 ) • D(ψ, α -1 (V )) = C(r 0 ) • D(φ, V ) ≤ C(r 0 )C(m, λ) • Area p (D, W ) ≤ C(r 0 )C(m, λ) • Area p (W , W ).
By Proposition 4.1 and Theorem 4.4, we may assume m → ∞ and thus sup

z∈α -1 (V ) dist (ψ(z), z) → 0.
Since α is uniformly continuous on C equipped with spherical metric. Therefore, the theorem follows.

The existence and uniqueness for shift locus

In this section, we discuss the existence and uniqueness of shift locus in S d for a given critical portrait and escaping rate. The main tool is quasiconformal surgery.

Theorem 4.6. Let Θ = {Θ 1 , • • • , Θ n } be arbitrary critical portrait and r > 0, then there exists an unique f in S d (r) such that Π(f ) = Θ.

Proof. Based on quasiconformal surgery, we shall prove the theorem by the following four steps.

Step I. Construct a topological polynomial F realizing Θ.

Start with the closed disk D, mark all of the points e 2πiθ with θ ∈ Θ i . Let z i be the center of gravity of the marked points, and join each of these points to z i by a straight line L θ . L Θ i := S θ∈Θ i L θ is a closed subset in D. Set r 0 := e dr and > 0 sufficiently small. Define the quotient map π :

C → C pinching L Θ i into a point z i . See figure 4.3. π(z) := 8 > > < > > : z if z ∈ C \ D r 0 -, z i if z ∈ L Θ i for 1 ≤ i ≤ n, homeomorphism otherwise. Then the interior of π(D) is a disjoint union of d topological disk D 1 , • • • , D d where D i T D j contains at most one point. For any θ ∈ Θ j , R (θ) := π({t = re 2πiθ , t > 1}) is a ray landing at z j . Then S 1≤j≤n {R (θ), θ ∈ Θ j } cuts D r 0 -into D 1 , • • • , D d pieces. Set D i Jordan domains such that D i ⊆ D i ⊆ D i .
For convenience, we may assume the above R (θ), ∂D i , ∂D i and ∂D i are C ∞ smooth except at the finite points z i .

Since D j is topological disk, there exists a conformal map F j :

D j → D r 0 -. Let v j = r 0 e 2πiσ d (Θ j ) , Z = {z 1 , • • • , z n } and Z j = D j T Z. We set F 0 be a differentiable branched z 1 z 2 v 2 v 1 D D r 0 - D r 0 D (r 0 -) d D r d 0 H π z 1 z 2 v 2 v 1 D j D j D j
• F 0 | ∂D r 0 -: z → z d . • F 0 = F j on ∂D j , for 1 ≤ j ≤ d. • F 0 (R (θ) T D r 0 -) = R v j := {te 2πiσ d (θ) , r 0 < t < (r 0 -) d } for θ ∈ Θ j . • F 0 | D j \D j
: is a diffeomorphism and its image is the set

D (r 0 -) d \ D r 0 -minus several R v i with z i ∈ ∂D i .
• F 0 sends the critical points z j to v j locally holomorphic. Now we can define the topological polynomial F as following

F (z) := 8 > > < > > : z d if z ∈ C \ D r 0 -, F i (z) if z ∈ D i for 1 ≤ i ≤ d, F 0 (z) otherwise.
Step II. Pulling back complex structure to construct polynomial f .

We define a new complex structure µ on C which is preserved by F as follows. Let µ 0 denote the standard complex structure on C. Set

µ(z) := 8 > > < > > : µ 0 (z) if z ∈ C \ D r 0 -, (F n+1 ) * (µ 0 )(z) if z ∈ S n≥0 F -n (H), µ 0 (z) otherwise. Since F -n (H) T F -m (H) = ∅, n = m, µ is well-defined. µ is F -invariant. Actually, for any z ∈ F -n (H), we have F (z) ∈ F -n+1 (H) and µ(z) = (F n+1 ) * (µ 0 )(z) = (F n • F ) * (µ 0 )(z) = F * • (F n ) * (µ 0 )(z) = F * µ(z
i : C \ f -(n+1) i P f i → C \ f -n i P f i is an unbranched covering.
Since h n conjugates between f 1 and f 2 on f -n i P f i , we may choose h n+1 to be the unique lift such that, after extended to f

-(n+1) 1 P f 1 , it satisfies h n+1 = h n on f -n P f 1 [Ha02, Proposition 1.30]. Since f i is conformal on C \ X f i (d -n r). We can extend h n+1 to C \ X f 1 (d -n r), such that h n • f 1 = f 2 • h n+1 on C. Then h n+1 is holomorphic on X f 1 (d -(n+1) r) and is a K-quasiconformal map.
Therefore there is a subsequence of {h n } which converges uniformly to a limit quasiconformal map h of C. Then

f 2 • h = h • f 1 and h is holomorphic in S n≥0 X f 1 (d -n r) which is the Fatou set. Since the measure of Julia set of f is zero, h is holomorphic on C. Since h fixes ∞, we have h = az + b. Note that a = h (∞) = Ψ f 1 (∞) • (Ψ -1 f 2 ) (∞) = 1 and b = 0 because f i are centered. Thus h = id and so f 1 = f 2 .

Proof of the results

In this section, we focus on a simple case, that is, all critical points escaping at the same rate (See theorem 4.7). During the proof we apply theorem 4.5 at the situation X 0 = ∅ in Assumption 4.6.1. For general cases, the technique is almost the same.

Let us start with a simple lemma.

Lemma 4.9. Let T be a topology space and S be a subset of T . Let h i : T → T , i ∈ {1, 2}, be maps such that the restrictions of h i on S have the same image, i.e., h 1 (S) = h 2 (S), and

h 1 | T \S = h 2 | T \S , then for any n ≥ 0, [ 0≤k≤n h -k 1 (S) = [ 0≤k≤n h -k 2 (S).
Proof. If n = 0, it follows obviously. We suppose

S 0≤k≤n h -k 1 (S) = S 0≤k≤n h -k 2 (S) by induction. Firstly, S 0≤k≤n+1 h -k 1 (S) ⊆ S 0≤k≤n+1 h -k 2 (S). For any x ∈ S 0≤k≤n+1 h -k 1 (S), h 1 (x) ∈ S 0≤k≤n h -k 1 (S) = S 0≤k≤n h -k 2 (S). If x ∈ T \ S then x ∈ h -1 2 • h 1 (x) and so x ∈ S 0≤k≤n+1 h -k 2 (S). If x ∈ S, then it is obviously true. Secondly, S 0≤k≤n+1 h -k 2 (S) ⊆ S 0≤k≤n+1 h -k 1 (S)
. It follows by the same arguments as above. The lemma is proved.

Let S d := S 0<r<∞ S d (r) and f Θ,r be the unique polynomial in theorem 4.6. We have the following theorem.

f r F r+δ f r+δ f δ,n b f δ

Surgery

Step I c-equi.via {φ δ,n } n≥0

Step II & Step III

Step IV Th.Alg.{η δ,n } n≥0

Step V Theorem 4.7. For any critical portrait Θ,

ψ δ := α δ z + β δ Step V converging ψ δ,n := η δ,n • φ -1 δ,n
R Θ : (0, ∞) → S d r → f Θ,r
is a simple arc in S d .

Proof. Given 0 < r < ∞ and critical portrait Θ, by Theorem 4.6, we only need to show that R Θ is continuous at r. Given δ 0 > 0 sufficiently small, for any |δ| ≤ δ 0 , let

f r (z) = z d + a d-2 z d-2 + • • • + a 1 z + a 0 and f r+δ (z) = z d + b δ,d-2 z d-2 + • • • + b δ,1 z + b δ,0
such that f r ∈ S d (r) and f r+δ ∈ S d (r + δ) with critical portrait Θ.

Step I. Surgery to construct topological polynomial F r+δ .

Let W δ,v i be a Jordan domain containing critical value v i := Φ -1 (e dr e 2πiθv i ) such that

• v δ,i := Φ -1 (e d(r+δ) e 2πiθv i ) ∈ W δ,v i . • W δ,v i T W δ,v i = ∅, for distinct critical values v i , v i . • diam(W δ,v i ) → 0 as δ → 0. Then every components of f -1 r (W δ,v i ) is Jordan domain. Let W δ,c i be one of them containing critical point c i . Let ζ δ : C → C be a quasi-conformal map such that ζ δ is identity outside all S j W δ,v j and sends v i to v δ,i on each W δ,v i . See figure 4.5. Set W δ := S c∈Crit(f ) W δ,c
and define a quasi-regular map

F r+δ (z) = 8 > < > : f r (z), if z ∈ C \ W δ ζ δ • f r (z), if z ∈ W δ .
Then it satisfies the following,

(1) F r+δ = f r on C \ W δ .

(2) Crit(F r+δ ) = Crit(f r ) and all the them escape to infinity at the same speed.

( Step II. F r+δ is c-equivalent to f r+δ via (φ δ,0 , φ δ,1 ).

) S i≥0 F -i r+δ (W δ ) = S i≥0 f -i r (W δ ) by Lemma 4.9. v 1 v δ,v 1 W δ,v 1 v 2 v δ,v 2 W δ,v 2 c 2 W δ,c 2 c 1 W δ,c 1 3
Let U δ := X fr (d(r + δ)) resp. U δ := X f r+δ (d(r + δ)). The critical values F (Crit(F )) resp. f r+δ are in the boundary of U δ resp. U δ . Set φ δ,0 := 8 > < > : Ψ -1 f r+δ • Ψ fr (z) if z ∈ U δ , g δ (z) if z ∈ C \ U δ , where g δ : C \ U δ → C \ U δ is a quasi-conformal map which coincides with Ψ -1 f r+δ • Ψ fr on C to ψ δ .
Applying the same method to sequences {1/ψ δ,n (1/z)} and {ψ -1 δ,n }, we know that {ψ δ,n } and {ψ -1 δ,n } uniformly converge on C equipped with spherical metric. It follows that the thurston sequence {f δ,n } uniformly converges to a polynomial b f δ on C, that is, for any > 0, there exists

N := N (δ, ) such that sup n≥N,z∈C dist (f δ,n (z), b f δ (z)) ≤ (4.42)
Moreover, b f δ and f r+δ conjugate by the affine map ψ δ by (4.40

), that is, b f δ • ψ δ (z) = ψ δ • f r+δ . (4.43) We denote by b f δ (z) = a δ,d z d + a δ,d-1 z d-1 + • • • + a δ,1 z + a δ,0 .
Step VI. Estimate the distance between f r and b f δ . Let V r := X fr (C 0 ), where C 0 is sufficient large number such that V r ⊆ V δ , for any |δ| ≤ δ 0 , and z i ∈ V r . By Step I(1) and Step IV(1), we know that η δ,n is holomorphic on V r . The theorem 4.5 guarantees the following crucial distortion, sup |δ|≤τ,n≥0,z∈Vr dist(η δ,n (z), z) → 0 as τ → 0, (4.44)

Thus there exists a neighborhood V r of ∞ such that V r ⊆ η δ,n (V r ), for any |δ| ≤ δ 0 and n ≥ 0, and sup

|δ|≤τ,n≥0,z∈V r dist (η -1 δ,n (z), z) → 0 as τ → 0, (4.45) Since dist (f δ,n (z), f r (z)) = dist (η δ,n • f r • η -1 δ,n+1 (z), f r (z)) ≤ dist (η δ,n • f r • η -1 δ,n+1 (z), f r • η -1 δ,n+1 (z)) + dist (f r • η -1 δ,n+1 (z), f r (z)).
Combining with (4.44) and (4.45), we have for any > 0, there exist τ := τ ( ) such that

f δ (z), f r (z)) ≤ sup |δ|≤τ,z∈V r dist ( b f δ (z), f δ,N (δ) (z)) + dist (f δ,N (δ) (z), f r (z)) ≤ 2 , (4.47)
where N (δ) = N (δ, ) is defined in (4.42). By Lemma 4.4, { b f δ } uniformly converges to f r on C. Therefore a δ,d → 1, a δ,d-1 → 0 as δ → 0.

(4.48)

Step VII. Estimate the distance between b f δ and f r+δ . By Step VI and (4.44), the sequence {η δ,n | Vr } n≥0 is holomorphic and locally uniformly bounded on V r \ {∞}. Thus it is a normal family and the entire sequence converges to a holomorphic map η δ | Vr , i.e., for any > 0, there exists N = N (δ, ) such that for n ≥ N sup n≥N,z∈Vr dist (η δ (z), η δ,n (z)) ≤ .

(4.49)

The new sequence {η δ | Vr } converges to id on V r as well. Because, by (4.44) and (4.49), for any > 0, there exists τ = τ ( ) such that

sup |δ|≤τ,z∈Vr dist (η δ (z), z) ≤ sup |δ|≤τ,z∈Vr dist (η δ (z), η δ,N (δ) (z)) + dist (η δ,N (δ) (z), z) ≤ 2 (4.50)
where N (δ) = N (δ, ) is defined in (4.49).

Note that φ δ,n (∞) = 1 in Step II. Then (4.39), (4.41), (4.49), (4.50), Weierstrass convergence theorem and the chain rule give This completes the proof.

α δ = ψ δ (∞) = lim n→∞ ψ δ,n (∞) = lim n→∞ η δ,n (∞) • (φ -1 δ,n ) (∞) = lim n→∞ η δ,n (∞) = η δ (∞) → 1 as δ → 0. ( 4
β δ = a δ,d-1 • a δ d • a δ,d →
Proof of Theorem 4.1. By theorem 4.6, the map P is well-defined and one-to-one. The continuity at (Θ, r) follows nearly the same as Theorem 4.7. We only need to change slightly in Step I. The surgery ξ δ sends v i to Ψ -1 (e d(r+δ) e 2πiθ v i ) ∈ W δ,v i , as Θ is close enough to Θ and |δ| is sufficiently small. We omit the proof.

Proof of Theorem 4.2. We know that J f is locally connected [START_REF] Yin | On the Julia set of semi-hyperbolic rational maps[END_REF]. If R Θ (t) lands at f , Theorem 1 in [Ki05] implies that the external rays of f with arguments in Θ i land at a common point c i which must be critical. By the unlinked property of critical portrait, we have c i = c j if i = j. Thus the local degree of f at critical point c i is #Θ i . Therefore, Θ is a critical portrait of f . For the sufficiency, we adopt exactly the same method as Theorem 4.7. Indeed, in dynamic plane R(θ i ) with θ i := σ d (Θ i ) lands at critical value v i . We can construct F δ , δ > 0, exactly the same in Step I of Theorem 4.7. Both of the distance dist (f, F δ ) and dist (F δ , f δ ) converge to 0 as δ → 0. Thus the parameter ray R Θ (t) lands at f . We omit the details.

Chapter 5

On the dynamics of a family of generated renormalization transformations

Introduction

The statistical mechanical models on hierarchical lattices have attracted many interests recently since they exhibit a deep connection between their limiting sets of the zeros of the partition functions and the Julia sets of rational maps in complex dynamics [BL,DSI, Qi, QL, QYG]. The well-known Yang-Lee theorem in statistical mechanics shows that the zeros of the partition function is dense in a line for many magnetic materials in a complex magnetic field plane. This means that the complex singularities of the free energy lie on this line, where the free energy is the logarithm of the partition function [YL]. By the works of Fisher and others [Fi], it was generally believed that the zeros of the partition function condense to some simple curve.

Until 1983, Derrida et al. showed that the zeros of the partition function condense to the Julia set of the renormalization transformation of so-called standard hierarchical lattices [DSI]. They proved that the singularities of the free energy lie on the Julia set of the rational map

z → z 2 + λ -1 2z + λ -2 2 .
(5.1)

This means that the distribution of the singularities of the free energy is not as simple as one desired. Henceforth, a lot of works related on the Julia sets of this renormalization transformation appeared [AY, BL, Ga, HL, Os, Qi, QL, QYG, WQYQG]. For the ideas formulated in renormalization transformation in statistical mechanics, see [Wi]. The proof of the connectivity of M 2 in [WQYQG] is based on constructing Riemann mapping from the capture domain to the unit disk D, which is tediously long. Here, we give a proof of Theorem 5.2 by using the methods of Teichmüller theory of the rational maps which was developed in [McS]. The proof is largely simplified and there are lots of additional results. For example, we show that the Julia set of U dλ is a quasicircle if and only if λ lies in the unbounded capture domain H 0 (Proposition 5.2) and each bounded capture domain contains exactly one center (Theorem 5.10).

If λ is large enough, then the Julia set of U dλ is a quasicircle (see Proposition 5.2). Hu and Lin observed that these circles becomes more and more "circular" as λ tends to ∞ in the case of d = 2 [HL]. In [Ga], Gao proved the Hausdorff dimension of the Julia set of U 2nλ tends to 1 for every n ≥ 2, which gave an affirmative answer of Hu and Lin proposed in 1989. In this chapter, we consider the asymptotic formula of the Hausdorff dimension of the Julia set J dλ of U dλ as the parameter λ tends to ∞.

Theorem 5.3. Let d ≥ 2. For large λ such that J dλ is a quasicircle, the Hausdorff dimension of J dλ is given by dim H (J dλ ) = 1 + 1 4 log d |λ| -2 d+1 + O(λ -3 d+1 ).

(5.4) Theorem 5.3 is a generation of [Os] in which the asymptotic formula of the Hausdorff dimension of J 2λ was calculated.

This chapter is organized as follows. In Section 5.2, we analyse the location of the critical points of U dλ and show that the Julia set of U dλ is always connected and prove Theorem 5.1. In section 5.4, we show that the parameter plane of U dλ can be decomposed into the non-escaping locus M d union infinitely many capture domains. In section 5.5, we give a complete classification of the quasiconformal conjugacy classes of U dλ . In section 5.6, we show that each bounded capture domain is simply connected and the unique unbounded capture domain is homeomorphic to the punctured disk and prove Theorem 5.2. We will prove the asymptotic formula (5.4) of Theorem 5.3 in section 5.7 but leave the complicated calculations to the last section as an appendix.

The location of critical points and the connected

Julia sets

Firstly, we give a splitting principle for U dλ . This principle is not exist if one considers U mnλ with m = n. This is the reason why we set m = n in this paper. For every λ ∈ C * , it is straightforward to verify that U dλ = T dλ • T dλ , where (5.8)

Since T dλ (1) = ∞, T dλ (∞) = 1 and 1, ∞ are both critical points of U dλ , it means that there exist two fixed immediate superattracting basins A dλ (1) and A dλ (∞) of U dλ with centers 1 and ∞ respectively. Under the iteration of T dλ , we have the following forward orbits:

ξ k → 1 → ∞ → 1 → ∞ → • • • and ω k → 1 -λ → 0 → (1 -λ) d → • • •
(5.9) for every 0 ≤ k ≤ d -1. Since the dynamical behaviors are determined by the critical forward orbits essentially, we only need to focus on the free critical orbit of 1 -λ (or equivalently, the forward orbit of 0) under the iteration of T dλ or U dλ . This is the reason why we define the non-escaping locus M d as in (5.3).

Lemma 5.1. Let U and V be two domains on C and assume that V is simply connected.

If f : U → V is a branched covering with only one critical value in V (counted without multiplicity), then U is also simply connected.

Proof. Let v be the unique critical value lying in V . Consider the unramified covering f : U \ f -1 (v) → V \ {v}. Since V \ {v} is an annulus with Euler characteristic 0, it follows that U \ f -1 (v) is also an annulus by the Riemann-Hurwitz formula. This means that U is a topological disk, which is simply connected as desired.

In order to prove a rational map has connected Julia set, one often needs to exclude the existence of Herman ring. The following lemma was proved in [Ya]. The proof of Lemma 5.2 relies on the quasiconformal surgery and the arguments are divided into two cases: Herman ring with period 1 and period at least two. However, the prove idea is different from [Mi2, Appendix A].

Theorem 5.4. The Julia set of T dλ is always connected for every d ≥ 2 and λ ∈ C * .

Proof. The proof idea is more or less similar to the case of quadratic rational maps in [Mi1, Lemma 8.2]. Note that the Julia set is connected if and only if each Fatou component is simply connected. By Sullivan's classification of the periodic Fatou components, every periodic Fatou component of T dλ is either a Siegel disk, a Herman ring, or an immediate basin for some attracting or parabolic point. By Lemma 5.2, it is known T dλ has no Herman ring.

By [START_REF] Milnor | Geometry and dynamics of quadratic rational maps, with an appendix[END_REF]Lemma 8.1], we know that if all the critical values of a rational map are contained in a single component of the Fatou set, then the Julia set is totally disconnected. However, the Julia set J dλ cannot be totally disconnected since T dλ has a superattracting periodic orbit of period 2. Therefore, the critical points 1 and 1 -λ lie in different Fatou components and each Fatou component of T dλ contains at most one critical value (∞ or 0 by (5.9)). Now we prove each Fatou component of T dλ is simply connected. Firstly, we assume that every periodic Fatou component of T dλ is simply connected. Note that the periodic orbit 1 ↔ ∞ is superattracting. There leaves only one critical point 1 -λ needing to consider. According to Lemma 5.1, the preimage of a simply connected region under a branched covering with only one critical value is again simply connected. This means every Fatou component of T dλ is simply connected by induction.

Then suppose that there exists a periodic Fatou component U of T dλ which is not simply connected and the period is p ≥ 1. This means that U is an attracting basin or a parabolic basin since T dλ has no Herman ring. Let z 0 be the attracting periodic point in U or parabolic periodic point on ∂U . We use V to denote a simply connected neighborhood or a simply connected petal of z 0 such that T •p dλ (V ) ⊂ V according to U is attracting or parabolic. Let V k be the component of T -kp dλ (V ) containing V . Then U = S k≥0 V k and V k+1 → T dλ (V k+1 ) → • • • → T •p-1 dλ (V k+1 ) → V k is a successive branched covering under T dλ with at most one critical value in each codomain since each Fatou component of T dλ contains at most one critical value. Suppose V k 0 is simply connected (at least k 0 = 0 is satisfied). By Lemma 5.1, we know that T •p-1 dλ (V k 0 +1 ), • • • , T dλ (V k 0 +1 ), V k 0 +1 are all simply connected since V k 0 is also. Inductively, it follows that each V k is simply connected and hence U is also simply connected. This contradicts the assumption that U is not simply connected.

Therefore, in any case, the Julia set of T dλ is always connected. This ends the proofs of Theorems 5.4 and 5.1.

The Julia set cannot be a Sierpińsk carpet

In this section, we will prove that if the parameter λ lies on the real axis, then the Julia set of U dλ can never be a Sierpińsk carpet by showing there always exist two Fatou components of U dλ whose boundaries are intersecting to each other.

Lemma 5.3. For every d ≥ 2 and λ ∈ R, there exist two Fatou components

V 1 , V 2 of U dλ such that V 1 ∩ V 2 = ∅.
Proof. If λ = 0, then U dλ degenerates to a parabolic polynomial U d0 (z) = ( z+d-1 d ) d whose Julia set J d0 is a Jordan curve. Let V 1 = A dλ (1) and V 2 = A dλ (∞) be the immediate superattracting basins of 1 and ∞ respectively. We have V 1 ∩ V 2 = J d0 = ∅.

In the following, we assume that λ ∈ R \ {0}. The dynamics of U dλ will be restricted on the real axis and the arguments will be divided into several cases. Let x ∈ R, by a direct calculation, we have

U dλ (x) = d 2 λ 2 (x -1) d-1 (x + λ -1) d-1 ((x + λ -1) d + (λ -1)(x -1) d ) d-1 ((x + λ -1) d -(x -1) d ) d+1 .
(5.10)

(1) Let λ > 0. If x ≥ 1, we have x-1 ≥ 0, x+λ-1 > 0, (x+λ-1) d +(λ-1)(x-1) d > 0 and (x + λ -1) d -(x -1) d > 0. This means that U dλ (x) ≥ 0 and U dλ is increasing on [1, +∞). Moreover, U dλ (x) = 0 if and only if x = 1. We claim that there exists at least one fixed point of U dλ lying in (1, +∞). Otherwise, we then have 1 < U dλ (x) < x for every

x > 1 since U dλ (1) = 1 and U dλ (1) = 0. This means that the interval (1, +∞) is contained in the attracting basin of 1, which is a contradiction since ∞ is a superattracting fixed point of U dλ .

Let 1 = x 0 < x 1 < • • • < x n < +∞ be the collection of all the fixed points of U dλ lying in [1, +∞), where n ≥ 1. It is easy to see U dλ (x) > x if x > x n . In particular, we have (x n , +∞) ⊂ A dλ (∞). Note that U dλ (x n ) ≥ 1. If U dλ (x n ) = 1, then x n is a parabolic fixed point of U dλ and A dλ (x n ) contains a small interval on the left of x n , where A dλ (x n ) is the immediate parabolic basin of x n . Let V 1 = A dλ (x n ) and V 2 = A dλ (∞). We have

x n ∈ V 1 ∩ V 2 . If U dλ (x n ) > 1, then
x n is a repelling fixed point of U dλ and x n-1 is an (or parabolic) attracting fixed point of U dλ . Moreover, [x n-1 , x n ) ⊂ A dλ (x n-1 ), where A dλ (x n-1 ) is the immediate attracting (or parabolic) basin of x n-1 . Let V 1 = A dλ (x n-1 ) and V 2 = A dλ (∞). We have

x n ∈ V 1 ∩ V 2 .
(2) Let λ < 0. If 0 ≤ x ≤ 1, then x -1 ≤ 0 and x + λ -1 < 0. If d ≥ 2 is even, then (x + λ -1) d + (λ -1)(x -1) d > 0, (x + λ -1) d -(x -1) d > 0 and U dλ (x) ≥ 0. If d ≥ 2 is odd, then U dλ (x) ≥ 0. This means that U dλ is increasing on [0, 1] for every at least d + 1 preimages under T dλ (counted with multiplicity, d in A dλ (1) and at least one elsewhere), which is impossible. The same argument also shows that ω k ∈ A dλ (1) if and only if ω l ∈ A dλ (1), where 0 ≤ k, l ≤ d -1. Then, A dλ (1) contains critical points {ω 1 , • • • , ω d , 1} of U dλ . This means that A dλ (1) is completely invariant under U dλ .

Since 1 -λ ∈ A dλ (∞), it means that T dλ : A dλ (∞) → A dλ (1) is d to 1. Therefore, ξ k ∈ A dλ (∞) for every 1 ≤ k ≤ d -1 since ξ 0 = ∞ ∈ A dλ (∞) and T dλ (ξ k ) = 1. Moreover, A dλ (∞) contains critical points {ξ 1 , • • • , ξ d , 1 -λ} of U dλ . This means that A dλ (∞) is also completely invariant under U dλ . Therefore, J dλ is a quasicircle since T dλ is hyperbolic and T dλ has exactly two Fatou components. This ends the proof of (5) ⇒ (1).

To finish, we prove (2) ⇒ (4). If ξ k ∈ A dλ (∞) for all 0 ≤ k ≤ d -1, then T dλ :

A dλ (∞) → A dλ (1) is d to 1. This means that 1 -λ ∈ A dλ by Riemann-Hurwitz formula.

The proof is complete.

Lemma 5.5. For every λ ∈ C * , we have 0 ∈ A dλ (∞) and 1 -λ ∈ A dλ (1).

Proof. If 0 ∈ A dλ (∞), then 1 -λ ∈ A dλ (1) by (5.9). Note that 1 lies also in A dλ (1). This means that T dλ has 2d -1 preimages in A dλ (1) for each point in A dλ (∞) by Riemann-Hurwitz formula, which is a contradiction. Moreover, 0 ∈ A dλ (∞) means 1 -λ ∈ A dλ (1) by (5.9).

Since 1 and ∞ are always periodic with period 2 under T dλ , the non-escaping locus Theorem 5.7. Let T dλ 0 , T dλ 1 ∈ R d be two different maps and let ϕ : C → C be a K -quasiconformal homeomorphism which conjugates T dλ 0 to T dλ 1 such that ϕ(λ 0 ) = λ 1 . Then there exists a holomorphic map t → λ t from an open disk D(0, r) (r > 1) into C * which maps 0 to λ 0 and 1 to λ 1 , such that for every t ∈ D(0, r), T dλ 0 is conjugate to T dλt by a K t -quasiconformal mapping ϕ t : C → C. Moreover, K t → 1 as t → 0.

The idea of the proof of Theorem 5.7 is standard in holomorphic dynamics. One can refer [Za,Theorem 5.1] for a proof in the similar situation. As an immediate corollary, we have A holomorphic family of rational maps f λ : Λ × C → C is quasiconformally constant if f λ 1 and f λ 2 are quasiconformally conjugate for any λ 1 and λ 2 in the same component of Λ. We call the family f λ has constant critical orbit relations if any coincidence f •n λ (c 1 ) = f •m λ (c 2 ) between the forward orbits of two critical points c 1 and c 2 of f λ persists under perturbation of λ. The following theorem was proved in [START_REF] Mcmullen | Quasiconformal homeomorphisms and dynamics. II-I. The Teichmüller space of a holomorphic dynamical system[END_REF]Theorem 2.7].

Theorem 5.8 ( [McS]). A holomorphic family f λ of rational maps with constant critical orbit relations is quasiconformally constant.

Proposition 5.2. The Julia set J dλ of T dλ is a quasicircle if and only if λ ∈ H 0 . Moreover, H 0 is unbounded and connected.

A more precise characterization of the structure of H 0 will be given in Theorem 5.10.

Proof. By the definition of H 0 and Lemma 5.4, it follows that if λ ∈ H 0 , then J dλ is a quasicircle. Conversely, if J dλ is a quasicircle, then 1 -λ ∈ A dλ (∞). This means that T dλ and T dλ 0 have the same critical orbit relations, where λ 0 ∈ H 0 . By Theorem 5.8, T dλ and T dλ 0 are quasiconformally conjugate to each other. By Corollary 5.2, it follows that λ ∈ H 0 and H 0 is connected.

To finish, we only need to show that H 0 is unbounded. Let α = λ -1 d+1 and ϕ α (z) = α d (z -1) be a linear transformation. By a straightforward calculation, we have

f α (z) := ϕ α • T dλ • ϕ -1 α = d-1 X i=0 C i d α i z d-i = 1 z d + C 1 d α z d-1 + • • • + C 1 d α d-1 z .
If α = 0 is small enough, then the Julia set of f α is a quasicircle since the Julia set of z → 1/z d is the unit circle. This means that J dλ is a quasicircle if λ is large enough.

By definition, the parameter λ ∈ topologically a strip and exp : U → W 2 \ γ is conformal in the interior of U , whose inverse is denoted by log : W 2 \ γ → U (see Figure 5.3). By the definition, {S 1 , • • • , S d n } is an IFS defined on U for large n since f •n is strictly expanding on W 1 in the Euclidean metric if n is large2 . For the convenient of the argument, we assume that f •n is expanding in the Euclidean metric for all n ≥ 1. The attractor J of {S 1 , • • • , S d n } is a closed set satisfying J = exp(J ). Moreover, J \ {z 1 } is the conformal image of J with two ends removed, where z 1 ∈ J ∩ γ is a fixed point of f . This means that the Hausdorff dimensions of J and J satisfy dim H (J ) = dim H (J).

Let F n | U := F d n i=1 S -1 i | S i (U ) be the lift of f •n under exp. Then each S i (U ) contains exactly one fixed point ζ i ∈ J of F n in its interior for 1 < i < d n and on its boundary for i = 1 and d n .

By Koebe distortion theorem there exist two sequences of numbers 0 < A n ≤ 1 ≤ B n , such that Then, we have

A n |F n (ζ i )| ≤ |S i (x) -S i (y)| |x -y| ≤ B n |F n (ζ i )| , ∀ 1 ≤ i ≤ d n ,
1 B s n,2 n ≤ d n X i=1 1 |F n (ζ i )| s n,2 ≤ d n X i=1 1 |F n (ζ i )| D ≤ d n X i=1 1 |F n (ζ i )| s n,1 = 1 A s n,1 n .
(5.16)

The d n -1 fixed points of f •n in the Julia set J are {z i = exp(ζ i ) : 1 ≤ i < d n }. In particular, z 1 = exp(ζ 1 ) = exp(ζ d n ). Since F n is conformally conjugate to f •n in the interior of each S i (U ), we have F n (ζ i ) = (f •n ) (z i ) for 1 ≤ i < d n . Therefore, by (5.16), we have As the parameter λ tends to ∞, the diameter of the Julia set J dλ of T dλ becomes larger and larger in the Euclidean metric and the shape of J dλ becomes more and more circular (see Figure 5.4). Therefore, one can make a scaling of J dλ (or equivalently, make a conjugate), such the new Julia set converges to the unit circle.

X z∈ Fix(f •n ) 1 |(f •n ) (z)| D = d n -1 X i=1 1 |(f •n ) (z i )| D = d n X i=1 1 |F n (ζ i )| D - 1 |F n (ζ d n )| D → 1 as n → ∞ since lim n→∞ A s n,1 n = lim n→∞ B

Specifically, define

J * dλ = {λ -d d+1 (z -1) : z ∈ J dλ }.

(5.17)

The following Lemma 5.8 has been proved in [START_REF] Qiao | Julia sets and complex singularities of free energies[END_REF]Theorem 4.3] as a special case.

Lemma 5.8. The scaled Julia set J * dλ converges to the unit circle in the Hausdorff topology as λ tends to ∞ and the Hausdorff dimension of J dλ tends to 1 as λ tends to ∞.

Although Lemma 5.8 is significant, however, we want to know further about the asymptotic formula of of the Hausdorff dimension of J dλ as λ tends to ∞. In order to calculate the Hausdorff dimension of J dλ , we do some setting first.

Recall that in Proposition 5.2, α = λ -1 d+1 . Then λα d = α -1 . Let ϕ α (z) = α d (z -1) be the linear transformation as before. We define a new rational map with parameter α as

f α (z) := ϕ α • T dλ • ϕ -1 α = d-1 X i=0 C i d α i z d-i = 1 z d + C 1 d α z d-1 + • • • + C 1 d α d-1 z .
(5.18)

This means that there exists a small ε > 0 such that f α : D ε × C → C is a holomorphic family of hyperbolic rational maps parameterized by D ε , where D ε := {z : |z| < ε}. Note that the Hausdorff dimension is invariant under a conformal isomorphism. This means that we only need to calculate the Hausdorff dimension of the Julia set J α of f α with α ∈ D ε since dim H (J α ) = dim H (J dλ ). We would like to remark that J α = J * dλ . Let E be a subset of C and (Λ, λ 0 ) a connected complex manifold with basepoint λ 0 . A family of maps h λ : E → C is called a holomorphic motion of E parameterized by Λ and with base point λ 0 if: (1) For each λ ∈ Λ, h λ is injective on E; (2) For each z ∈ E, h λ (z) is a holomorphic function of λ ∈ Λ; and (3) h λ 0 is identity on E (see [Ly], [MSS] or [START_REF] Mcmullen | Complex Dynamics and Renormalization[END_REF]Chap. 4]).

Proof of Theorem 5.3. By (5.18), it follows that the Julia set J α is the unit circle if α = 0. For z ∈ J 0 = T, we have f 0 (z) = z -d . Note that f α is a holomorphic family of hyperbolic rational maps with parameter α ∈ D ε . There exists a holomorphic motion φ α : J 0 → C of J 0 parameterized by D ε and with base point 0 such that φ α (J 0 ) = J α and

f α • φ α (z) = φ α • f 0 (z) = φ α (z -d ) (5.19)
for all z ∈ J 0 , see [START_REF] Mcmullen | Complex Dynamics and Renormalization[END_REF]Chap. 4]. Since every point on J 0 moves holomorphically, we can write φ α (z) in power series of α as φ α (z) = z (1 + u 1 (z)α + u 2 (z)α 2 + O(α 3 )), (5.20)

where z ∈ J 0 .

In the following, we adopt the notation q := -d since the negative sign is boring in the expressions during the calculation. Meantime, we assume that d ≥ 3 first. If α is

Figure 1 :

 1 Figure 1: The carpet F 5,1 .

Figure 2 :

 2 Figure 2: The non-escaping loci M 2 and M 3

Figure 1

 1 Figure 1.1: The standard Sieipiński carpet S 3 .

A

  C * -Cylinder A is a set of the form A = {z ∈ C; r ≤ |z| ≤ R} with 0 < r < R < +∞. The metric on A induced by the length element |dz|/|z| which is the flat metric. Equipped with this metric, A is isometric to a finite cylinder of circumference 2π and length log(R/r). The boundary components {z ∈ C; |z| = r} and {z ∈ C; |z| = R} are called the inner and outer boundary components of A, respectively.

Figure 1 . 2 :

 12 Figure 1.2: Every path in Γ(C, C ) must intersect with C 0 .

Figure 1 . 3 :

 13 Figure 1.3: Every path in Γ(C, C ) must intersect with C 0 .

Figure 1 . 4 :

 14 Figure 1.4: The weak tangent W Fn,p (0).

g( 0 )

 0 = 0 and g(∞) = ∞.

  are round circles with the same diameter and equidistributed clockwise in the annuli bounded by ∂D and O 1 .

Figure 1 . 5 :

 15 Figure 1.5: The carpet F 7,1,2 .

Figure 2

 2 Figure2.1: The Julia set of f (z) = z 3 + λ/z 3 and an enlargement of a parabolic Fatou component, where λ ≈ 0.02772313 such that J f is a Sierpiński carpet containing a parabolic periodic point. The peripheral circles of J f are not uniform quasicircles but they are uniformly relatively separated.

  Then there are two positive constants C 1 (d, m) and C 2 (d, m) depending only on d and m, such that

Figure 2

 2 Figure 2.2: The pull back of a simply connected domain U under the rational map f with degree 4, where f (W ) = U and ∂W= C 1 ∪ C 2 ∪ C 3 . The complement of W consists of 3 simply connected components V 1 , V 2 and V 3 . In particular, f (V 1 ) = f (V 2 ) = C \ U and f (V 3 ) = C.Moreover, W contains 4 critical points of f and V 3 \ W (the white annulus) contains two.

Figure 2 . 3 :

 23 Figure 2.3: Sketch illustration of the mapping relation.

R

  c (±θ ) and R c (±q(θ )). The two dynamical rays R c (θ) and R c (0) are contained in different components of C \ H. Moreover, all the dynamical rays R c (±q n (θ)), where n ≥ 2, are contained in H by the definition of R and θ . This means that the collection of their landing pointsS n≥2 P •n c (c) are contained in H.Therefore, the critical value c (which is the landing point of R c (±θ)) and the β fixed point of P c are not contained in the ω-limit set of the origin.Step 3. Construct the semi-hyperbolic rational map f λ whose Julia set is quasisymmetrically equivalent to a round carpet. Consider the McMullen map f λ (z) = z d + λ/z d , where λ ∈ C \ {0} and d ≥ 3. The free critical points of f λ are 2d-th unit roots of λ. They are either escaping to ∞ or have bounded orbits at the same time. The non-escaping locus of f λ is defined as Λ d := {λ ∈ C \ {0} : The free critical orbits of f λ are not attracted by ∞}. See left picture in Figure 2.4 for the case when d = 3. According to [Ste, Theorem 9], there exists exactly one copy M of the Mandelbrot set of order one in Λ d ∩ {λ ∈ C * : | arg(λ)| < π/

  to be a portrait of z if all R(θ i ) land at z. Denote by α(T ) := z the base point and v(T ) := n the valence of T . Obviously, we have 3 ≤ v(T ) ≤ v(z). Let T be a portrait of z. Each connected components of C \ S θ∈T R(θ) is called a sector of T based at z. Evidently, any sector S of T is bounded by two rays R(θ a ), R(θ b ) with θ a , θ b ∈ T . Let I(S) be the segment of R/Z \ {θ a , θ b } disjoint with T . Then there is a one-to-one correspondence betweens sectors based at z and the segments of R/Z \ T , characterized by the property that R(t) is contained in S if and only if t is contained in I(S). Denote the correspondence by I : S → I(S).

Figure 3

 3 Figure 3.1: Portraits T, T with distinct base points

( 1 )

 1 The map σ d : t → dt mod Z carries T bijectively onto the portraitT : = {σ d (θ 1 ), • • • , σ d (θ v(T ) )} of f (α(T )) preserving cyclic order. Define the portrait map to beσ d : T → T .(2) Let S be a sector of T bounded by R(θ a ) and R(θ b ). Then the sector mapσ d : S → S ,where S is the sector of T bounded by R(σ d (θ a )) and R(σ d (θ b )), is well defined and one-to-one.

( 3 )

 3 l(σ d (S)) = d l(S) mod Z. Moreover, the integer n 0 := d l(S) -l(σ d (S)) is the number of critical points, counting multiplicity, of f contained in S.

( 4 )

 4 Figure 3.2: Sector maps

  3), suppose S is bounded by R(θ a ), R(θ b ). Let γ ab (t) be a segment of equipotential curve {z ∈ C : G f (z) = 1} which lies in S with γ(0) = x a and γ(1) = x b , where {x a } := γ ab T R(θ a ) and {x b } := γ ab T R(θ b ). Let Q be the close domain bounded by R(θ a ), γ ab and R(θ b ). See figure 3.2. Consider the image f (∂Q). It starts at f (z) and goes along the rays R(σ(θ a )) until it arrives at f (x a ), then it rotates d l(S) angles, parameterized by angles of external rays, along the equipotential curve {z ∈ C : G f (z) = d} to f (x b ), finally it turns to f (z) along R(σ(θ b )) and stops. Let G d := {z ∈ C : G f (z) < d}. Let Q be the domain σ d (S) T G d . By the arguments above, it is easy to see that f (γ ab ) surround ∂G d in n 0 times and overlap ∂G d T ∂Q one time more. Thus, l(σ d (S)) + n 0 = d l(S). Moreover, z ∈ ∂G d \ ∂Q has n 0 preimages in γ ab and z ∈ ∂G d T ∂Q has n 0 + 1 preimages in γ ab . The winding number of points in G d \ f (∂Q ) are w(z) = 8 < :

Proof.Figure 3 . 3 :

 33 Figure 3.3: Constructing regulated arc

  (η ∪ η ) T U consists at most four internal rays and all of the internal rays hit only at the center point c(U ). It is impossible for them to bounded a domain W , a contradiction. The regulated arc is denoted by [x, y]. The open arc (x, y) is defined by [x, y] \ {x, y}, and similarly the semi-open arc [x, y) and (x, y].

\

  {x, y} = ∅ and p ∈ W p \ I I. (3.12) See figure 3.4. By the denseness of E in J f , W p T E is not empty. Choose a point z in W p T E. If z is in I, then we are done. If not, there exists an arc γ zp in W p joining z and p, because W p is arcwise connected.

Figure 3

 3 Figure 3.4: illustrating the proof of lemma 3.8

  for any distinct Fatou component U and U which intersect W z . By the definition of F , We know that F | Wz is injective. Therefore, F | Wz is a homeomorphism by the domain invariance theorem. The claim follows. Secondly, consider point z in the finite set Crit(F ). Let W be sufficiently small topological disk around F (z) and φ : W → D F (z) → 0 the topological parameterization. Let W be one of the component F -1 W containing z. Since F : W \ {z} → W \ {F (z)} is an unbranched covering by the claim. The Riemann Hurwitz formula implies W is a topological disk around z. Denote by δ := deg(F | W \{z} ). Consider the following communicate diagram,

Lemma 3. 1 .

 1 For any U and p ∈ ∂U , the left resp. right supporting ray of U at p exists and is unique. Let R(θ) be a ray land at p, then R(θ) is the left resp. right supporting ray of U at p if and only if F (R(θ)) is the left resp. right supporting ray of F (U ) at F (p) Proof. Firstly, there are at least one and at most finite many rays landing at p by [DH84] and Theorem 3.1. Thus it exists and is unique by definition. Let R(θ ) be the right (left) supporting ray of U at p. L θθ := R(θ) S {p} S R(θ ) bounds a domain V containing U . The map F | V is locally homeomorphic at p. So F (R(θ)) and F (R(θ )) are rays supporting F (U ). Since F preserves the orientation. F (R(θ)), F (U ) and F (R(θ )) are in the same cyclic order around F (p) as R(θ), U and R(θ ) around p.

  then c ∈ ∂U and the intersection is exactly either a point {c} or one ray together with the landing point c. The latter happens if and only if Θ(c)T Θ(U ) = ∅.

  for distinct critical Fatou component U, U , then the intersection is exactly either a point {p} := ∂U T ∂U or one ray together with the landing point p. The latter happens if and only if Θ(U ) T Θ(U ) = ∅.

  c A Given critical portrait Ò A, one can construct a critical diagram D ⊆ D as follows. See figure 3.5. Start with the unit circle R/Z, for each Ò Θ i , mark all of the points e 2πiθ with θ ∈ Ò Θ i . Let b z i be the center of gravity of the marked points, and join each of these points e 2πiθ to b z i by a straight line segment l θ . Then we obtain a closed set D i := S l θ in the unit disk. It follows easily by Conditions (2) that distinct D i and D j will not cross each other. Let D := S 1≤i≤d D i be critical diagram associated to Ò A. The Condition (1) implies that D \ D are d simply connected domains W 1 , • • • , W d . Denote by I i the interior of W i T ∂D. Then {I i } 1≤i≤d is a partition of R/Z, each elements of which consists of finite open intervals with total length 1/d by Condition (3).

  b

Figure 3 . 5 :

 35 Figure 3.5: An example of critical diagram D

  passes through one critical Fatou component, consisting of two internal rays.(2.3) [α(θ i ), α(θ i+1 )] passes through two critical Fatou component U and U , consisting of four internal rays.In fact, if [α(θ 1 ), α(θ 2 )] passes through more than two critical Fatou component. Let U be one of them with U T {α(θ i ), α(θ i+1 )} = ∅. Then the supporting properties imply that there exists a external ray in R(U ) contained in Y , impossible. Assume [x, y] \ Y = ∅, otherwise, (2) follows. Let γ(t) := [x, y] with γ(0) = x and γ(1) = y. Set t 1 := inf 0≤t≤1 {t : γ(t) ∈ C \ Y } and t 2 := sup 0≤t≤1 {t : γ(t) ∈ C \ Y }. Then γ(t i ) ∈ ∂Y , i ∈ {0, 1} and so [γ(t 0 ), γ(t 1 )] ⊆ L θ i θ i+1 . We have [x, y] T ∂Y = [γ(t 0 ), γ(t 1 )] and [x, y] = [x, γ(t 0 )] S [γ(t 0 ), γ(t 1 )] S [γ(t 1 ), y]. Thus [x, y] ⊆ Y .

  thus p is a critical point. There exists at least a external ray in R(p) contained in Y . Otherwise, consider the section S of C \ R(p) containing U, U , F | S is locally homeomorphic at p, thus it can not paster [p, c(U )] and [p, c(U )] together. This contradicts the choice of R(θ i ) and R(θ i+1 ). Therefore F | [c(U ),c(U )] is one-to-one.

Figure 3 . 6 :

 36 Figure 3.6: An example of partition corresponding to critical diagram in figure 3.5. Here the critical Fatou components are U 1 , U 2 , U 3 and U 4 . There exists a critical points c 0 with deg F (c 0 ) = 2.

  Now we pay attention to the two regulated arc [ξ, ξ N ] and [ξ N , ξ 2N ]. Both of them are contained in H. Their relations are in one of the following five possibilities. See figure 3.7.

( C1 )

 C1 If f c has a parabolic or attracting Fatou cycle of period p ≥ 1. Then there exists a unique point z in the boundary of critical value Fatou component U such that f p c (z) = z. Let S be the sector containing U and bounded by supporting rays of U at z. Then set I c := {θ ∈ R/Z : R(θ) ⊆ S}. Obviously, I c = R/Z if and only if exactly one ray lands at z. (C2) In other cases, we have c ∈ J fc . Then there is a unique sector S based at c containing critical point 0. Set I c := R/Z \ {θ ∈ R/Z : R(θ) ⊆ S}. Evidently, I c is a single angle if and only if only one ray lands at critical value c. For any f c , f c in F, we say f c ≺ f c if and only if I c ⊇ I c . If I c = R/Z, denote by [η c , ξ c ] := I c . Let I c S I c := σ -1 2 (I c ) with I c := [η c , ξ c ] and I c := [η c , ξ c ], where {η c , η c } := σ -1 2 (η c ) and {ξ c , ξ c } := σ -1 2 (ξ c ). The above [•, •] are measured in positive cyclic order and we distinguish it from the notation of regulated arc by acting on distinct categories. Evidently, I c and I c are symmetric respect to origin with length |I c | = |I c | = 1 2 |I c |. Lemma 3.11 (Properties of characteristic arc). For any f c ∈ F, then (1) If f c is in case (C2), then (1.1) The rays R(η c ), R(η c ), R(ξ c ), R(ξ c ) land at critical point 0. (1.2) If I c is not a single point, let S c resp. S c be the sectors bounded by R(η c ) and R(ξ c ) resp. R(η c ) and R(ξ c ) and S c the sectors bounded by R(η c ) and R(ξ c ) avoiding the critical point. Then (S c S S c ) T S c = ∅ and f maps S c resp. S c conformally onto S c . Denote by H c := S c S S c .

Figure 3

 3 Figure 3.8: illustrating the proof of Lemma 3.11, Left: case (C1). Right: case (C2)

  If we give additional condition that diam U i , diam V i and diam E are less than 0 . Then there exist positive constants C 1 (d, m) and C 2 (d, m) such that (4.1) and (4.2) hold by replacing Shape e , Λ e and C i (d, m) with Shape, Λ and C i (d, m) respectively.

Lemma 4. 4 .

 4 Let {f n } be a sequence of rational maps with constant degree d ≥ 1. Let U ⊆ C be a non-empty open set and {f n } converge uniformly to a map g on U as n → ∞, then g is a rational map and deg g ≤ d. Moreover, deg g = d implies that {f n } converges uniformly to g on C as n → ∞.

  d 0 +1 , • • • , a d } by the claim and hence g is rational map with deg(g) = d 0 ≤ d. Moreover, deg(g) = d implies d 0 = d and {f n } converges uniformly to g on C. The lemma is finished.

Theorem 4. 3 .

 3 Let φ : V → C be an univalent map on open set V .

  with a, b ∈ C * such that ψ a,b := α b • φ • β a fixes 0 and ∞. By (1) and (2), we haveD(φ, V ) = D(ψ a,b , β -1 a (V )) ≥ 1 2π log |ψ a,b (0)ψ a,b (∞)| = 1 2π log |φ (z)φ (w)||z -w| 2 |φ(z) -φ(w)| 2 δ := D(φ, V ), λ 0 = |φ (0)|, λ 1 = |φ (1)| and λ ∞ = |φ (∞)|. Apply (4.6) for the pairs (0, 1) and (1, ξ) as ξ → ∞, we get |log λ i λ j | ≤ 2πδ for i = j. It follows that |log λ i | ≤ 3πδ for i ∈ {0, 1, ∞} For any z ∈ V \ {∞}, choose w n → ∞ as n → ∞. By (4.6), we have |log |φ (z)λ ∞ || ≤ 2πδ. Thus |log |φ (z)|| ≤ 5πδ.(4) Without loss of generality, we can assume φ fix 0, 1, ∞ on V by (1). Then (3) implies φ (z) = 1 on V . Therefore φ = id.(5) It follows directly by (3). (6) See Theorem 8.1 (b) in[CT15].

  Notations C * := C \ {0, ∞}, D t := {z ∈ C : |z| < t}, A(z 0 ; r 1 , r 2 ) := {z ∈ C : r 1 < |z -z 0 | < r 2 } and A(r 1 , r 2 ) := {z ∈ C : r 1 < |z| < r 2 } for z 0 ∈ C and 0 < r 1 < r 2 . Let A ⊆ C be annulus. Let Γ Height be the family of all locally rectifiable curves in A joining the two components of C \ A and Γ Width the family of all locally rectifiable closed curves in A separating the two components of C \ A. Let ρ : A → [0, ∞] be a non-negative Borel measure function on A. The ρ-area of A is Area (ρ, A) := ZZ A ρ 2 (z)dxdy. The ρ-length of a locally rectifiable curve γ in A is Length (ρ, γ) := Z γ ρ(z)|dz|.

  Lemma 4.6. Let W C be an open set with #C \ W ≥ 2. Suppose E is a measurable set with E W . Then Area p (E, W ) := sup φ Area (ρ * , φ(E)) < ∞, where the supermum is taken over all univalent maps φ : W → C * . Proof. Without loss of generality, assume W ⊆ C * . Let d = dist e (E, ∂W ) > 0. Let φ : W → C * be any univalent map. For any z ∈ E, consider the univalent map φ : B e (z, d) → C * . By the classic Koebe's 1/4-Theorem, we have B e (φ(z), |φ (z)|d 4 ) ⊆ φ(B e (z, d)) ⊆ φ(W ) ⊆ C * . Then |φ(z)| = dist e (φ(z), 0) ≥ |φ (z)d 4 .

4.5. 3

 3 Univalent maps off m-nested and λ-scattered nested disc system Theorem 4.4. Let Y be a finite set in C. Let {(D x , D x , D x )} x∈Y be m-nested and λscattered disk system in open set W ⊆ C. Let D := S x∈Y D x and D := S x∈Y D x . Then For any sufficient large m, there exists C(m, λ) > 0 with C(m, λ) → 0 as m → ∞, such that D(φ, C \ W ) ≤ C(m, λ) • Area p (D, W ) for any univalent map φ : C \ D → C. Proof. Firstly, we define a partial order on the finite set Y . For any x, y ∈ Y , x ≺ y ⇐⇒ either x = y or D x D y . Let I 1 be the set of all maximal elements in Y . Inductively, I k , k ≥ 2, is set of all maximal elements in Y \ S 1≤i≤k-1 I i . Then Y = I 1 [ We denote by D k := S x∈I k D x and D k := S x∈I k D x . Obviously, it follows that

Claim 4 .

 4 Let x ∈ I k with 1 ≤ k ≤ n. Then for any curve γ ⊆ D x with endpoints γ(0), γ(1) ∈ ∂D x , there exists a curve e γ in D x with e γ(0) = γ(0) and e γ(1) = γ(1) such that Length (ρ k , γ) ≥ Length (ρ k-1 , e γ). (4.16) Proof. If x ∈ I k , let e γ := γ, by definition of ρ k , it follows. Thus we assume x ∈ I k . Since ρ k (z) = ρ 0 (z)(1 -e -πm ) -k > ρ k-1 (z) by Claim 1. We also assume γ T D x = ∅. Let h x : D x → C be a univalent branch of the map 1 2π log • χ A (See figure 4.1). Let H x = h x (D x \ D x ). Then ρ 0 (z) = |h x (z)| and mod H x = mod D x \ D x ≥ m > 5log2 2π .

Figure 4 . 1 :

 41 Figure 4.1: Illustrating the proof of Claim 4

  1. Then {B(x, δ x )} x∈K covers the compact set K. Thus there exists finite set Σ such that K ⊆ S x∈Σ B(x, δ x ). Let 3δ be the Lebesgue number of the finite open covering and η := max {η x , x ∈ Σ}. Then any B(x, 2δ) with x ∈ K is contained in a ball B(ξ, δ ξ ) with ξ ∈ Σ. Therefore, (1)(2)(4)(5)(6) hold by Lemma 4.1 and Lemma 4.3.

Figure 4

 4 Figure 4.2: A sketch of the open sets U x , O x , E x , U x (r x ) and U x (r x ). The shaded domains are pullback disks.

Proposition 4. 1 .

 1 Let W be an open topological disk in C such that X ⊆ W . Then there exists uniform λ ∈ (0, 1) such that, for any m > 0 and any integer n ≥ 0, we have• m-nested system {(D x , D x , D x )} x∈Yn and λ-scattered {D x } x∈Yn in W , where Y n is a subset of {x ∈ X; n(x) ≤ n}, • pullback disk system {B x } x∈X induced by disks {B(x, δ)} x∈X 0 , where δ := δ(m) is independent on n, Given m > 0, construct {(D x , U x , O x , E x , U x (r x ), U x (r x )} x∈X in W as above.Let Y n := {x ∈ X ; n(x) ≤ n}, D x := U x (r x ) and D x := U x (r x ). The equations (4.30), (4.33) and (4.34) imply that {(D x , D x , D x )} x∈Yn is m-nested. For the λ-scattered property, by pre-composing a Möbius transformation, we may assume D is bounded in C and W ⊆ C. Recall that D := S x∈X D x . Let V x be the union of all D y ( = D x ) contained in D x and h : W → C * arbitrary univalent map.

  Area e D x ≤ C • Area e E x . (4.36) for some constant C > 1 independent of x, m and n. Let W be another open topological disk in C such that D ⊆ W ⊆ W . Applying the Koebe distortion therorem to h, we get a constant C 3 ≥ 1 such that max ξ∈W |h (ξ)| ≤ C 3 • min ξ∈W |h (ξ)|. (4.37) Since h preserves modulus of annulus, we have mod h(W ) \ h(W ) = mod W \ W . There exists constant C 4 independent of h such that diam e h(W ) ≤ C 4 • dist e (∂h(W ), ∂h(W )).

  3(1)(6), equation (4.35), Theorem 4.4 and the monotonicity of Area p (D, W ) on D, we have, for any z

Figure 4 . 3 :

 43 Figure 4.3: Graph of the map π, domains and rays explaining the construction of topological polynomial F .

Figure 4 . 4 :

 44 Figure 4.4: Relation of maps in the proof of Theorem 4.7

Figure 4

 4 Figure 4.5: Perturbation F r+δ of f r : z → z 3 -(1.081921 -0.087513i) z + 0.130061 + 1.446914i

  sup |δ|≤τ,n≥0,z∈V r dist (f δ,n (z), f r (z)) ≤ . (4.46) By (4.42) and (4.46), for any > 0, there exist τ := τ ( ) ≤ δ 0 such that sup |δ|≤τ,z∈V r dist ( b

0 as δ → 0 Thus both ψ δ and ψ - 1 δ

 01 uniformly converge to identity on C. Since f r+δ conjugates b f δ by ψ δ in (4.43), we have sup |δ|≤τ,z∈C dist (f r+δ (z), b f δ (z)) → 0 as τ → 0. Therefore, by Lemma 4.4 and (4.47), we have sup |δ|≤τ,z∈C dist (f r (z), f r+δ (z)) → 0 as τ → 0.

U

  dλ (z) = (z + λ -1) d + (λ -1)(z -1) d (z + λ -1) d -(z -1) d ! dand T dλ (z) = z + λ -1 shows that the set of all critical points of T dλ is {1, 1 -λ}, and both with multiplicity d -1. Note thatU -1 dλ (∞) = T -1 dλ (1) = d-1 [ k=0 {ξ k } and U -1 dλ (0) = T -1 dλ (1 -λ) = follows that ξ k and ω k are critical points of U dλ with multiplicity d -1, where 0 ≤ k ≤ d -1. In particular, ξ 0 = ∞. Therefore, the set of all critical points of U dλ is Crit(U dλ ) = {1, 1 -λ, ∞} ∪ d-1 [ k=1 {ξ k } ∪ d-1 [ k=0 {ω k }.

  Lemma 5.2 ( [Ya, Corollary 3.2]). The renormalization transformation U dλ has no Herman ring.

M

  d associated to T dλ can be defined as M d = {λ ∈ C * : T •2n dλ (0) → 1 and T •2n+1 dλ (0) → 1 as n → ∞} ∪ {0}. (5.11) Definition 5.1. Define H 0 := {λ ∈ C * : 0 ∈ A dλ (1)}. For every n ≥ 1, defineH n := {λ ∈ C * : T •n dλ (0) ∈ A dλ (1) and T •n-1 dλ (0) ∈ A dλ (∞)}.(5.12)Each component of H n is called a capture domain of depth n, where n ≥ 0.Proposition 5.1. The parameter space of T dλ has the following decomposition:C = M d (By definitions of the non-escaping locus and H n , we have M d ∩ ( S n≥0 H n ) = ∅. We need to show that two capture domains with different depths are disjoint and each λ ∈ C \ M belongs to H n for some n ≥ 0. First, suppose that λ ∈ H m ∩ H n for m = n. Without loss of generality, assume that m > n ≥ 0. By Definition 5.1, we have T •n dλ (0) ∈ A dλ (1) andT •m-1 dλ (0) ∈ A dλ (∞). This means that T •m-1 dλ (0) ∈ A dλ (1) and hence T •m dλ (0) ∈ A dλ (∞), which contradicts T •m dλ (0) ∈ A dλ (1). Therefore H m ∩ H n = ∅ for m = n.In the case of Corollary 5.1, W is called a hyperbolic component. Otherwise, W is called a queer component. It was generally believed that queer components do not exist. But if they do, then every T dλ admits an invariant line field on its Julia set and the Julia set has positive Lebesgue area. See Figures 5.1 and 5.2 for various Julia sets of J dλ .

Figure 5 . 1 :

 51 Figure 5.1: Julia sets of T 2λ with λ 1 ≈ 1.319448 + 1.633170i and λ 2 ≈ 1.5 + 0.866025i. The critical orbit 1 ↔ ∞ captures the critical orbit 1 -λ 1 → 0 → a → b → 1 and disjoint with the critical orbit 1 -λ 2 → 0 → c → 1 -λ 2 .

Figure 5 . 2 :

 52 Figure 5.2: Julia sets of T 2λ with λ 3 ≈ 2.046736 + 1.589069i and λ 4 = 4.0. T 2λ 3 has a Siegel disk with periodic 4 and J 2λ 4 is a quasicircle.

Corollary 5. 2 .

 2 Quasiconformal conjugacy classes in R d are either single points or open and connected. In particular, the conjugacy classes on ∂M d are single points.

S

  n≥0 H n if and only if the critical orbit1 -λ → 0 → (1 -λ) d → • • • tends to the attracting periodic cycle 1 → ∞ → 1. A point λ is called a center of a hyperbolic component W ⊂ M d if the critical point 1 -λ is periodic.On the other hand, λ is called a center of a capture domain of S n≥1 H n if the critical point 1 -λ is eventually mapped to 1.

Figure 5

 5 Figure 5.3: Sketch illustration of the construction of the IFS.

  x, y ∈ U , (5.15) and lim n→∞ A n = lim n→∞ B n = 1. See also[START_REF] Milnor | Geometry and dynamics of quadratic rational maps, with an appendix[END_REF] Theorem 2.7].By Theorem 5.12, the Hausdorff dimensionD = dim H (J ) = dim H (J) satisfies 0 ≤ s n,1 ≤ D ≤ s n,2 ≤ 2, where

  lim n→∞ |F n (ζ d n )| = +∞. The proof is completed.

Figure 5

 5 Figure 5.4: The Julia sets of T 2λ , both are quasicircles, where λ = 30 and 1000, respectively. It can be seen that the Julia set becomes more circular as the parameter λ becomes more larger (compare the right picture in Figure 5.2).Figure ranges: [-10, 16] × [-13, 13] and [-125, 125] × [-125, 125].

  Figure 5.4: The Julia sets of T 2λ , both are quasicircles, where λ = 30 and 1000, respectively. It can be seen that the Julia set becomes more circular as the parameter λ becomes more larger (compare the right picture in Figure 5.2).Figure ranges: [-10, 16] × [-13, 13] and [-125, 125] × [-125, 125].

  c kn , δ/2), D(c kn , δ)) and g = f •kn , we know that there exists a constant C 1 (d) > 0 such that Shape(Y kn , c) = max w∈∂Y kn |w -c| dist(c, ∂Y kn ) Now we estimate the relative distance of ∂U kn and ∂U kn by (2.20) and (2.21).

			≤ C 1 (d),	(2.20)
	where d is the constant appeared in Theorem 2.5.
	Similarly, there exists a constant C 2 (d) > 0 such that
	Shape(X kn , c) =	max w∈∂X kn |w -c| dist(c, ∂X kn )	≤ C 2 (d).	(2.21)

  1 branched covering, there exist δ -1 distinct choices of z p . By Lemma 3.1, all the p external rays supports the Fatou cycle in the same direction.Secondly, for critical Fatou component

  Recently, Qiao considered the generalized diamond hierarchical Potts model and For the connectivity of Mandelbrot set M d , Wang et al. proved that M 2 is connected in [WQYQG, Theorem 1.1]. We now generate this result to all M d , where d ≥ 2. Theorem 5.2. The non-escaping locus M d is connected for d ≥ 2.

The map f is expanding in the hyperbolic metric in a neighborhood of J.

Rigidité quasi-symétrique, tapis de Julia et le débarquement de dynamique resp. paramètres rayonsQuasisymmetric rigidity, carpet Julia sets and the landing of dynamic resp. parameter raysJinsong ZENG

Acknowledgements

Contents

Let λ 0 = Φ -1 (c) ∈ M ∩ R + , where c = c(θ) ∈ (-2, 1/4) is the real parameter on the boundary of the Mandelbrot set determined in Step 2. By the symmetry of McMullen maps, all 2d free critical points of f λ 0 are non-recurrent and they have infinite forward orbits. This means that f λ 0 is semi-hyperbolic (and not sub-hyperbolic). Let B ∞ be the immediate attracting basin of ∞ of f λ 0 . Then JP Φ(λ 0 ) ∩B ∞ = {z λ 0 }, where z λ 0 is the image of the β fixed point of J Pc under the quasiconformal conjugacy stated above [QXY, Lemma 4.1]. Note that B ∞ is the unique periodic Fatou component of f λ 0 , it follows that the ω-limit sets of the critical points of f λ 0 are disjoint with the periodic Fatou component of f λ 0 by the construction of P c .

By [START_REF] Qiu | Fatou components and Julia sets of singularly perturbed rational maps with positive parameter[END_REF]Lemma 4.4], the Julia set of f λ 0 is a Sierpiński carpet. By Theorem 2.2, the peripheral circles of J f λ 0 are uniform quasicircles and uniformly relatively separated. By Bonk's criterion ( [Bon, Corollary 1.2]), the Julia set of f λ 0 is quasisymmetrically equivalent to a round carpet, as required.

General cases

Proposition 3.5. Let {Π i } 1≤i≤d be the partition of C induced by the critical portrait of f . If x, y ∈ J f have the same itinerary respect to {Π i } 1≤i≤d , then either x = y or x, y are in the boundary of a Fatou component, which is mapped to a siegel disk. Firstly, U is finite. If not, since there is no wandering Fatou components and the number of periodic Fatou components is finite. Infinite many elements in U will eventually be mapped onto a periodic one. This contradicts (3.16).

Secondly, any U ∈ U is mapped to a siegel disk. If not, let (x , y ) = U T [x, y]. If there exists N ≥ 0 such that orbits of x N and y N avoid the finite set X := S 1≤i≤d (J f T ∂Π i ), then x N and y N have the same itinerary. Lemma 3.2 gives x N = y N . This contradicts (3.16). Thus there exist N and a periodic point ξ ∈ X such that x N = ξ or y N = ξ. Suppose x N = ξ. Let ξ ∈ Θ(U 0 ) and p the period of ξ. Then F p fixes x N and iterates y N to at least two distinct segments of ∂U 0 \ Θ(U 0 ). By properties of supporting rays, x n , y n must be separated by Θ(U 0 ) for some n, a contradiction. Finally, U consists of only one Fatou component. If not, let U = U ∈ U. Let M, N be integers such that F M (U ) = F M +N (U ), F M (U ) = F M +N (U ). Then

By Lemma 3.7, ξ := ∂F M (U )

T F M [c(U ), c(U )] is periodic. Since F N | ∂F M (U ) conjugates a irrational rotation. Thus ξ can not be periodic, a contradiction. The proof is completed.

Proof of Theorem 3.2. The theorem follows immediately by Propositions 3.5.

Application to core entropy

Consider a quadratic polynomial family F := {f c = z 2 +c : f c has no Siegel disks and J fc is locally connected }. As an application of Theorem 3.2, we shall prove the monotonicity of core entropy.

Assumption

Let f be a rational map with degree d ≥ 2 and no empty Fatou set. Denote by Orb(S) := S k≥0 f k (S) for set S ⊆ C. Let X 0 := X 0 S X 0 be a finite set in C such that

• X 0 contains no periodic points.

Thus X 0 are disjoint with the immediate rotation domains, such as periodic Siegel disks and Herman rings, and there exists δ 0 > 0 such that

(2) For any x ∈ X 0 , B(x, δ 0 ) T Orb(c) = ∅ for any critical point c / ∈ X 0 .

(3) For any x ∈ X 0 , B(x, δ 0 ) \ {x} is disjoint with critical orbits and Julia set , thus every component of f -n B(x, δ 0 ) is open disk and in Fatou set.

(4) For any x ∈ X 0 , B(x, δ 0 )

T Orb(f B(x, δ 0 )) = ∅. Because each points in X 0 is iterated into either periodic rotation domains or converging to periodic points. Hence

Shrinking Lemma

Lemma 4.7 (Shrinking lemma). Let K := Orb(X 0 ) S X 0 . Then for any > 0, there exist 0 < δ ≤ and integer η ≥ 1, such that, for all x ∈ K and n ≥ 1,

(1)

Recall that m 0 is a constant in subsection 4.2.1 and

an unique normalized quasiconformal map ψ :

Step III. f realizes the given critical portrait Θ and r.

Since µ ψ | C\Dr 0 = µ 0 , ψ is holomorphic on C \ D r 0 . By post-composing suitable Möbius transformation, we can assume

Then the critical value are contained in ψ(∂D r 0 ) with the same equipotential log r 0 = dr. Thus

. By the definition of F and external ray, it follows that ψ(R (θ)) is the external ray R(θ) of f landing at the critical point ψ(z i ), for any θ ∈ Θ i . Thus Π(f ) = Θ.

Step IV. The uniqueness of f . Assume that f 1 and f 2 are polynomials in S d (r) with the same critical portrait Θ. The idea is to construction quasiconformal conjugacy h between f 1 and f 2 which is conformal in the basin of f . Since J f has Lebesgue measure 0, we can argue that h : C → C is conformal and so h is an affine map.

Denote by

Recall that G f i are the Green functions measuring the escape rate of points to ∞ and Ψ f i are the Böttcher coordinates of f i . Then the following diagram commutes,

Extend h 0 to C \ X f 1 (r) so that h 0 is a K-quasiconformal map on C. Inductively, for

the boundary of U δ . We can define φ δ,1 via the following lift

We choose the lift that agree with φ δ,0 on U δ . Since φ δ,0

we know that φ δ,1 on C \ U δ is homeomorphic and so is isotopic to φ δ,0 rel ∂U δ by the Alexander's trick. Therefore, globally we have

(2) φ δ,0 and φ δ,1 are isotopic rel U δ which contains P F r+δ := S i≥1 F i r+δ (Crit(F r+δ )).

(3) φ δ,0 = φ δ,1 on U δ and maps it onto U δ holomorphic.

Step III. From c-equivalence to isotopies {H n } n≥0 and {φ δ,n } n≥0 .

Let H 0 : C × I → C to be an isotopy rel U δ such that H 0 (•, 0) = φ δ,0 , H 0 (•, 1) = φ δ,1 , and

So the map φ δ,1 | C\F -1 r+δ (U δ ) can be considered as a lift of H 0 (•, 0)| C\U δ by the non-branched covering maps

By the homotopy lifting theorem for covering maps, the isotopy

Repeating this argument we get quasi-conformal map φ δ,n and isotopies H n between C and C rel.

Step IV. Apply Thurston Algorithm to obtain {η δ,n } n≥0 and polynomials {f δ,n } n≥0 .

Let η δ,0 = id. Then η δ,0 • F r+δ defines a complex structure on C by pulling back the standard complex structure on C. The uniformiaztion theorem guarantees the existence of an unique homeomorphism η δ,1 :

Recursively, there exist quasi-conformal map η δ,n and polynomial f δ,n , for n ≥ 0, such that

Step V. The uniformly convergence of f δ,n and ψ δ,n .

to be quasi-conformal map. Then, combining with Step V (3) and Step VI (1), we have

It follows that {ψ δ,n } is a normal family. For any sequence {ψ δ,k i } there is a subsequence locally uniform converging on C to a

, which is the Fatou set. Thus ψ δ is holomorphic on C since the measure of the Julia set of f r is zero. Combining with the conditions that ψ δ (z δ,i ) = z i i ∈ {1, 2} and ψ δ (∞) = ∞, we know that ψ δ is the affine map

which sending z δ,i to z i . Since ψ δ is independent of any locally uniform converging subsequence in {ψ δ,n }, we know that the entire sequence {ψ δ,n } locally uniform converges on proved that the family of rational maps

are actually the renormalization transformation of the generalized diamond hierarchical Potts model [Qi, Theorem 1.1], where m, n ≥ 2 are both integers and λ ∈ C * := C \ {0} is a complex parameter. The standard diamond lattice (m = n = 2) and the diamond-like lattice (m = 2 and n ∈ N) are the special cases of (5.2).

In this chapter, we will consider the case for d := m = n ≥ 2. For simplicity, we use

We not only study the properties of the Julia sets of U dλ , but also consider the connectivity of the non-escaping locus of the parameter space of this renormalization transformation. If λ = 0, then U dλ degenerates to a parabolic polynomial U d0 (z) = ( z+d-1 d ) d whose Julia set is a Jordan curve. For the connectivity of the Julia sets of U dλ , we have following Theorem.

Theorem 5.1. The Julia set of U dλ is always connected for every d ≥ 2 and λ ∈ C * .

Note that Qiao and Li proved that the Julia set of U dλ is connected for d = 2 and λ ∈ R [QL]. We would like to remark that if m = n, then there exists parameter λ ∈ C * such the Julia set of U mnλ defined in (5.2) is disconnected (see [START_REF] Qiao | Julia sets and complex singularities of free energies[END_REF]Figure 3.1] for example).

The Mandelbrot set of quadratic polynomials f c (z) = z 2 + c is defined by

Douady and Hubbard showed that M is connected [DH]. For higher degree polynomials with only one critical point, there are associated Multibrot sets. For rational maps, one way to study the parameter space is to consider the connectedness locus, which consists of all parameters such the corresponding Julia set is connected. However, the connectedness locus makes no sense in our case since every Julia set is connected. For λ = 0, then 1 and ∞ are two superattracting fixed points of U dλ . The non-escaping locus M d associated to this family is defined by

Obviously, "non-escaping" here means the collection of those parameters such that the orbit of 0 cannot be attracted by 1 and ∞. Note that 0 is a critical value of U dλ . The non-escaping locus M d can be identified as the complex plane cutting out infinitely many simply connected domains, which will be called "capture domains" later (see Figure 2 and Proposition 5.1). There exist many small copies of the Mandelbrot set M in M d which correspond to the renormalizable parameters. d ≥ 2. Moreover, U dλ (x) = 0 if and only if x = 1. By a straightforward calculation, we have 0 < U dλ (0) < 1. Now we divide the arguments into two cases.

If there exists no fixed point of U dλ in (0, 1), then we have 0 < x < U dλ (x) < 1 for every 0 < x < 1. This means that 0 lies in the immediate attracting basin of 1. By Lemma 5.4(5), we know that J dλ is a quasicircle. In particular, A dλ (1) ∩ A dλ (∞) = J dλ = ∅. If there exists at least one fixed point of U dλ in (0, 1), we denote all of them by 0 < x 1 <

where n ≥ 1. By a completely similar argument as the case λ > 0, one can show that the fixed point x n is contained in the boundaries of two different Fatou components. Therefore, the proof is complete.

Theorem 5.5. For every d ≥ 2 and λ ∈ R, the Julia set J dλ is not a Sierpińsk carpet.

Proof. Note that if J dλ is a Sierpiński carpet, then the closure of any two Fatou components of U dλ cannot be intersecting to each other. But this contradicts Lemma 5.3. The proofs of Theorems 5.5 and ?? are finished.

By computer experiments, it is shown that A dλ (1) ∩ A dλ (∞) = {z 0 } for λ ∈ C, where z 0 is a repelling fixed point of U dλ . Therefore, the Julia set J dλ can never be a Sierpińsk carpet for any λ ∈ C (see Figures 5.1 and 5.2).

Decomposition of the parameter space

In this section, we divide the parameter space of T dλ into the non-escaping locus M d union countably many capture domains. Recall that A dλ (1) and A dλ (∞) are the immediate superattracting basins of 1 and ∞ respectively.

Lemma 5.4. For each λ ∈ C * , the following conditions are equivalent:

(1) The Julia set

In particular,

Proof. We first prove (1) ⇒ (2)(3)(4)(5). If J dλ is a quasicircle, the Fatou set of T dλ consists of two simply connected Fatou components A dλ (1) and A dλ (∞) whose common boundary is J dλ . Since T dλ permutes 1 and ∞, by (5.9), it follows that (2) holds and

Therefore, (3)(4)(5) hold. By (5.9), we have (3) ⇒ (4) ⇒ (5). Now we prove (5) ⇒ (1). Suppose that 0 ∈ A dλ (1). By (5.6), we have

. This contradicts the choice of the integer k. So we have λ ∈ H k in this case. The proof is complete.

See Figure 2 for the non-escaping loci M 2 and M 3 . There some capture domains are also clearly visible (blank regions).

Quasiconformal conjugacy classes

Let R d be the collection of all T dλ , where λ ∈ C * . In this section, we give a complete characterization of the quasiconformal conjugacy classes in R d .

Definition 5.2. Let Λ be a complex manifold. A holomorphic family of rational maps parameterized by Λ is a holomorphic map f λ : Λ × C → C such that f λ (z) is a rational map for fixed λ ∈ Λ and depends holomorphically on λ ∈ Λ for fixed z ∈ C.

The parameter λ ∈ Λ is called a J-stable parameter of a holomorphic family of rational maps f λ if the total number of attracting cycles of f λ is constant in a neighborhood of λ.

Theorem 5.6. The boundary ∂M d is the set of parameters such that T dλ are not J-stable in R d .

Proof. By [START_REF] Mcmullen | Complex Dynamics and Renormalization[END_REF]Theorem 4.2], T dλ 0 is J-stable if and only if both critical sequences {T •k dλ (1λ)} k≥0 and {T •k dλ (1)} k≥0 are normal for λ in a neighborhood of λ 0 . Since {T •k dλ (1)} n≥0 lies in a finite orbit 1 ↔ ∞, we only need to consider the orbit of 1 -λ. If λ 0 ∈ H n for some n ≥ 0, the orbit of 1 -λ 0 will be attracted by the cycle 1 ↔ ∞. For λ close to λ 0 , the orbit of 1 -λ still converges to the cycle 1 ↔ ∞. By Montel's theorem,

On the other hand, if λ 0 ∈ ∂M d , then {T •k dλ 0 (1 -λ)} k≥0 omits the attracting basin of 1 ↔ ∞. However, there are arbitrary small perturbation of λ 0 such that {T •k dλ (1 -λ)} k≥0 converges to the cycle 1 ↔ ∞. This means that T dλ is not J-stable on ∂M d .

Corollary 5.1. Let W be a component in the interior of M d . If there exists λ 0 ∈ W such that 1 -λ 0 converges to an attracting cycle, then every λ ∈ W also has this property. Proof. By Theorem 5.6, every T dλ ∈ W is J-stable. This means that there exists a small neighborhood of λ such the number of attracting cycles is constant. Since 1-λ 0 converges to an attracting cycle, this means that the constant is 2. The corollary follows.

Lemma 5.6. Every hyperbolic component in M d and capture domain in H n has a center, where n ≥ 1. Meanwhile, H 0 has no center.

It will be proved in next section that every hyperbolic component in M d and capture domain in H n has exactly one center, where n ≥ 1 (Theorem 5.10).

Proof. Let W be a hyperbolic component in M d . For every λ ∈ W , let m(λ) be the multiplier of the attracting periodic orbit of T dλ other than 1 ↔ ∞. It can be checked as in [START_REF] Douady | Systèmes dynamiques holomorphes[END_REF]Theorem 4,p. 46] and [?, Theorem 2.1, p. 134] that the multiplier mapping λ → m(λ) defined from W to D is proper and holomorphic. This means that W has at least one center.

Let W be a component of H n , where n ≥ 1. Then for every λ ∈ W , T •n dλ (0) ∈ A dλ (1) and n is the smallest integer satisfying this property. Let ψ λ : A dλ (1) → D be the unique Böttcher map define on the immediate basin of 1 such that ψ λ • U dλ = (ψ λ (z)) d , ψ λ (1) = 0 and ψ λ (1) = 1. By the definition of ψ λ , it follows that ψ λ depends holomorphically on λ ∈ W . Define a map m : W → D by m(λ) = ψ λ (T •n dλ (0)). It is clearly that m is holomorphic. We then prove m is proper. Let λ k ∈ W be a sequence converging to λ ∈ ∂W as n → ∞. Suppose that there exists a subsequence of λ k , denote also by λ k , such that m(λ k ) converges to an interior point w ∈ D. Since the family of univalent mappings

. Hence T dλ is hyperbolic. This is a contradiction since λ ∈ ∂W . Finally, by the definition of H 0 and Lemma 5.4, A dλ (1) contains only one critical point 1 (counted without multiplicity). Note that A dλ (1) lies in a superattracting periodic Fatou component and T dλ (1 -λ) = 0 = 1, it follows that the orbit of 1 -λ is disjoint with the orbit 1 ↔ ∞. The proof is complete. Now we give a complete characterization of the quasiconformal conjugacy classes in

Theorem 5.9. Quasiconformal conjugacy classes in R d can be listed as follows:

(1) Hyperbolic components in the interior of M d with the center removed.

(2) Capture components of H n with the center (if any) removed, where n ≥ 0.

(3) Centers of hyperbolic or capture domains.

(4) Queer components in the interior of M d .

(5) Single points on the boundary of M d .

Proof. By Corollary 5.2, the five cases stated in the theorem are disjoint to each other and (4)(5) are indeed quasiconformal conjugacy classes. (1)(2) are quasiconformal conjugacy classes by Theorem 5.8. As every queer component is a conjugacy class, one can get a proof in [Za,Theorem 3.4] by a word for word analysis.

Simply connectivity of the capture domains

In this section, we prove that the non-escaping locus M d is connected. This amounts to showing that H 0 is homeomorphic to the punctured disk D * := D \ {0} and each of the component of H n is homeomorphic to the unit disk for n ≥ 1.

One way to do this is to follow the standard way of Douady-Hubbard's parameterization of the hyperbolic components of the quadratic Mandelbrot set [Do]. This method was developed by Roesch to study the parameter space of the cubic Newton maps [START_REF] Roesch | Topologie locale des méthodes de Newton cubiques[END_REF][START_REF] Roesch | On capture zones for the family f (z) = z 2 + λ/z 2 , in "Dynamics on the Riemann Sphere: A Bodil Branner Festschrift[END_REF] and Qiu, Roesch, Wang and Yin to study the parameter space of the McMullen maps [QRWY]. Moreover, this parameterized method was generated and then used in the proof of M 2 is connected [WQYQG, Theorem 1.1].

However, to prove H 0 is homeomorphic to the punctured disk D * and each of the component of H n is homeomorphic to the unit disk for n ≥ 1, it would be much easier to use the methods of Teichmüller theory of the rational maps which was developed in [McS] (in which, a different proof of the connectivity of the Mandelbrot set was given).

We first recall some definitions in [McS]. By definition, the Teichmüller space Teich(T dλ ) of T dλ consists of all pairs (T dλ , [ϕ]), where ϕ : C → C is a quasiconformal mapping which conjugates T dλ to T dλ . Here Proof. Let W be a component of H n with all centers removed. Then the forward orbit of 1 -λ under T dλ is infinite for λ ∈ W . By Theorem 5.9, W denotes a single quasiconformal conjugacy class.

For any basepoint λ ∈ W , it follows that the critical point 1 -λ belongs to the attracting basin of the cycle 1 → ∞ → 1. In particular, T •n dλ (0) ∈ A dλ (1) and T •n dλ (0) = 1. Define the Green function on A dλ (1) by

Note that G dλ can be extended to the Fatou set of T dλ by pulling back.

Let γ be the equipotential of G dλ passing through the critical point 1-λ (In particular, it is homeomorphic to the figure 8

Then b J dλ is the closure of the grand orbits of all periodic points and critical points of T dλ . The complement U := C \ b J dλ consists of countably many annuli with finite modulus which lie in a same grand orbit. By [START_REF] Mcmullen | Quasiconformal homeomorphisms and dynamics. II-I. The Teichmüller space of a holomorphic dynamical system[END_REF]Theorem 6.2], we have

where M 1 (J dλ , T dλ ) denotes the unit ball in the space of all T dλ -invariant Beltrami differentials supported on J dλ . Note that every hyperbolic rational map carries no invariant line fields on the Julia set, it follows that M 1 (J dλ , T dλ ) is trivial since T dλ is hyperbolic when λ ∈ W ⊂ H n .

Since W denotes a single quasiconformal conjugacy class, we have

by [START_REF] Mcmullen | Quasiconformal homeomorphisms and dynamics. II-I. The Teichmüller space of a holomorphic dynamical system[END_REF]Theorem 6.1]. Note that every quasiconformal self-conjugacy ψ of T dλ fixes the grand orbits of the critical points 1 and 1 -λ and hence fixes the boundaries of each annulus of U . Moreover, ψ is the identity on J dλ . Therefore, [ψ] ∈ Mod(T dλ ) is identity on b J dλ and it is possibly a power of a Dehn twist in the annuli of U . This means that Mod(T dλ ) is a subgroup of Z.

By Lemma 5.6, each W cannot be simply connected is a component of H n for n ≥ 1. On the other hand, W is not simply connected if W = H 0 by Proposition 5.2. So Mod(T dλ ) = Z. This means that W is homeomorphic to a punctured disk. This means that each W contains exactly only one center if W = H 0 . The proof is complete.

Proof of Theorem 5.2. This is a direct corollary of Proposition 5.1 and Theorem 5.10.

Proof of the asymptotic formula

By Proposition 5.2, if the parameter λ lies in the unbounded capture domain H 0 , then the Julia set J dλ is a quasicircle. In this case, J dλ moves holomorphically in H 0 and its Hausdorff dimension depends real analytically on λ by a classic result of Ruelle. The following Theorem 5.11 is a weak version of [START_REF] Ruelle | Repellers for real analytic maps[END_REF]Corollary 6].

Theorem 5.11. Let f λ : Λ × C → C be a holomorphic family of hyperbolic rational maps parameterized by Λ, where Λ is a complex manifold. Then the Hausdorff dimension of the Julia set of f λ depends real analytically on λ ∈ Λ.

Let Ω be a closed subset of R n . A map S : Ω → Ω is called a contraction on Ω if there exists a real number c ∈ (0, 1) such that |S(x) -S(y)| ≤ c|x -y| for all x, y ∈ Ω. A finite family of contractions {S 1 , S 2 , • • • , S m } defined on Ω ⊂ R n , with m ≥ 2, is called an iterated function system or IFS in short.

To compute the Hausdorff dimension of J dλ with λ ∈ H 0 , we need the following result (see [START_REF] Falconer | Fractal geometry: Mathematical Foundations and Applications[END_REF]Theorem 9.1, Propositions 9.6 and 9.7]).

Theorem 5.12 ( [Fa]). Let {S 1 , . . . , S m } be an IFS on a closed set Ω ⊂ R n such that

(1) There exists a unique non-empty compact set

The non-empty compact set J appeared in Theorem 5.12(1) is called the attractor of the IFS {S 1 , . . . , S m }.

Let f be a rational map with degree at least two. We use Fix(f ) to denote the set of all the fixed points in the Julia set of f . Lemma 5.7. Let f be a hyperbolic rational map whose Julia set J is a quasicircle. Then the Hausdorff dimension D := dim H (J) of J is determined by 1 lim n→∞ A n (D) = 1, where

(5.14)

The notation Fix(f •n ) in (5.14) denotes the collection of all the repelling periodic points of f with period n (the period is not necessary the smallest). The Julia set of a hyperbolic rational map can be seen as the limit of a sequence of IFS which are defined in terms of the inverse branches of the iterations of the rational map.

Proof. Let d ≥ 2 be the degree of f . Since f is hyperbolic and the Julia set J of f is a quasicircle, there exist a pair of closed annular neighborhoods W 1 , W 2 of J and a quasiconformal mapping φ :

ε} is a closed annular neighborhood of the unit circle and ε > 0 is small enough. Without loss of generality, we only consider the first case since the completely similar argument can be applied to the second one.

In order to define IFS, it is more convenient to lift J and f under the exponential map. Hence we assume further that J separates 0 and ∞. Define a curve γ :

and denote it by U . Then U is 1 The statement and the proof of this lemma were not correct in the previous version. We would like to thank Peter Haïssinsky for pointing out to us. See [WBKS, §4, (4.2)] for the same statement.

small enough, we can expand f α in (5.18) in power series of α as f α (z) = z q -qz q+1 α + q(q + 1) 2 z q+2 α 2 + O(α 3 ).

(5.21) Substituting (5.20) and (5.21) into (5.19), then comparing the terms to the second order in α, we obtain the following equations:

u 2 (z q ) -qu 2 (z) = q(q -1) 2 u 2 1 (z) -q(q + 1)zu 1 (z) + q(q + 1) 2 z 2 .

(5.23)

For each non-zero integer l ∈ Z, the functional equation

has the formal solution

(5.25)

Note that the solution (5.25) is convergent if |z| = 1. This means that the solution of (5.22) is

(5.26) Therefore, the equation (5.23) can be reduced to u 2 (z q ) -qu 2 (z) = -q (q + 1)

(5.27) By (5.24) and (5.25), the solution of u 2 is

(5.28)

For each n ≥ 1, the collection of the fixed points of f •n α on the Julia set J α forms the finite set Fix(f •n α ) = ¨φα (e 2πit j ) :

(5.29) By (5.19) and the chain rule, we have (f •n α ) (φ α (e 2πit j )) =

Q n-1 m=0 f α (φ α (e 2πiq m t j )). The calculation in Appendix ( §5.8) shows that for every D > 0 and all sufficiently large n, the following holds:

.

(5.30)

Let D α := dim H (J α ) be the Hausdorff dimension of J α . One can write the corresponding (5.14) of f α in Lemma 5.7 as

.

(5.31)

Fix some large n, when α is small enough, (5.31) is equivalent to

(5.32) By Theorem 5.11 and Lemma 5.8, D α depends real analytically on α in a small neighborhood of the origin and D 0 = 1. This means that in a small neighborhood of 0, the Hausdorff dimension of J α can be written as

(5.33) Substituting (5.33) into (5.32) and comparing the corresponding coefficients, we have a 10 = a 01 = a 20 = a 02 = 0 and a 11 = 1/(4 log |q|).

(5.34)

This means that

(5.35)

Note that q = -d and α = λ -1 d+1 . This ends the proof of Theorem 5.3 in the case of d ≥ 3.

If d = 2, then (5.21) can be written as f α (z) = z q -qz q+1 α. Following the calculation process of d ≥ 3 and carefully omitting some corresponding terms, it can be checked that Theorem 5.3 still holds for d = 2. The proof is complete.

Appendix

This section will devote to proving (5.30). From (5.21), we have f α (z) = qz q-1 -q(q + 1)z q α + q(q + 1)(q + 2) 2 z q+1 α 2 + O(α 3 ).

(5.36) Substituting (5.20) into (5.36), we have f α (φ α (z)) = qz q-1 + qz q-1 [(q -1)u 1 (z) -(q + 1)z] α + qz q-1 (q + 1)(q + 2) 2 z 2 + (q -1)(q -2) 2 u 2 1 (z) -q(q + 1)zu 1 (z) + (q -1)u 2 (z)

(5.37)

Define σ := σ(t) = e 2πit ∈ T. Then σσ = 1. For 0 ≤ m ≤ n -1, by (5.37), we have

where A m = q 2 (q -1) u 1 (σ q m ) -q 2 (q + 1) σ q m (5.39) and B m = q 2 (q + 1)(q + 2) 2 σ 2q m + q 2 (q -1)(q -2) 2 u 2 1 (σ q m ) -q 3 (q + 1)σ q m u 1 (σ q m ) + q 2 (q -1)u 2 (σ q m ).

(5.40)

For every D > 0, by (5.38), we have

+ O(α 3 ).

(5.41) Lemma 5.9. Let m, m 1 , m 2 ∈ N. If n ≥ 1, then:

(1) q m ≡ 0 mod q n -1.

(2) q m 1 + q m 2 ≡ 0 mod q n -1.

(3) q m 1 -q m 2 ≡ 0 mod q n -1 if and only if m 1 -m 2 = kn for some k ∈ Z.

Proof. Since (q, q n -1) = 1, it means that (q m , q n -1) = 1 for m ≥ 0. Then (1) follows.

To prove (2), it suffices to show that q m + 1 ≡ 0 mod q n -1 for m ≥ 0 since q n -1 is relative prime to q m for m ≥ 0 by (1). Set m = kn + r, where k ≥ 0 and 0 ≤ r ≤ n -1. We have q m + 1 = q kn+r -q r + q r + 1 ≡ q r + 1 ≡ 0 mod q n -1

The proof of (3) is similar to that of (2). Since q n -1 is relative prime to q m for m ≥ 0, we need to find out the condition on m such that q m -1 ≡ 0 mod q n -1 for fixed n ≥ 1. Set m = kn + r, where k ≥ 0 and 0 ≤ r ≤ n -1. We have q m -1 = q kn+r -q r + q r -1 ≡ q r -1 mod q n -1.

This means that q m -1 ≡ 0 mod q n -1 if and only if r = 0 since |q r -1| < |q n -1|.

Following [WBKS, § 2], it is convenient to introduce the average notation

where G is a continuous function defined on the interval [0, 1) and t j = j/(q n -1) is defined in (5.29).

In order to prove (5.30), we only need to prove for every D > 0 and sufficiently large n, the following holds

.

(5.43)

For each n ≥ 1 and any k ∈ Z, it is straightforward to verify the average in (5.42) has the following useful property:

(5.44)

Lemma 5.10. For 0 ≤ m, m 1 , m 2 ≤ n -1, we have σ q m n = 0, u 1 (σ q m ) n = 0, σ q m 1 +q m 2 n = 0, σ q m 1 u 1 (σ q m 2 ) n = 0, u 1 (σ q m 1 )u 1 (σ q m 2 ) n = 0 and u 2 (σ q m ) n = 0.

Proof. By (5.26) and (5.28), the average property (5.44) and Lemma 5.9(1)(2), the equations stated in the Lemma can be verified directly.

As an immediate corollary of Lemma 5.10, from (5.39) and (5.40), we have

By (5.41) and Corollary 5.3, we have

(5.45) By (5.39) and (5.40), we have

q -1) 2 u 1 (σ q m 1 )u 1 (σ q m 2 ) n + q 4 (q + 1) 2 σ q m 1 -q m 2 n -q 4 (q 2 -1) u 1 (σ q m 1 )σ -q m 2 + u 1 (σ q m 2 )σ q m 1 n .

(5.46) Since 0 ≤ m 1 , m 2 ≤ n -1, it follows that m 1 -m 2 = kn for k ∈ Z if and only if m 1 = m 2 . By Lemma 5.9(3), we have

This means that X 0≤m 1 ,m 2 ≤n-1 σ q m 1 -q m 2 n = n.

(5.48)

Similarly, by Lemma 5.9(3), we have

(5.49)

(5.50) Moreover, by Lemma 5.9(3), we have ¬ u 1 (σ q m 1 )u 1 (σ

σ q k 1 +m 1 -q k 2 +m 2 n q k 1 +k 2 = 8 < :

( 1 q m 1 -m 2 + 1 q n-(m 1 -m 2 ) )

q 2+n (q 2 -1)(q n -1) if m 1 > m 2 ,

( 1 q m 2 -m 1 + 1 q n-(m 2 -m 1 ) )

q 2+n (q 2 -1)(q n -1) if m 1 ≤ m 2 .

(5.51) This means that (similar to the reduction process of (5.50)) X 0≤m 1 ,m 2 ≤n-1 u 1 (σ q m 1 )u 1 (σ q m 2 ) n = nq 2 (q -1) 2 .

(5.52) By substituting (5.48), (5.50) and (5.52) into (5.46), we have X 0≤m 1 ,m 2 ≤n-1

A m 1 A m 2 n = nq 4 .

(5.53) By (5.45) and (5.53), it follows that (5.43) holds. The proof of (5.30) is completed.

Résumé

Cette thèse est constituée de cinq parties distinctes.

La première partie est consacrée au problème de rigidité quasi-symétrique associé à un nouveau modèle de tapis de Sierpinski, qui ne sont pas quasisymétriquement équivalent aux tapis de Sierpinski usuels.

La seconde partie est une discussion portant sur la géométrie quasi-symétrique des ensembles de tapis de Julia, incluant en outre le quasi-cercle uniforme, ainsi que certaines propriétés de séparation uniforme.

Lors de la troisième partie, nous déterminerons une condition permettant de savoir quand deux rayons externes d'un polynôme tendent vers un même point. Comme application, nous montrerons également la monotonie de l'entropie associée à une famille de polynômes quadratiques.

La quatrième partie est inspirée du travail récent de Cui Guizhen et Tan Lei. En utilisant des outils classiques (module d'anneau et chirurgie quasi-conforme), nous étudierons la convergence de certains rayons en campagne locus espace des paramètres.

Enfin, la dernière partie pore sur la famille des transformations de renormalisations générées. Plus précisément, cette partie abordera la connexité de ces ensembles de Julia, et le lieu de confinement dans l'espace des paramètres, ainsi que la formule asymptotique de la dimension d'Hausdorff des ensembles de Julia.

Mots clés

rigidité quasisymétrique, sierpinski usuels, tapis de Julia, rayons externes, rayons paramètres, transformations de renormalisations générées.

Abstract

The thesis consists of five parts.

The first part is concerned with the quasisymmetric rigidity of a new Sierpinski carpet, which are not quasisymmetrically equivalent to the standard Sierpinski carpets.

The second part discusses the quasisymmetrically geometry of the carpet Julia sets, including the uniformly quasicircle and uniformly separated properties.

The third part is to determine when two external rays of a polynomial land at the same point. As an application, we also show the monotonicity of core-entropy on a family of quadratic polynomials.

In the fourth part, following Cui and Tan's work, we use the classic tools modulus of annulus and quasiconformal surgery to study the landing of some parameter rays in shift locus parameter space.

The last part discusses a family of generated renormalization transformations. Specifically, it is on the connectivity of its Julia sets and the non-escaping locus in its parameter space, the asymptotic formula of the Hausdorff dimention of the Julia sets.

Key Words

Quasisymmetric rigidity, sierpinski carpets, carpet Julia sets, external rays, parameter rays, generated renormalization transformations.