Wenjuan Peng

Tan Lei

Combinatorial rigidity of multicritical maps 1

In this paper, we combine the KSS nest constructed in [KSS] and the analytic method in [AKLS] to prove the combinatorial rigidity of multicritical maps.

Introduction

Rigidity is one of the fundamental and remarkable phenomena in holomorphic dynamics. The general rigidity problem can be posed as Rigidity problem [L]. Any two combinatorially equivalent rational maps are quasi-conformally equivalent. Except for the Lattès examples, the quasi-conformal deformations come from the dynamics of the Fatou set.

In the quadratic case, the rigidity problem is equivalent to the famous hyperbolic conjecture. The MLC conjecture asserting that the Mandelbrot set is locally connected is stronger than the hyperbolic conjecture (cf. [DH]). In 1990, Yoccoz [Hu] proved MLC for all parameter values which are at most finitely renormalizable. Lyubich [L] proved MLC for infinitely renormalizable quadratic polynomials of bounded type. In [KSS], Kozlovski, Shen and van Strien gave the proof of the rigidity for real polynomials with all critical points real. In [AKLS], Avila, Kahn, Lyubich and Shen proved that any unicritical polynomial f c : z → z d + c which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid, which implies that the connectedness locus (the Multibrot set) is locally connected at the corresponding parameter values. The rigidity problem for the rational maps with Cantor Julia sets is totally solved (cf. [YZ], [Z]). In [Z], Zhai took advantage of a length-area method introduced by Kozlovski, Shen and van Strien (cf. [KSS]) to prove the quasi-conformal rigidity for rational maps with Cantor Julia sets. Kozlovski and van Strien proved that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate (cf. [KS]).

In the following, we list some other cases in which the rigidity problem is researched (see also [Z]). (i) Robust infinitely renormalizable quadratic polynomials [START_REF] Mcmullen | Complex Dynamics and Renormalizations[END_REF]. (ii) Summable rational maps with small exponents [GS]. (iii) Holomorphic Collet-Eckmann repellers [PR]. (iv) Uniformly weakly hyperbolic rational maps [Ha].

In [PT], we have discussed the combinatorial rigidity of unicritical maps. In this paper, we will give a proof of the combinatorial rigidity of multicritical maps (see the definition in section 2).

In the proof, we will exploit the powerful combinatorial tool called "puzzle" and a sophisticated choice of puzzle pieces called the KSS nest constructed in [KSS] (see Theorem 7.11). To get the quasi-conformal conjugation, we adapt the analytic method in [AKLS] (see Lemma 3.2).

The paper is organized as follows. In section 2, we introduce the definition of the multicritical maps which we study in this paper and present the results of this paper, Theorems 2.1, 2.2, 2.3. In section 3, we apply the Spreading Principle appeared in [KSS] to prove Theorem 2.1. In section 4, we resort to the quasiconformal surgery to prove Theorem 2.2. Proof of Theorem 2.3 (a) is given in section 5. We reduce Theorem 2.3 (b) to Main Proposition in section 6. The proof of Main Proposition is presented in section 7. In subsection 7.1, we reduce Main Proposition to Proposition 7.3. The proof of Proposition 7.3 is given in subsection 7.2.

Statement

The Set up. V = ⊔ i∈I V i is the disjoint union of finitely many Jordan domains in the complex plane C with disjoint and quasi-circle boundaries, U is compactly contained in V, and is the union of finitely many Jordan domains with disjoint closures;

f : U → V is a proper holomorphic map with all critical points contained in K f := {z ∈ U | f n (z) ∈ U ∀n}, with each V-component containing at most one connected component of K f containing critical points.

Denote by Crit(f ) the set of critical points of f and by P := n≥1 c∈Crit(f ) {f n (c)} the postcritical set of f .

Let intK f denote the interior of K f . For x ∈ K f , denote by K f (x) the component of K f containing the point x. We call a component of K f a critical component if it contains a critical point. The map f maps each component of K f onto a component of K f . A component K of K f is called periodic if f p (K) = K for some p ≥ 1, wandering if f i (K) ∩ f j (K) = ∅ for all i = j ≥ 0.

Two maps in the set-up (f : U → V), ( f : Ũ → Ṽ) are said to be c-equivalent (combinatorially equivalent), if there is a pair of orientation preserving homeomorphisms h 0 , h 1 : V → Ṽ such that

       h 1 (U) = Ũ and h 1 (P) = P h 1 is isotopic to h 0 rel ∂V ∪ P h 0 • f • h -1 1 | Ũ = f h 1 | V\U is C 0 -qc (an abbreviation of quasi-conformal) for some C 0 ≥ 1, (1)
in particular

V ⊃ U h 1 -→ Ũ ⊂ Ṽ f ↓ ↓ f V -→ h 0 Ṽ commutes.
This definition is to be compared with the notion of combinatorial equivalence introduced by McMullen in [START_REF] Mcmullen | Self-similarity of Siegel disks and Hausdorff dimension of Julia sets[END_REF]. Notice that this definition is slightly different from the definitions of combinatorial equivalence in [AKLS] and [KS], since we define it without using the external rays and angles. We say that f and f are qc-conjugate off K f if there is a qc map H :

V\K f → Ṽ\K f so that H • f = f • H on U\K f , i.e. U\K f H -→ Ũ\K f f ↓ ↓ f V\K f -→ H Ṽ\K f commutes.
We say that f and f are qc-conjugate off intK f if there is a qc map H :

V → Ṽ so that H • f = f • H on U\intK f , i.e. U\intK f H -→ Ũ\intK f f ↓ ↓ f V\intK f -→ H Ṽ\intK f commutes.
We say that f and f are qc-conjugate if there is a qc map

H ′ : V → Ṽ so that H ′ • f = f • H ′ on U, i.e. V ⊃ U H ′ -→ Ũ ⊂ Ṽ f ↓ ↓ f V -→ H ′ Ṽ commutes.
Theorem 2.1. Let f, f be two maps in the set-up. Suppose f and f are qc-conjugate off K f by a qc map H. Assume that f satisfies the following property ( * ):

For every critical component K f (c) of K f , c ∈ Crit(f )
, and every integer n ≥ 1, there exists a puzzle piece Q n (c) containing c such that:

(i) For every critical component

K f (c), the pieces {Q n (c)} n≥1 form a nested sequence with n Q n (c) = K f (c) (the depth of Q n (c) may not equal to n). (ii) For each n ≥ 1, the union c∈Crit(f ) Q n (c) is a nice set. (iii) There is a constant C, such that for each pair (n, K f (c)) with n ≥ 1, a critical component K f (c), the map H| ∂Qn(c) admits a C-qc extension inside Q n (c).
Then the map H extends to a qc map from V onto Ṽ which is a conjugacy off intK f . See Definition 1 (1) and (2) in the next section for the definitions of a puzzle piece, the depth of it and a nice set.

Theorem 2.2. Let f be a map in the set-up. Then for any component

D of intK f , f i (D) ∩ f j (D) = ∅ for all i = j ≥ 0.
Theorem 2.3. Let f, f be two maps in the set-up. Then the following statements hold.

(a) If f and f are c-equivalent, then they are qc-conjugate off

K f . (b) Suppose H : V\K f → Ṽ\K f is a qc conjugacy off K f . Assume that for every critical component K f (c), c ∈ Crit(f ), satisfying that f l (K f (c)) is a critical periodic component of K f for some l ≥ 1,
there are a constant M c and an integer N c ≥ 0 such that for each n ≥ N c , the map H| ∂Pn(c) admits an M c -qc extension inside P n (c), where P n (c) is a puzzle piece of depth n containing c. Then the map H extends to a qc conjugacy off intK f . Furthermore, if for every component K of K f with non-empty interior, H| ∂K extends to a qc conjugacy inside K, then f and f are qc-conjugate by an extension of H.

Proof of Theorem 2.1

Suppose that f and f are qc-conjugate off K f by a C 0 -qc map H. Starting from the property ( * ), we will prove that H admits a qc extension across K f which is a conjugacy off intK f . Definition 1. (1) For every n ≥ 0, we call each component of f -n (V) a puzzle piece of depth n for f . Similarly, we call each component of f -n ( Ṽ) a puzzle piece of depth n for f . Denote by depth(P ) the depth of a puzzle piece P .

We list below three basic properties of the puzzle pieces.

(a) Every puzzle piece is a quasi-disk and there are finitely many puzzle pieces of the same depth.

(b) Given two puzzle pieces P and Q with depth(P ) > depth(Q), either P ⊂⊂ Q or P ∩ Q = ∅.

(c) For x ∈ K f , for every n ≥ 0, there is a unique puzzle piece of depth n containing x. Denote the piece by P n (x). Then P n+1 (x) ⊂⊂ P n (x) and ∩ n≥0 P n (x) is exactly the component of K f containing x.

(2) Suppose X ⊂ V is a finite union of puzzle pieces (not necessarily of the same depth). We say X is nice if for any z ∈ ∂X and any n ≥ 1, f n (z) / ∈ X as long as f n (z) is defined, that is, for any component P of X, for any n ≥ 1, f n (P ) is not strictly contained in X. For example, if X has a unique component, obviously it is a nice set.

( Given an open set X consisting of finitely many puzzle pieces, let

D(X) = {z ∈ V | ∃k ≥ 0, f k (z) ∈ X} = ∪ k≥0 f -k (X). For z ∈ D(X)\X, let k(z) be the minimal positive integer such that f k(z) (z) ∈ X. Set L z (X) := Comp z (f -k(z) (Comp f k(z) (z) (X))).
Obviously,

f k(z) (L z (X)) = Comp f k(z) (z) (X).
Lemma 3.1. Suppose X is a finite union of puzzle pieces. The following statements hold.

(1) For any

z ∈ D(X)\X, L z (X), f (L z (X)), • • • , f k(z)-1 (L z (X)) are pairwise disjoint.
(2) Suppose X is nice and z ∈ D(X)\X. Then for all

0 ≤ i < k(z), f i (L z (X)) ∩ X = ∅. In particular, if X ⊃ Crit(f ), then L z (X) is conformally mapped onto a component of X by the iterate of f k(z) . Proof. (1) Assume there exist 0 ≤ i < j < k(z) with f i (L z (X)) ∩ f j (L z (X)) = ∅. Then f i (L z (X)) ⊂⊂ f j (L z (X)) and f k(z)-j (f i (L z (X))) ⊂⊂ f k(z)-j (f j (L z (X))) = f k(z) (L z (X)) = Comp f k(z) (z) (X). So f k(z)-j+i (z) ∈ X. But 0 < k(z) -j + i < k(z)
. This is a contradiction with the minimality of k(z).

(2) Assume there is some 0

≤ i 0 < k(z) with f i 0 (L z (X)) ∩ X = ∅. We can show that f i 0 (L z (X)) ∩ X ⊂⊂ f i 0 (L z (X)).
In fact, when i 0 = 0, this is due to the minimality of k(z); when i 0 = 0, it is because z ∈ X. Let P be a component of X with

P ⊂⊂ f i 0 (L z (X)). So f k(z)-i 0 (P ) ⊂⊂ f k(z)-i 0 (f i 0 (L z (X))) = Comp f k(z) (z) (X). It contradicts the condition that X is nice.
The corollary below follows directly from the above lemma.

Corollary 3.2. Suppose X is a finite union of puzzle pieces. The following statements hold.

(i) For any

z ∈ D(X)\X, {L z (X), f (L z (X)), • • • , f k(z)-1 (L z (X)
)} meets every critical point at most once and

deg(f k(z) : L z (X) → Comp f k(z) (z) (X)) ≤ ( max c∈Crit(f ) deg c (f )) #Crit(f ) (ii) Suppose X is nice and z ∈ D(X)\X. Then L w (X) = L z (X) for all w ∈ L z (X) and L w ′ (X) ∩ L z (X) = ∅ for all w ′ ∈ L z (X).
(iii) Suppose X is nice and z ∈ D(X)\X. Then for all

0 < i < k(z), f i (L z (X)) = L f i (z) (X). Let K be a critical component of K f and c 1 , c 2 , • • • , c l be all the critical points on K. Then P n (c 1 ) = P n (c 2 ) = • • • = P n (c l ) and deg(f | Pn(c 1 ) ) = (deg c 1 (f ) -1) + • • • + (deg c l (f ) -1) + 1
for all n ≥ 0. We can view K as a component containing one critical point of degree deg(f | Pn(c 1 ) ). Hence in the following, we assume that each V-component contains at most one critical point. Now we will combine the property ( * ) and the Spreading Principle appeared in [KSS] to prove Theorem 2.1.

Proof of Theorem 2.1. First fix n ≥ 1. We shall repeat the proof of the Spreading Principle in [KSS] to get a qc map H n from V onto Ṽ.

Set W n := c∈Crit(f ) Q n (c). Then by Lemma 3.1 (2), each component of D(W n ) is mapped conformally onto a component of W n by some iterate of f .

For every puzzle piece P , we can choose an arbitrary qc map φ P : P → P with φ P | ∂P = H| ∂P since H is a qc map from a neighborhood of ∂P to a neighborhood of ∂ P and ∂P, ∂ P are quasi-circles (see e.g. [CT], Lemma C.1). Note that by the definition of W n , there are finitely many critical puzzle pieces not contained in W n . So we can take C ′ n to be an upper bound for the maximal dilatation of all the qc maps φ P , where P runs over all puzzle pieces of depth 0 and all critical puzzle pieces not contained in W n .

Given a puzzle piece P , let 0 ≤ k ≤ depth(P ) be the minimal nonnegative integer such that f k (P ) is a critical puzzle piece or has depth 0. Set τ (P ) = f k (P ). Then f k : P → τ (P ) is a conformal map and so is f k : P → τ ( P ), where P is the puzzle piece bounded by H(∂P ) for f and τ ( P ) = f k ( P ). Given a qc map q : τ (P ) → τ ( P ), we can lift it through the maps f k and f k , that is, there is a qc map p : P → P such that f k • p = q • f k . Notice that the maps p and q have the same maximal dilatation, and if q| ∂τ (P ) = H| ∂τ (P ) , then p| ∂P = H| ∂P .

Let Y 0 = V denote the union of all the puzzle pieces of depth 0. Set X 0 = ∅. For j ≥ 0, we inductively define X j+1 to be the union of puzzle pieces of depth j + 1 such that each of these pieces is contained in Y j and is a component of D(W n ); set Y j+1 := (Y j ∩ f -(j+1) (V))\X j+1 . We have the following relations: for any j ≥ 0,

Y j = (Y j \f -(j+1) (V)) ⊔ X j+1 ⊔ Y j+1 , Y j+1 ⊂⊂ Y j , X j ′ ∩ X j = ∅ for any j ′ = j .
Given any component Q of Y j+1 , we claim that τ (Q) is either one of the finitely many critical puzzle pieces not contained in W n , or one of the finitely many puzzle pieces of depth 0. In fact, for such

Q, either Q ∩ D(W n ) = ∅ or Q ∩ D(W n ) = ∅.
In the former case, since Crit(f ) ⊂ W n ⊂ D(W n ), Q is mapped conformally onto a puzzle piece of depth 0 by the iterate of f depth (Q) . So τ (Q) is a puzzle piece of depth 0. In the latter case, if

Q ∩ D(W n ) ⊂⊂ D(W n ), then Q is compactly contained in a component of D(W n ), denoted by Q ′ , and Q ′ ⊂⊂ X j ′ for some j ′ < j + 1. But Q ⊂⊂ Y j ⊂⊂ Y j-1 ⊂⊂ • • • ⊂⊂ Y 0 and X j ∩Y j = ∅, X j-1 ∩Y j-1 = ∅, • • • , X 0 ∩Y 0 = ∅. This is a contradiction. Hence Q ∩ D(W n ) ⊂⊂ Q. If there is a critical point c ∈ Q ∩ D(W n ), then the component of W n containing c is compactly contained in Q and τ (Q) = Q. Otherwise, τ (Q) must be a critical puzzle piece not contained in W n .
Define H (0) = φ P on each component P of Y 0 . For each j ≥ 0, assuming that H (j) is defined, we define H (j+1) as follows:

H (j+1) =        H (j) on V\Y j H on Y j \f -(j+1) (V) the univalent pullback of φ on each component of X j+1 the univalent pullback of φ τ (Q) on each component Q of Y j+1 ,
where the map φ is the qc-extension obtained by the assumption ( * ).

Set C n = max{C 0 , C ′ n , C}. The {H (j) } j≥0 is a sequence of C n -qc maps. Hence it is precompact in the uniform topology.

By definition, H (j) = H (j+1) outside Y j . Thus, the sequence {H (j) } converges pointwise outside

j Y j = {x ∈ K f | f k (x) / ∈ W n , k ≥ 0}.
This set is a hyperbolic subset, on which f is uniformly expanding, and hence has zero Lebesgue measure, in particular no interior. So any two limit maps of the sequence {H (j) } j≥0 coincide on a dense open set of V, therefore coincides on V to a unique limit map. Denote this map by H n . It is C n -qc. By construction, H n coincides with H on V\(( j X j ) ∪ ( j Y j )) is therefore C 0 -qc there; and is C-qc on j X j . It follows that the maximal dilatation of H n is bounded by max{C 0 , C} except possibly on the set j Y j . But this set has zero Lebesgue measure. It follows that the maximal dilatation of H n is max{C 0 , C}, which is independent of n.

The sequence H n : V → Ṽ has a subsequence converging uniformly to a limit qc map

H ′ : V → Ṽ, with H ′ | V\K f = H. Therefore H ′ is a qc extension of H. On the other hand, H • f = f • H on U\K f . So H ′ • f = f • H ′ holds
on U\intK f by continuity. Therefore H ′ is a qc-conjugacy off intK f . This ends the proof of Theorem 2.1.

Proof of Theorem 2.2

In this section let f : U → V be a map in the set-up. Let q ≥ 1 denote the number of components of

V. Enumerate the V-components by V 1 , V 2 , • • • , V q . Lemma 4.1. Let W be an open round disk centered at 0 with radius > 1 containing V. The map f : U → V extends to a map F on V so that -on each component V i of V, the restriction F | V i : V i → W is a quasi-regular branched covering; -every component of U is a component of F -1 (V); -the restriction F on F -1 (V) is holomorphic. Proof. Part I. Fix any component V i of V such that V i ∩ U = ∅. We will extend f | V i ∩U
to a map F on V i with the required properties. It will be done in three steps. Refer to Figure 1 for the construction of F on V i .

E V1 D u1 u2 V1 V2 v1 v2 D W Ψ1 Ψ2 F G Figure 1 The construction of F on V 1 . In this figure, q = 2, i = 1.
Step 1. The first step is to construct a Blaschke product G : D → D of degree d i , where D denotes the unit disk and d i is determined below.

For every component V j of V, we define

q ij = #{U a component of U | U ⊂ V i , f (U ) = V j } Set q i = max j q ij . Then q i ≥ 1 and #{components of U ∩ V i } = j q ij ≤ q • q i .
We construct a Blaschke product G : D → D, as well as a set D which is the union of q Jordan domains in D with pairwise disjoint closures, as follows:

• If V i does not contain critical points of f , then set G(z) = z q i , and choose D to be a collection of q Jordan domains compactly contained in D\{0} with pairwise disjoint closures. Set

d i = q i . Note that each component of D has exactly q i preimages. So #{components of G -1 (D)} = q • q i ≥ #{components of U ∩ V i } .
• Otherwise, by assumption in the set-up, the set Crit(f ) intersects exactly one component

U of U ∩ V i . Set d i = q i + deg f | U -1.
Choose G so that it has degree d i , and has two distinct critical points u 1 and u 2 such that deg

u 1 (G) = q i , deg u 2 (G) = deg(f | U ) and G(u 1 ) = G(u 2 ). 2 Set v i = G(u i ), i = 1, 2. Now choose D
to be a collection of q Jordan domains compactly contained in D\{v 1 } with pairwise disjoint closures and with v 2 ∈ D. Note that the preimage of any D-component not containing v 2 has d i components, whereas the preimage of the D-component

containing v 2 has d i -deg(f | U ) + 1 = q i components. So #{components of G -1 (D)} = (q -1)d i + q i = (q -1)(q i + deg f | U -1) + q i = q • q i + (q -1)(deg f | U -1) > q • q i ≥ #{components of U ∩ V i } .
In both cases G : D\G -1 (D) → D\D is a proper map with a unique critical point.

Step 2. Make U, V 'thick'.

In W, take q Jordan domains with smooth boundaries

V j , j = 1, • • • , q, such that each V j is compactly contained in W, V j ⊂ V j for each j = 1, • • • , q,
and all of the V j have pairwise disjoint closures. Denote V = ∪ q j=1 V j . In V i , take U to be a union of #{components of G -1 (D)} (which is greater than the number of U-components in V i ) Jordan domains with smooth boundaries with the following properties:

-U is compactly contained in V i ; -(U ∩ V i ) is compactly contained in U; -each component of U contains at most one component of (U ∩ V i ); -the components of U have pairwise disjoint closures.
There exists a qc map Ψ 2 :

W → D such that Ψ 2 ( V) = D.
Let now U be any component of U. There is a unique component U of U containing U . Also f (U ) = V j ⊂ V j for some j, and Ψ 2 ( V j ) is a component, denoted by D(U ), of D. See the following diagram:

U ⊂ U G -1 (D(U )) ↓ f ↓ G V j ⊂ V j Ψ 2 -→ D(U )
There is a qc map Ψ 1 :

V i → D so that Ψ 1 ( U) = G -1 (D) and, for any component U of V i ∩ U, the set Ψ 1 ( U ) is a component of G -1 (D(U )). Then we can define a quasiregular branched covering F : V i \ U → W\ V of degree d i to be Ψ -1 2 • G| D\G -1 (D) • Ψ 1 .
Step 3. Glue. Define at first contained in E. Then F maps ∂ E homeomorphically onto ∂ V j for some j. Define F to be a conformal map from E onto V j by Riemann Mapping Theorem and F extends homeomorphically from E onto V j .

F = f on V i ∩ U. For each component E of U not containing a component of U,
Notice that the map F is defined everywhere except on a disjoint union of annular domains, one in each component of U. Furthermore F maps the two boundary components of each such annular domain onto the boundary of V j \V j for some j, and is a covering of the same degree on each boundary component.

This shows that F admits an extension as a covering of these annular domains. As all boundary curves are smooth and F is quasi-regular outside the annular domains, the extension can be made quasi-regular as well.

Part II. We may now extend F to every V component intersecting U following the same procedure as shown in Part I. Assume that V i is a V-component disjoint from U. We define F : V i → W to be a conformal homeomorphism and we set d i = 1. We obtain a quasi-regular map F : V → W as an extension of f :

U → V. By construction, F is holomorphic on F -1 (V).
Lemma 4.2. There is an integer d so that for the map g : z → z d , the map F has an extension on W\V so that F : W\V → g(W)\W is a quasi-regular branched covering, coincides with g on ∂W and is continuous on W. In particular

F -1 (W) = V and F is holomorphic on F -2 (W) = F -1 (V). Proof. Set d = q i=1 d i ,
where the d i 's are defined in the proof of Lemma 4.1. See Figure 2 for the proof of this lemma.

V1 V2 W W g(W) F P Ψ1 Ψ2 Figure 2
The domain V is defined as in the proof of the previous lemma. Now take a Jordan domain W with smooth boundary such that W ⊂⊂ W ⊂⊂ g(W).

Let

P (z) = (z -a 1 ) d 1 (z -a 2 ) d 2 • • • (z -a q ) dq ,
where a 1 , a 2 , • • • , a q ∈ C are distinct points.

Note that for each 1 ≤ i ≤ q, we have P (a i ) = 0, and a i is a critical point of P whenever d i > 1.

Take r > 0 small enough and R > 0 large enough such that {0 < |z| ≤ r} ∪ {R ≤ z| < ∞} contains no critical value of P . Obviously P : P -1 ({r ≤ |z| ≤ R}) → {r ≤ |z| ≤ R} is a holomorphic proper map of degree d.

Note that P -1 ({|z| ≤ R} is a closed Jordan domain, and the set P -1 ({|z| ≤ r}) consists of q disjoint closed Jordan domains, each containing exactly one of the a i 's in the interior.

There exist qc maps Φ 1 :

W → P -1 ({|z| ≤ R}), Φ 2 : g(W) → {|z| ≤ R} such that for i = 1, • • • , q, the set Φ 1 ( V i ) is equal to the component of P -1 ({|z| ≤ r})
containing a i , and Φ 2 ( W) = {|z| ≤ r}, and that

P (Φ 1 (z)) = Φ 2 (g(z)), z ∈ ∂W. Set F = Φ -1 2 • P • Φ 1 on W\ V. Fix any i = 1, • • • , q. Both maps F : ∂ V i → ∂ W and F : ∂V i → ∂W are coverings of degree d i .
We may thus extend as before F to a qusiregular covering map from V i \V i onto W\W.

This ends the construction of F .

Proof of Theorem 2.2. Extend the map F in Lemma 4.2 to C by setting F = g on C\W.

This F is quasi-regular, and is holomorphic on (C\W)∪F -2 (W). So every orbit passes at most twice the region W\F -2 (W). By Surgery Principle (see Page 130 Lemma 15 in [Ah]), the map F is qc-conjugate to a polynomial h. The set K F can be defined as for h and the two dynamical systems F | K F and h| K h are topologically conjugate.

Theorem 2.2 holds for the pair (h, K h ) (in place of (f, K f ) ) by Sullivan's nowandering-domain theorem. It follows that the result also holds for (F, K F ). But K f is an F -invariant subset of K F with every component of K f being a component of K F , and with F | K f = f | K f . So the theorem holds for the pair (f, K f ).

Proof of Theorem 2.3 (a)

We just repeat the standard argument (see for example Appendix in [START_REF] Mcmullen | Self-similarity of Siegel disks and Hausdorff dimension of Julia sets[END_REF]).

Assume that f, f are c-equivalent. Set U = U 1 , and U n = f -n (V). The same objects gain a tilde for f . For t ∈ [0, 1], let h t : V → Ṽ be an isotopy path linking

h 0 to h 1 .
Then there is a unique continuous extension (t, z)

→ h(t, z), [0, ∞[×V → Ṽ such that 0) each h t : z → h(t, z) is a homeomorphism, 1) h t | ∂V∪P = h 0 | ∂V∪P , ∀t ∈ [0, +∞[, 2) for n ≥ 1, t > n and x ∈ V\U n we have h t (x) = h n (x), 3) for t ∈ [0, 1] the following diagram commutes: . . . . . . U 2 h t+2 -→ Ũ2 ↓ f ↓ f U 1 h t+1 -→ Ũ1 ↓ f ↓ f V ht -→ Ṽ.
Set then Ω = n≥1 V\U n = V\K f , and Ω = Ṽ\K f . Then there is a qc map

H : Ω → Ω such that H(x) = h n (x) for n ≥ 1 and x ∈ V\U n and that H • f | Ω∩U = f • H| Ω∩ Ũ, i.e. H realizes a qc-conjugacy from f to f off K f . The qc constant of H is equal to C 0 , the qc constant of h 1 on V\U. 6 Proof of Theorem 2.3 (b)
Main Proposition. Let f be a map as in the set-up. Assume that for every

critical component K f (c), c ∈ Crit(f ), satisfying that f l (K f (c)) is a critical periodic component of K f for some l ≥ 1,
there are a constant M c and an integer N c ≥ 0 such that for each n ≥ N c , the map H| ∂Pn(c) admits an M c -qc extension inside P n (c), where P n (c) is a puzzle piece of depth n containing c. Then f satisfies the property ( * ) stated in Theorem 2.1.

We will postpone the proof of Main Proposition in the next section. Here we combine this proposition and Theorem 2.1 to give a proof of Theorem 2.3 (b).

Proof of Theorem 2.3 (b). Combining Main Proposition and Theorem 2.1, we have a qc conjugacy off K f admits a qc extension across K f which is a conjugacy off intK f .

Proof of Main Proposition

In this section, we always assume f is a map in the set-up with the assumption that each V-component contains at most one critical point.

Reduction of Main Proposition

Definition 2. (1) For x, y ∈ K f , we say that the forward orbit of x combinatorially accumulates to y, written as x → y, if for any n ≥ 0, there is j ≥ 1 such that f j (x) ∈ P n (y).

Clearly, if x → y and y → z, then x → z.

Let Forw(x) = {y ∈ K f | x → y} for x ∈ K f .
(2) Define an equivalence relation in Crit(f ) as follows:

for c 1 , c 2 ∈ Crit(f ), c 1 ∼ c 2 ⇐⇒ c 1 = c 2 , or c 1 → c 2 and c 2 → c 1 . Let [c] denote the equivalence class containing c for c ∈ Crit(f ). It is clear that [c] = {c} if c → c. (3) We say that [c 1 ] accumulates to [c 2 ], written as [c 1 ] → [c 2 ], if ∃ c ′ 1 ∈ [c 1 ], ∃ c ′ 2 ∈ [c 2 ] such that c ′ 1 → c ′ 2 .
It is easy to check that if

[c 1 ] → [c 2 ], then ∀ c ′′ 1 ∈ [c 1 ], ∃ c ′′ 2 ∈ [c 2 ] such that c ′′ 1 → c ′′ 2 .
It follows from this property that if

[c 1 ] → [c 2 ], [c 2 ] → [c 3 ], then [c 1 ] → [c 3 ]. (4) Define D(f ) := Crit(f )/ ∼. Define a partial order ≤ in D(f ): [c 1 ] ≤ [c 2 ] ⇐⇒ [c 1 ] = [c 2 ] or [c 2 ] → [c 1 ].
We can decompose the quotient D(f ) as follows. Let D 0 (f ) be the set of elements in D(f ) which are minimal in the partial order

≤, that is, [c] ∈ D 0 (f ) if and only if [c] doesn't accumulate to any element in D(f )\{[c]}. For every k ≥ 0, assume D k (f ) is defined, then D k+1 (f ) is defined to be the set of elements in D(f ) which are minimal in the set D(f ) \ (D k (f ) ∪ D k-1 (f ) ∪ • • • ∪ D 0 (f )) in the partial order ≤.
For the construction above, we can prove the properties below.

Lemma 7.1. (P1) There is an integer

M ≥ 0 such that D(f ) = M k=0 D k (f ). (P2) For every k ≥ 0, given [c 1 ], [c 2 ] ∈ D k (f ), [c 1 ] = [c 2 ], we have [c 1 ] → [c 2 ] and [c 2 ] → [c 1 ]. (P3) Let [c 1 ] ∈ D s (f ), [c 2 ] ∈ D t (f ) with s < t. Then [c 1 ] → [c 2 ]. (P4) For every k ≥ 1, every [c] in D k (f ) accumulates to some element in D k-1 (f ).
Proof. (P1) holds because D(f ) is a finite set and D i (f ) ∩ D j (f ) = ∅ for i = j.

(P2) and (P3) follow directly from the minimal property of the elements in D k (f ) for every 0 ≤ k ≤ M .

(P4) Let k = 1. If there is some element [c 1 ] ∈ D 1 (f ) such that it doesn't accumulate to any element in D 0 (f ), then combining with (P2) and (P3), we have [c 1 ] doesn't accumulate to any element in

D(f )\{[c 1 ]}. Consequently, [c 1 ] ∈ D 0 (f ). But [c 1 ] ∈ D 1 (f ) and by (P1), D 0 (f ) ∩ D 1 (f ) = ∅.
We get a contradiction. So any element in D 1 (f ) will accumulate to some element in D 0 (f ). Now we suppose k ≥ 2 and (P4) holds for

D 1 (f ), D 2 (f ), • • • , D k-1 (f ). Assume (P4) is not true for D k (f ), that is, there is some [c k ] ∈ D k (f ) such that [c k ] doesn't accumulate to any element in D k-1 (f ). If [c k ] doesn't accumulate to any element in ∪ k-2 j=0 D j (f ), then by (P2) and (P3), we conclude that [c k ] ∈ D 0 (f ) which contradicts the condition that [c k ] ∈ D k (f ) and the fact that D k (f ) ∩ D 0 (f ) = ∅ by (P1). Let 0 ≤ i ≤ k -2 be an integer satisfying that [c k ] won't accumulate to any element in ∪ k-1 j=i+1 D j (f ) and [c k ] accumulates to some element in D i (f ). Then [c k ] won't accumulate to any element in ∪ M j=i+1 D j (f )\{[c k ]} and hence [c k ] ∈ D i+1 (f ). But notice that i + 1 ≤ k -1 and [c k ] ∈ D k (f ). A contradiction.
Combining with the transitive property stated in Definition 2 (3), we can consecutively apply (P3) above and prove the following.

Corollary 7.2. For every k ≥ 1, every [c] in D k (f ) accumulates to some element in D 0 (f ).

We will deduce Main Proposition from the following result.

Proposition 7.3. Assume that for every critical component K f (c), c ∈ Crit(f ), satisfying that f l (K f (c)) is a periodic critical component of K f for some l ≥ 1, there are a constant M c and an integer N c ≥ 0 such that for each n ≥ N c , the map H| ∂Pn(c) admits an M c -qc extension inside P n (c), where P n (c) is a puzzle piece of depth n containing c. Then for every c ∈ [c 0 ] and every integer n ≥ 1, there is a puzzle piece K n (c) containing c with the following properties.

(i) For every c ∈ [c 0 ], the pieces

{K n (c)} n≥1 is a nested sequence. (ii) For each n ≥ 1, c∈[c 0 ] K n (c) is a nice set. (iii) There is a constant M = M ([c 0 ]
), such that for each n ≥ 1 and each c ∈ [c 0 ], H| ∂Kn(c) admits an M -qc extension inside K n (c).

We will postpone the proof of Proposition 7.3 to the next subsection. Here we prove the following lemma and then use it and Proposition 7.3 to prove Main Proposition.

Lemma 7.4. Let [c 1 ] and [c 2 ] be two distinct equivalence classes. Suppose that for each i = 1, 2, W i is a nice set consisting of finitely many puzzle pieces such that each piece contains a point in [c i ].

(1)

If [c 1 ] → [c 2 ] and [c 2 ] → [c 1 ], then W 1 ∪ W 2 is a nice set containing [c 1 ] ∪ [c 2 ]. (2) Suppose [c 2 ] → [c 1 ] and min P 2 a comp. of W 2 depth(P 2 ) ≥ max P 1 a comp. of W 1 depth(P 1 ),
i.e., the minimal depth of the components of W 2 is not less than the maximal depth of those of W 1 . Then W 1 ∪ W 2 is nice.

Before proving this lemma, we need to give an assumption for simplicity. Notice that given two critical points c, c ′ , if c → c ′ , then there is some integer n(c, c ′ ) depending on c and c ′ such that for all j ≥ 1, for all n ≥ n(c, c ′ ), f j (c) ∈ P n (c ′ ). Since #Crit(f ) < ∞, we can take n 0 = max{n(c, c ′ ) | c, c ′ ∈ Crit(f )}. Without loss of generality, we may assume that n 0 = 0, that is to say we assume that ( * * )

for any two critical points c, c ′ , for all j ≥ 1,

f j (c) ∈ P 0 (c ′ ) if c → c ′ .
In the following paragraphs until the end of this article, we always assume ( * * ) holds.

Proof of Lemma 7.4. (1) According to Definition 2 (3) and the assumption ( * * ), we know that

[c 1 ] → [c 2 ] ⇐⇒ ∀c ′ 1 ∈ [c 1 ], ∀c ′ 2 ∈ [c 2 ], c ′ 1 → c ′ 2 ⇐⇒ ∀c ′ 1 ∈ [c 1 ], ∀c ′ 2 ∈ [c 2 ], ∀n ≥ 0, ∀j ≥ 0, f j (c ′ 1 ) ∈ P n (c ′ 2 ) ⇐⇒ ∀ a puzzle piece P ∋ c ′ 1 , ∀n ≥ 0, ∀j ≥ 0, f j (P ) ∩ P n (c ′ 2 ) = ∅.
In particular, for any

c ′ 1 ∈ [c 1 ], any c ′ 2 ∈ [c 2 ], for the component P 1 of W 1 containing c ′
1 and the component P 2 of W 2 containing c ′ 2 , for any j ≥ 0, f j (P 1 ) ∩ P 2 = ∅. It is equivalent to say that for any component P of W 1 , for any j ≥ 0, f j (P ) ∩ W 2 = ∅.

Similarly, from the condition that [c 2 ] → [c 1 ], we can conclude that for any component Q of W 2 , for any j ≥ 0,

f j (Q) ∩ W 1 = ∅. Hence W 1 ∪ W 2 is a nice set.
(2) On one hand, from the proof of (1), we know that

[c 2 ] → [c 1 ] implies that for any component Q of W 2 , for any j ≥ 0, f j (Q) ∩ W 1 = ∅.
On the other hand, for any component P of W 1 , for any j ≥ 0, we have depth(f j (P )) = depth(P )j ≤ max

P 1 a comp. of W 1 depth(P 1 ) -j ≤ min P 2 a comp. of W 2 depth(P 2 )
and then f j (P ) can not be strictly contained in W 2 .

Hence W 1 ∪ W 2 is nice.

Now we can derive Main Proposition from Proposition 7.3.

Proof of Main Proposition. (i) follows immediately from Proposition 7.3 (i).

(ii) For every [c] ∈ D(f ) and every ĉ ∈ [c], let {K n (ĉ)} n≥1 be the puzzle pieces obtained in Proposition 7.3.

Given [c 0 ] ∈ D k (f ), 0 ≤ k < M , let A k ([c 0 ]) = {[c] ∈ D k+1 (f ) | [c] → [c 0 ]}. Clearly, #A k ([c 0 ]) < ∞. Recall that D(f ) = ⊔ M i=0 D i (f ). For every [c 0 ] ∈ D M (f ), set Q n (c) = K n (c) for each c ∈ [c 0 ]. Now consider [c 0 ] ∈ D M -1 (f ). If A M -1 ([c 0 ]) = ∅, then set Q n (c) = K n (c) for each c ∈ [c 0 ]. Otherwise, there exists a subsequence {l n } n≥1 of {n} such that min c ′ ∈[c 0 ] depth(K ln (c ′ )) ≥ max [c ′ ]∈A M -1 ([c 0 ]) depth(Q n (c ′ ))
because min c ′ ∈[c 0 ] depth(K n (c ′ )) increasingly tends to the infinity as n → ∞.

We repeat this process consecutively to D M -2 (f ), • • • , D 0 (f ) and then all Q n (c) are defined. Combining the properties (P2), (P3) stated in Lemma 7.1 and Lemma 7.4, we easily conclude that ∪ c∈Crit(f ) Q n (c) is a nice set for every n ≥ 1.

(iii) Since #D(f ) < ∞, we can take the constant C = max{ M (

[c]) | [c] ∈ D(f )}.

Proof of Proposition 7.3

First, we need to introduce a classification of the set Crit(f ) and several preliminary results.

Definition 3. (i) Suppose c → c. For c 1 , c 2 ∈ [c], we say that P n+k (c 1 ) is a child of P n (c 2 ) if f k (P n+k (c 1 )) = P n (c 2 ) and f k-1 : P n+k-1 (f (c 1 )) → P n (c 2 ) is conformal.
c is called persistently recurrent if for every n ≥ 0, every c ′ ∈ [c], P n (c ′ ) has finitely many children. Otherwise, c is said to be reluctantly recurrent.

It is easy to check that if c is persistently recurrent, then so is every c ′ ∈ [c] and this is also true for a reluctantly recurrent c.

(ii) Let

Crit n (f ) = {c ∈ Crit(f ) | c → c ′ for any c ′ ∈ Crit(f )}, Crit e (f ) = {c ∈ Crit(f ) | c → c and ∃ c ′ ∈ Crit(f ) such that c → c ′ }, Crit r (f ) = {c ∈ Crit(f ) | c → c and c is reluctantly recurrent}, Crit p (f ) = {c ∈ Crit(f ) | c → c and c is persistently recurrent}. Then Crit(f ) = Crit n (f )⊔Crit e (f )⊔Crit r (f )⊔Crit p (f ) is a classification of Crit(f ).
In this section, we will use sometimes the combinatorial tool -the tableau defined by Branner-Hubbard in [BH]. The reader can also refer to [QY] and [PQRTY] for the definition of the tableau.

For x ∈ K f , the tableau T (x) is the graph embedded in {(u, v) | u ∈ R -, v ∈ R} with the axis of u pointing upwards and the axis of v pointing rightwards (this is the standard R 2 with reversed orientation), with vertices indexed by -N × N, where N = {0, 1, • • • }, with the vertex at (-m, 0) being P m (x), the puzzle piece of depth m containing x, and with f j (P m (x)) occupying the (-m + j, j)th entry of T (x). The vertex at (-m + j, j) is called critical if f j (P m (x)) contains a critical point. If f j (P m (x)) contains some y ∈ K f , we call the vertex at (-m + j, j) is a y-vertex.

All tableau satisfy the following three basic rules (see [BH], [QY], [PQRTY]).

(Rule 1). In T (x) for x ∈ K f , if the vertex at (-m, n) is a y-vertex, then so is the vertex at (-i, n) for every 0 ≤ i ≤ m.

(Rule 2). In T (x) for x ∈ K f , if the vertex at (-m, n) is a y-vertex, then for every 0 ≤ i ≤ m, the vertex at (-m + i, n + i) is a vertex being P -m+i (f i (y)).

(Rule 3) (See Figure 3). Given x 1 , x 2 ∈ K f . Suppose there exist integers m 0 ≥ 1, n 0 ≥ 0, i 0 ≥ 1, n 1 ≥ 1 and critical points c 1 , c 2 with the following properties.

(i) In T (x 1 ), the vertex at (-

(m 0 +1), n 0 ) is a c 1 -vertex and (-(m 0 +1-i 0 ), n 0 + i 0 ) is a c 2 -vertex.
(ii) In T (x 2 ), the vertex at (-m 0 , n 1 ) is a c 1 -vertex and (-(m 0 + 1), n 1 ) is not critical.

If in T (x 1 ), for every 0 < i < i 0 , the vertex at (-

(m 0 -i), n 0 + i) is not critical, then in T (x 2 ), the vertex at (-(m 0 + 1 -i 0 ), n 1 + i 0 ) is not critical. T (x1) : n0 n0 + i0 -(m0 -i0) -(m0 + 1 -i0) -m0 -(m0 + 1) T (x2) : n1 n1 + i0 -(m0 -i0) -(m0 + 1 -i0) -m0 -(m0 + 1) c1 c2 c1 c2 Figure 3
Recall that in subsection 7.1, we make the assumption ( * * ). Here, we translate that assumption in the language of the tableau. It is equivalent to assume that for c, c ′ ∈ Crit(f ), c ′ -vertex appears in T (c) iff c ′ ∈ Forw(c).

Lemma 7.5. 1. Let K f (c) be a periodic component of K f with period p. Then the following properties hold.

(1)

f i (K f (c ′ )) ∈ {K f (c), f (K f (c)), • • • , f p-1 (K f (c))}, ∀c ′ ∈ Forw(c), ∀i ≥ 0. (2) Forw(c) = [c]. (3) c ∈ Crit p (f ). 2. Let c ∈ Crit p (f ) with K f (c) non-periodic.
Then the following properties hold.

(1)

Forw(c) = [c]. (2) For every c ′ ∈ [c], c ′ ∈ Crit p (f ) with K f (c ′ ) non-periodic.
Proof. 1. Notice that K f (c) is periodic iff there is a column in T (c)\{0-th column} such that every vertex on that column is a c-vertex. According to this and using the tableau rules, it is easy to check that the statements in Point 1 are true. 2.

(1) This property is the same as Lemma 1 in [QY]. For self-containedness, we introduce the proof here.

Assume there is some c ′ ∈ Crit(f ) with c → c ′ but c ′ → c. In the following, all the vertices we discuss are in T (c). One may refer to Figure 4 for the proof.

T (c) : 0 m i n i + m i n i + m i + t i -t i -n i -(n i + t i ) c1(i) ∈ W1(i) ∩ [c] c ′ / ∈ [c] c2(i) ∈ [c] Figure 4
If there exists a column such that every vertex on it is a c ′ -vertex, then c ′ → c because c → c. Hence there are infinitely many c ′ -vertices {(-n i , m i )} i≥1 such that (-(n i + 1), m i ) is not critical and lim i→∞ n i = ∞.

By the tableau rule (Rule 2) and the assumption ( * * ), we can see that there are no vertices being critical points in [c] on the diagonal starting from the vertex (-n i , m i ) and ending at the 0-th row. Since c → c, from the vertex (0, n i + m i ), one can march horizontally t i ≥ 1 steps to the right until the first hit of some c 2 (i)-vertex in [c]. Then by (Rule 1), there are no vertices being critical points in [c] on the diagonal from the vertex (-t i , n i + m i ) to the vertex (0, n i + m i + t i ). Therefore, there are no vertices being critical points in [c] on the diagonal from the vertex (-(n i + t i ), m i ) to the vertex (0, n i + m i + t i ), denote this diagonal by I.

If there exists a point c ∈ Forw(c)\[c] on the diagonal I, then by the assumption ( * * ), every vertex, particularly the end vertex (0, n i + m i + t i ) of I, can't be a ĉ-vertex for any ĉ ∈ [c]. This contradicts the choice that the vertex (0,

n i + m i + t i ) is a c 2 (i)-vertex for c 2 (i) ∈ [c].
Consequently, there are no critical points in Forw(c) on the diagonal I. Combining with the assumption ( * * ), we know that there are no critical points on the diagonal I.

Follow the diagonal from the vertex (-(n i + t i ), m i ) left downwards until we reach a critical vertex W 1 (i) (such W 1 (i) exists since the 0-th column vertex on that diagonal is critical). Let c 1 (i) be the critical point in W 1 (i). Then c 1 (i) ∈ [c] follows from the fact that (0,

n i + m i + t i ) is a c 2 (i)-vertex for c 2 (i) ∈ [c]
and the assumption ( * * ) Therefore, W 1 (i) is a child of P 0 (c 2 (i)). Notice that the depth of W 1 (i) is greater than n i . As c 2 (i) lives in the finite set [c] and n i → ∞ when i → ∞, some point in [c] must have infinitely many children. This is a contradiction with the condition that c ∈ Crit p (f ).

(2) follows directly from Point 1 (1) and Point 1 (2).

Set

Crit

per (f ) = {c ∈ Crit p (f ) | K f (c) is periodic}. Lemma 7.6. (i) If c ∈ Crit n (f ) ∪ Crit p (f ), then [c] ∈ D 0 (f ); if c ∈ Crit e (f ), then [c] ∈ D 0 (f ).
(ii) For every c 0 ∈ Crit(f ), one of the following cases will occur. Case 1. (ii) Suppose that Case 1 and Case 2 do not happen.

Forw(c 0 ) ∩ (Crit n (f ) ∪ Crit r (f )) = ∅. Case 2. Forw(c 0 ) ⊂ Crit p (f ).
Let c ∈ Forw(c 0 ) with c / ∈ Crit p (f ). Notice that Crit(f ) = Crit n (f ) ∪ Crit r (f ) ∪ Crit p (f ) ∪ Crit e (f ). So c ∈ Crit e (f ) and then by (i), [c] ∈ D k (f ) for some k ≥ 1. It follows from Corollary 7.2 that [c] = {c} accumulates to some element [c] ∈ D 0 (f ).
Since Case 1 does not happen, we conclude that for every Recall that in section 3, for an open set X consisting of finitely many puzzle pieces, we define the set D(X) and L z (X) for z ∈ D(X)\X. The following is a property about L z (X) when X consists of a single piece.

[ĉ] ∈ D 0 (f ) with [c 0 ] → [ĉ],
Lemma 7.7. Let P be a puzzle piece and the set {x 1 , • • • , x m } ⊂ V be a finite set of points with each x i ∈ D(P )\P for 1 ≤ i ≤ m. Let f k i (L x i (P )) = P for some k i ≥ 1. Then (1) for every 1 ≤ i ≤ m, every 0 ≤ j < k i , either

f j (L x i (P )) = L xs (P ) for some 1 ≤ s ≤ m, or f j (L x i (P )) ∩ L xt (P ) = ∅ for all 1 ≤ t ≤ m;
(2) ∪ m i=1 L x i (P ) P is a nice set. Proof. (1) (by contradiction). Assume that there are integers 1 ≤ i 0 ≤ m, 0 ≤ j 0 < k i 0 , and there is some

L x i 1 (P ) for 1 ≤ i 1 ≤ m, such that f j 0 (L x i 0 (P )) = L x i 1 (P ) and f j 0 (L x i 0 (P )) ∩ L x i 1 (P ) = ∅.
Then either f j 0 (L x i 0 (P )) ⊂⊂ L x i 1 (P ) or f j 0 (L x i 0 (P )) ⊃⊃ L x i 1 (P ).

We may assume f j 0 (L x i 0 (P )) ⊂⊂ L x i 1 (P ). The proof of the other case is similar to this case.

On one hand , since f k i 0 -j 0 : f j 0 (L x i 0 (P )) → P and

f k i 1 : L x i 1 (P ) → P , we have k i 0 -j 0 > k i 1 .
On the other hand, we know that k i 0 -j 0 is the first landing time of the points in f j 0 (L x i 0 (P )) to P because f j 0 (L x i 0 (P )) = L f j 0 (x i 0 ) (P ) by Corollary 3.2 (3), while from the assumption that f j 0 (L x i 0 (P )) ⊂⊂ L x i 1 (P ), we have k i 1 is also the first landing time of the points in f j 0 (L x i 0 (P )) to P . So k i 0j 0 = k i 1 . A contradiction.

(2) For any q ≥ 1 (as long as depth(f q (P )) ≥ 0), depth(f q (P )) < depth(P ) < depth(L xs (P ))

for every 1 ≤ s ≤ m. So f q (P ) can not be strictly contained in ∪ m i=1 L x i (P ) for all q ≥ 1.

Fix 1 ≤ i ≤ m. For 1 ≤ j < k i , by (1), we know that f j (L x i (P )) is not strictly contained in ∪ m i=1 L x i (P ). Since P is a single puzzle piece and then it is nice, by Lemma 3.1 (2), we have f j (L x i (P )) ∩ P = ∅. When j ≥ k i , notice that as long as depth(f j (L x i (P ))) ≥ 0, we have depth(f j (L x i (P ))) ≤ depth(P ) < depth(L xs (P ))

for every 1 ≤ s ≤ m which implies that f j (L x i (P )) is not strictly contained in ∪ m i=1 L x i (P ) P .

Lemma 7.8. Let Q, Q ′ , P, P ′ be puzzle pieces with the following properties.

(a) Q ⊂⊂ Q ′ , c 0 ∈ P ⊂⊂ P ′ for c 0 ∈ Crit(f ). (b) There is an integer l ≥ 1 such that f l (Q) = P, f l (Q ′ ) = P ′ . (c) (P ′ \P ) ∩ (∪ c∈Forw(c 0 ) ∪ n≥0 {f n (c)}) = ∅. Then for all 0 ≤ i ≤ l, (f i (Q ′ )\f i (Q)) ∩ Forw(c 0 ) = ∅. Proof. If f l-1 (Q ′ )\f l-1 (Q) contains some c ∈ Forw(c 0 ), notice that f (f l-1 (Q)) = P and deg(f : f l-1 (Q ′ ) → P ′ ) = deg c (f ), then f (c) ∈ P ′ \P . It contradicts the condition (c).
For the case l = 1, the lemma holds. Now assume that l ≥ 2. We first prove that

(f l-2 (Q ′ )\f l-2 (Q)) ∩ Forw(c 0 ) = ∅. If (f l-1 (Q ′ )\f l-1 (Q)) ∩ (Crit(f )\Forw(c 0 )) = ∅, then f : f l-1 (Q ′ )\f l-1 (Q) → P ′ \P. If f l-2 (Q ′ )\f l-2 (Q) contains some c ′ ∈ Forw(c 0 ), notice that f (f l-2 (Q)) = f l-1 (Q) and deg(f : f l-2 (Q ′ ) → f l-1 (Q ′ )) = deg c ′ (f ), so f (c ′ ) ∈ f l-1 (Q ′ )\f l-1 (Q) and then f 2 (c ′ ) ∈ P ′ \P which contradicts the condition (c). Hence under the assumption that (f l-1 (Q ′ )\f l-1 (Q)) ∩ (Crit(f )\Forw(c 0 )) = ∅, we come to the conclusion that (f l-2 (Q ′ )\f l-2 (Q)) ∩ Forw(c 0 ) = ∅. Otherwise, f l-1 (Q ′ )\f l-1 (Q) contains some c 1 ∈ Crit(f )\Forw(c 0 ). Since c 1 ∈ Forw(c 0 ), c 1 ∈ Forw(c) for any c ∈ Forw(c 0 ). By the assumption ( * * ), we conclude that (f l-2 (Q ′ )\f l-2 (Q)) ∩ Forw(c 0 ) = ∅.
Continue the similar argument as above, we could prove the lemma for all 0 ≤ i ≤ l -3.

The analytic method we will use to prove Proposition 7.3 is the following lemma on covering maps of the unit disk.

Lemma 7.9. (see [AKLS] Lemma 3.2)

For every integer d ≥ 2 and every 0 < ρ < r < 1 there exists L 0 = L 0 (ρ, r, d) with the following property. Let g, g : (D, 0) → (D, 0) be holomorphic proper maps

In the construction of the sequence in Theorem 7.11, the operators Γ, A, B are used. As they can be read off from the dynamical degree on the boundary of the puzzle pieces, and H preserves this degree information, Theorem 7.11 is valid for the pair of sequences ( Kn (c 0 ), Kn (c 0 )) n≥1 as well, with the same constant S, and probably a different ∆0 as a lower bound for mod( Kn (c 0 )\ Kn (c 0 )). Recall that for each i ≥ 1, p i denotes the integer such that f p i (K i (c 0 )) = K i-1 (c 0 ). We have f p i ( Ki (c 0 )) = Ki-1 (c 0 ), and f p i : K i (c 0 ) → K i-1 (c 0 ) and f p i : Ki (c 0 ) → Ki-1 (c 0 ) are proper holomorphic maps of degree S.

Fix now n ≥ 1. Set v n = c 0 , and then, for i = n -1, n -2, • • • , 1, set consecutively v i = f p i+1 +...+pn (c 0 ). Since (K i (c 0 )\K - i (c 0 )) ∩ orb f ([c 0 ]) = ∅, all the critical values of f p i+1 | K i+1 (c 0 ) , as well as v i , are contained in K - i (c 0 ), 1 ≤ i ≤ n -1. Let ψ i : (K i (c 0 ), v i ) → (D, 0) be a bi-holomorphic uniformization, i = 1, • • • , n. For i = 2, • • • , n, let g i = ψ i-1 • f p i • ψ -1
i . These maps fix the point 0, are proper holomorphic maps of degree at most S, with the critical values contained in

ψ i-1 (K - i-1 (c 0 )). Let ψ i (K - i (c 0 )) = Ω i . Since mod(D \ Ω i ) = mod(K i (c 0 ))\K - i (c 0 )) ≥ ∆ 0 > 0 and Ω i ∋ ψ i (v i ) = 0, 1 ≤ i ≤ n, these domains are contained in some disk D s with s = s(∆ 0 ) < 1. So the critical values of g i are contained in Ω i-1 ⊂ D s , 2 ≤ i ≤ n.
The corresponding objects for f will be marked with tilde. The same assertions hold for gi . Then all the maps g i and gi satisfy the assumptions of Lemma 7.9, with d ≤ S, and ρ = max{s, s}.

(D, 0) ψn ←- (K n (c 0 ), v n ) ( Kn (c 0 ), ṽn ) ψn -→ (D, 0) g n ↓ ↓ f pn f pn ↓ ↓ gn (D, 0) ψ n-1 ←-(K n-1 (c 0 ), v n-1 ) ( Kn-1 (c 0 ), ṽn-1 ) ψn-1 -→ (D, 0) g n-1 ↓ ↓ f p n-1 f p n-1 ↓ ↓ gn-1 . . . . . . . . . . . . g 3 ↓ ↓ f p 3 f p 3 ↓ ↓ g3 (D, 0) ψ 2 ←- (K 2 (c 0 ), v 2 ) ( K2 (c 0 ), ṽ2 ) ψ2 -→ (D, 0) g 2 ↓ ↓ f p 2 f p 2 ↓ ↓ g2 (D, 0) ψ 1 ←- (K 1 (c 0 ), v 1 ) ( K1 (c 0 ), ṽ1 ) ψ1 -→ (D, 0)
Note that each of ψ i , ψi extends to a homeomorphism from the closure of the puzzle piece to D.

Let us consider homeomorphisms η i : T → T given by η i = ψi • H| ∂K i (c 0 ) • ψ -1 i . They are equivariant with respect to the g-actions, i.e., η i-1

• g i = gi • η i .
Due to the qc extension of H| ∂K 1 (c 0 ) , we know that η 1 extends to an L 1 -qc map D → D. Then η 1 is a L 1 -quasisymmetric map. Fix some r with ρ < r < 1. We conclude that η 1 extends to an L-qc map ξ 1 : D → D which is the identity on D r , where L depends on L 1 , ρ and r.

Let L 0 = L 0 (ρ, r, S) be as in Lemma 7.9, and let L ′ = max{L, L 0 }. For i = 2, 3, • • • , n, apply consecutively Lemma 7.9 to the following left diagram (from below to top):

T ηn -→ T g n ↓ ↓ gn T η n-1 -→ T g n-1 ↓ ↓ gn-1 . . . . . . T η 2 -→ T g 2 ↓ ↓ g2 T η 1 -→ T , we get (D, 0) ξn -→ (D, 0) g n ↓ ↓ gn (D, 0) ξ n-1 -→ (D, 0) g n-1 ↓ ↓ gn-1 . . . . . . (D, 0) ξ 2 -→ (D, 0) g 2 ↓ ↓ g2 (D, 0) ξ 1 -→ (D, 0)
so that for i = 2, . . . , n, the map η i admits an L ′ -qc extension ξ i : D → D which is the identity on D r . The desired extension of H| ∂Kn(c 0 ) inside K n (c 0 ) is now obtained by taking ψ-1 n • ξ n • ψ n . Now we show that for c ∈ [c 0 ]\{c 0 }, for each n ≥ 1, H| ∂Kn(c) admits an L′ -qc extension inside K n (c) with the constant L′ independent of n.

Fix

n ≥ 1. Let f qn (K n (c)) = K n (c 0 ). Since K n (c 0 ))\K - n (c 0 ) ∩ orb f ([c 0 ]) = ∅, all the critical values of f qn | Kn(c) are contained in K - n (c 0 ). Let ϕ n : (K n (c 0 ), f qn (c)) → (D, 0) and λ n : (K n (c), c) → (D, 0) be bi-holomorphic uniformizations. Set π n = ϕ n •f qn •λ -1
n . This map fixes the point 0, are proper holomorphic maps of degree at most δ b , with the critical values contained in ϕ n (K - n (c 0 )). Since mod(D \ ϕ n (K - n (c 0 ))) = mod(K n (c 0 )\K - n (c 0 )) ≥ ∆ 0 > 0 and ϕ n (f qn (c)) = 0 belongs to ϕ n (K - n (c 0 )), the set ϕ n (K - n (c 0 )) is contained in the disk D s (here s is exactly the number defined for the case of c 0 in this proof). So the critical values of π n are contained in

ϕ n (K - n (c 0 )) ⊂ D s . (D, 0) λn ←- (K n (c), c) ( Kn (c), c) λn -→ (D, 0) π n ↓ ↓ f qn f qn ↓ ↓ πn (D, 0) ϕn ←-(K n (c 0 ), f qn (c)) ( Kn (c 0 ), f qn (c)) φn -→ (D, 0)
Let Kn (c), c ∈ Crit( f ) be the puzzle piece bounded by H(∂K n (c)). The corresponding objects for f will be marked with tilde. The same assertions hold for πn . Then all the maps π n and πn satisfy the assumptions of Lemma 7.9, with d ≤ δ b , and ρ = max{s, s}.

Note that each of ϕ n , φn , λ n , λn extends to a homeomorphism from the closure of the puzzle piece to D.

Let us consider homeomorphisms α n : T → T and β n : T → T given by α n = φn • H| ∂Kn(c 0 ) • ϕ -1 n and

β n = λn • H| ∂Kn(c) • λ -1 n . Then β n • π n = π n • α n .
Due to the L ′ -qc extension of H| ∂Kn(c 0 ) , we know that α n extends to an L ′ -qc map D → D. We still fix the number r with ρ < r < 1. We can extend α n to be an L-qc map µ n : D → D which is the identity on D r , where L depends on L ′ , ρ and r.

Let L0 = L0 (ρ, r, δ b ) be as in Lemma 7.9, and let L′ = max{ L, L0 }. We apply Lemma 7.9 to the following left diagram: (1) Every K n (c) is a pullback of P , that is f sn (K n (c)) = P for some s n ≥ 1, deg(f sn : K n (c) → P ) ≤ δ b+1 , and all critical values of the map f sn : K n (c) → P are contained in T .

(2) H| ∂Kn(c) admits a qc extension inside K n (c) with the maximal dilatation independent of n.

(3) For every n ≥ 1, c∈[c 0 ] K n (c) is a nice set. Since there are finitely many puzzle pieces of the same depth, we can take a subsequence {u n } n≥1 such that f un (P un (c 0 )) = P for some fixed puzzle piece P of depth 0.

Given c → c 0 , c ∈ Crit(f ), let f vn (L c (P un (c 0 )) = P un (c 0 ). Then

deg(f vn+un : L c (P un (c 0 )) → P ) = deg(f vn : L c (P un (c 0 )) → P un (c 0 )) • deg(f un : P un (c 0 ) → P ) ≤ δ b • δ For c 0 ∈ Crit(f ), we set K n (c 0 ) = P un (c 0 ) and s n = u n . For c → c 0 , c ∈ Crit(f ), set K n (c) = L c (P un (c 0 )) and s n = v n + u n . Now suppose c 0 ∈ Crit r (f ) and c → c 0 , c ∈ Crit(f ).
Since T (c 0 ) is reluctantly recurrent, there exist an integer n 0 ≥ 0, c 1 , c 2 ∈ [c 0 ] and infinitely many integers l n ≥ 1 such that {P n 0 +ln (c 2 )} n≥1 are children of P n 0 (c 1 ) and then deg(f ln :

P n 0 +ln (c 2 ) → P n 0 (c 1 )) = deg c 2 (f ) ≤ δ. Suppose f kn (L c (P n 0 +ln (c 2 ))) = P n 0 +ln (c 2 ) for k n ≥ 1. Then deg(f ln+kn : P n 0 +ln+kn (c) → P n 0 (c 1 )) = deg(f kn : L c (P n 0 +ln (c 2 )) → P n 0 +ln (c 2 )) • deg(f ln : P n 0 +ln (c 2 ) → P n 0 (c 1 )) ≤ δ b • δ.
Take a strictly increasing subsequence of l n , still denoted by l n such that {P n 0 +ln (c 2 )} n≥1 is a nested sequence of puzzle pieces containing c 2 and then {P n 0 +ln+kn (c)} n≥1 is a nested sequence of puzzle pieces containing c. Set P = P n 0 (c 1 ). For c = c 2 , set s n = l n and K n (c) = P n 0 +ln (c); for c = c 2 , set s n = k n + l n and K n (c) = P n 0 +kn+ln (c). Now fix c 0 ∈ Crit n (f ) ∪ Crit r (f ). Take a topological disk T ⊂⊂ P such that T contains all the puzzle pieces of depth n 0 +1. For each c = c 0 or c → c 0 , c ∈ Crit(f ), each n ≥ 1, all critical values of f sn | Kn(c) are contained in the union of puzzle pieces of depth n 0 + 1 because Crit(f ) ⊂ K f and f (K f ) = K f . Consequently, the set T contains all the critical values of the map f sn | Kn(c) .

(2) We will use Lemma 7.9 to construct the qc extension. Fix c = c 0 or c → c 0 and n ≥ 1. Let P and Kn (c) be the puzzle pieces for f bounded by H(∂P ) and H(∂K n (c)) respectively. Since H preserves the degree information, for the map f , we also have the similar statement as for f in (1), more precisely, f sn ( Kn (c)) = P , deg( f sn : Kn (c) → P ) ≤ δ b+1 , and all critical values of the map f sn : Kn (c) → P contained in T , where T is a topological disk in P containing all puzzle pieces for f of depth n 0 + 1 in P .

Let mod(P \T ) = ∆ 1 and mod( P \ T ) = ∆1 . Let ι n : (P, f sn (c)) → (D, 0) and

θ n : (K n (c), c) → (D, 0) be bi-holomorphic uniformizations. Let h n = ι n • f sn • θ -1
n . Then h n fixes the point 0, is a proper holomorphic map of degree at most δ b+1 , with all the critical values contained in ι n (T ).

Since mod

(D \ ι n (T )) = mod(P \T ) = ∆ 1 > 0 and ι n (T ) ∋ ι n (f sn (c)) = 0, we have ι n (T ) ⊂ D t with t = t(∆ 1 ) < 1. So the critical values of h n are contained in ι n (T ) ⊂ D t .
The corresponding objects for f will be marked with tilde. The same assertions hold for hn . Then the maps h n and hn satisfy the assumptions of Lemma 7.9, with d ≤ δ b+1 , and ρ = max{t, t}.

(D, 0) θn ←-(K n (c), c) ( Kn (c), c) θn -→ (D, 0) h n ↓ ↓ f sn f sn ↓ ↓ hn (D, 0) ιn ←-(P, f sn (c)) ( P , f sn (c)) ιn -→ (D, 0)
Note that each of ι n , ιn , θ n , θn extends to a homeomorphism from the closure of the puzzle piece to D.

Let us consider homeomorphisms κ n : T → T and σ n : T → T given by κ

n = ιn • H| ∂P • ι -1 n and σ n = θn • H| ∂Kn(c) • θ -1 n respectively. Then κ n • h n = hn • σ n .
Notice that H| ∂P has a K 1 -qc extension in P for some K 1 ≥ 1. The number K 1 is independent of n because the choice of P does not depend on n. Fix some r with ρ < r < 1. We conclude that κ n extends to a K-qc map ω n : D → D which is the identity on D r , where K depends on K 1 , ρ and r.

Let K 0 = K 0 (ρ, r, δ b+1 ) be as in Lemma 7.9 and let K ′ = max{K, K 0 }. Apply Lemma 7.9 to the following left diagram : T (c0) :

Kn(c0) K - n (c0) c2 f qn Kn(c1) K - n (c1)
Figure 5

Again by the equation ( 4), we know that all critical values of the map f rn : K n (c 0 ) → K n (c 1 ) are contained in K - n (c 1 ). Using the similar method as in the proof of Lemma 7.12 (1), we could obtain the uniformly qc extension of H| Kn(c 0 ) inside K n (c 0 ). We omit the details here.

Subcase 2. Suppose Forw(c 0 ) ⊂ Crit per (f ). If f l (K f (c 0 )) is periodic for some l ≥ 1, then there is some critical periodic component in the periodic cycle of it. By the condition of Proposition 7.3, there is an integer N c 0 such that P Nc 0 +n (c 0 ) has an M c 0 extension, where M c 0 is independent of n. Set K n (c 0 ) := P Nc 0 +n (c 0 ). It is done.

Now we suppose K f (c 0 ) is wandering. For each ĉ ∈ Forw(c 0 ), by the condition of Proposition 7.3, there are a constant M ĉ and an integer N ĉ such that the map H extends to an M ĉ-qc extension inside P n (c) for all n ≥ N ĉ. Set N := max{N ĉ, ĉ ∈ Forw(c 0 )}.

We claim that

Claim 1. There exist a point c 1 ∈ Forw(c 0 ), a topological disk Z ⊂⊂ P N (c 1 ) and a nested sequence of puzzle pieces containing c 0 , denoted by {K n (c 0 )} n≥1 , satisfying that for every n ≥ 1, f wn (K n (c 0 )) = P N (c 1 ) for some Let p be the period of K f (c). Then in T (c), for every 0 < j < p, either (-(N + m nj), j) is not critical or (-(N + m n + 1j), j) is critical. Using (Rule 3) several times, we conclude that there are no critical vertices on the diagonal starting Since c 1 (n) belongs to the finite set Forw(c 0 ) and m n → ∞ as n → ∞, we could find a subsequence of n, say itself, such that {P N +mn+kn+ln (c 0 )} n≥1 form a nested sequence and c 1 (n) ≡ c 1 . Set w n = N + m n + k n + l n .

Similarly to the proof of Lemma 7.13 (1), one take a topological disk Z ⊂⊂ P N (c 1 ) such that all the puzzle pieces of depth N + 1 are contained in Z and particularly, all of the critical values of f mn+kn+ln | P N +mn+kn+ln (c 0 ) are contained in Z.

Using the similar method in the proof of Lemma 7.13 (2), we could show that H| ∂Kn(c 0 ) admits a qc extension inside K n (c 0 ) whose maximal dilatation is independent of n.

In Case 3, we will first draw the similar conclusion to Lemma 7.13 (1). Take arbitrarily a point c ∈ Crit e (f ) ∩ Forw(c 0 ) such that Frow(c) contains some point in Crit p (f ). In T (c 0 ), let {(0, t n )} n≥1 be all the c-vertices on the 0-th row with 1 ≤ t 1 < t 2 < • • • .

Since c 0 ∈ Crit e (f ) and then c 0 → c 0 , by the assumption ( * * ), the c 0 -vertex will not appear in T (c 0 )\{0-th column}, in particular, for each n ≥ 1, there are no c 0 -vertices on the diagonal starting from the vertex (-(t n -1), 1) and ending at the vertex (-1, t n -1). Denote that diagonal by J n . Since c → c, by the assumption ( * * ), there are no c-vertices on the diagonal J n .

We claim that for every n ≥ 1, the diagonal J n meets every point in Crit(f )\{c 0 , c} at most once. In fact, if not, then there is some n ′ and some c ′ ∈ Crit(f ) such that the diagonal J ′ n meets c ′ at least twice. By the assumption ( * * ), we can conclude that c 0 → c ′ → c ′ → c. See Figure 6. By the condition of Case 3 and c ′ → c ′ , we know that c ′ ∈ Crit p (f ) and then Forw(c ′ ) = [c ′ ] by Lemma 7.5. Thus c → c ′ and then c → c. This contradicts c ∈ Crit e (f ).

T (c0) :

-1

-(t n ′ -1) -t n ′ t n ′ -1 t n ′ c ′ c ′ c Figure 6
By the argument above, one easily find that {P tn (c 0 )} n≥1 is a nested sequence of puzzle pieces containing c 0 and deg(f tn : P tn (c 0 ) → P 0 (c)) ≤ δ b-1 . Let T 2 be a topological disk compactly contained in P 0 (c) such that T 2 contains all the puzzle pieces of depth 1 in P 0 (c). Notice that all the critical values are contained in the union of all the puzzle pieces of depth 1 in P 0 (c), so they are also contained in T 2 .

Set K n (c 0 ) := P tn (c 0 ) and then we get similar objects as in Lemma 7.13 (1). Using the similar method in the proof of Lemma 7.13 (2), we could also show that H| ∂Kn(c 0 ) admits a qc extension inside K n (c 0 ) whose maximal dilatation is independent of n.

(2) Since c 0 ∈ Crit e (f ) and then [c 0 ] = {c 0 }, we know that ∪ c∈[c 0 ] K n (c) = K n (c 0 ) and obviously K n (c 0 ) is nice. Summarizing Lemmas 7.12, 7.13 and 7.14, we have proved Proposition 7.3.

A An application of Theorem 2.3

Cui and Peng proved the following result in [CP] (see Theorem 1.1 in [CP]).

Theorem A.1. Let U be a multiply-connected fixed (super)attracting Fatou component of a rational map f . Then there exist a rational map g and a completely invariant Fatou component V of g, such that (1) (f, U ) and (g, V ) are holomorphically conjugate, i.e., there is a conformal map mapping from U onto V and conjugating f to g, (2) each Julia component of g consisting of more than one point is a quasi-circle which bounds an eventually superattracting Fatou component of g containing at most one postcritical point of g. Moreover, g is unique up to a holomorphic conjugation.

We call (g, V ) is a holomorphic model for (f, U ).

To show the uniqueness of the model (g, V ), they divided the proof into two parts. First they proved the following proposition (see also Proposition 1.3 in [CP]).

  ) Let A be an open set and z ∈ A. Denote the component of A containing z by Comp z (A).

  take a Jordan domain E with smooth boundary compactly a new map that maps each sector homeomorphically onto D\[ 1 2 , 1[, and agrees with z → z qi elsewhere. This gives a branched covering Ĝ from D onto D with two critical points and two critical values. Use Ĝ to pull back the standard complex structure of D turn Din a Riemann surface. Uniformize D by a map φ : D → D. Then G = Ĝ • φ suites what we need.

  Case 3. For any c ∈ Forw(c 0 ), either c ∈ Crit p (f ) or, c ∈ Crit e (f ) and Forw(c) contains a critical point in Crit p (f ), and the latter c always exists. Proof. (i) By the definitions of Crit n (f ) and Crit e (f ), we easily see that if c ∈ Crit n (f ), [c] = {c} ∈ D 0 (f ) and if c ∈ Crit e (f ), [c] = {c} ∈ D 0 (f ). If c ∈ Crit p (f ), then by the previous lemma, we know that Forw(c) = [c] and then [c] ∈ D 0 (f ).

  every point in [ĉ] belongs to Crit p (f ). Note that [c 0 ] → [c] → [c] and [c] ∈ D 0 (f ). Hence every point in [c] belongs to Crit p (f ), particularly, c ∈ Crit p (f ).

  map β n admits an L′ -qc extension ν n : D → D which is the identity on D r . The desired extension ofH| ∂Kn(c) inside K n (c) is obtained by taking λ-1 n • ν n • λ n . (2) The set ∪ c∈[c 0 ] K n (c) is nice follows directly from Lemma 7.7. Lemma 7.13. Given c 0 ∈ Crit n (f ) ∪ Crit r (f ).Then there exist a puzzle piece P of depth n 0 , a topological disk T ⊂⊂ P , and a nested sequence of puzzle pieces containing c, denoted by {K n (c)} n≥1 , for each c ∈ Crit(f ) with c = c 0 or c → c 0 , satisfying the following properties.

Proof. ( 1 )

 1 Suppose c 0 ∈ Crit n (f ) and then [c 0 ] = {c 0 }. In T (c 0 )\{0-th column}, every vertex is non-critical. So for each n ≥ 1, deg(f n : P n (c 0 ) → P 0 (f n (c 0 ))) = deg c 0 (f ) ≤ δ.

  so that the map σ n admits a K ′ -qc extension ζ n : D → D which is the identity on D r . The desired extension of H| ∂Kn(c) inside K n (c) is now obtained by taking θ-1 n • ζ n • θ n .

  w n ≥ 1, deg(f wn : K n (c 0 ) → P N (c 1 )) ≤ δ b and all critical values of the map f wn | Kn(c 0 ) are contained in the set Z.Proof. Suppose c ∈ Forw(c 0 ). Refer to the following figure for the proof.Since K f (c 0 ) is wandering, in T (c 0 ), there are infinitely many vertices {(-(N + m n ), k n )} n≥1 such that (-(N + m n ), k n ) isthe first vertex being c on the m n -th row, (-(N + m n + 1), k n ) is not critical and lim n→∞ m n = ∞. Then f kn (L c 0 (P N +mn (c))) = P N +mn (c) and deg(f kn : P N +mn+kn (c 0 ) → P N +mn (c)) ≤ δ b .

  + mn + kn)-(N + mn + kn + ln) c c1(n) ∈ Forw(c0)from the vertex at (-(N + m n + 1), k n ) to the vertex at (-(N + 1), m n + k n ). From the vertex (-N, m n + k n ), march horizontally l n ≥ 1 steps until the first hit of some c 1 (n) vertex for some c 1 (n) ∈ Forw(c 0 ). Then there is no critical vertex on the diagonal starting from the vertex (-(N + m n + k n + l n -1), 1) to the vertex (-(N + 1), m n + k n + l n -1). Therefore deg(f mn+kn+ln : P N +mn+kn+ln (c 0 ) → P N (c 1 (n))) ≤ δ b .

2010 Mathematics Subject Classification: 37F10, 37F20

One way to construct such a map G is as follows: Consider the map z → z qi together with a preimage x ∈]0, 1[ of 1/2. Cut D along [x, 1[, glue in deg(f | U ) consecutive sectors to define a new space D . Define

Acknowledgment. The authors would like to thank Cui Guizhen, Jeremy Kahn and Shen Weixiao for inspiring discussions.

of degree at most d, with critical values contained in D ρ . Let η, η ′ : T → T be two homeomorphisms satisfying g • η ′ = η • g, where T denotes the unit circle. Assume that η admits an L-qc extension ξ : D → D which is the identity on D r . Then η ′ admits an L ′ -qc extension ξ ′ : D → D which is the identity on D r , where L ′ = max{L, L 0 }.

In the following, we will discuss Crit per (f ), Crit p (f ), Crit n (f ) ∪ Crit r (f ) and Crit e (f ) successively.

For any c ∈ Crit per (f ), by the condition of Proposition 7.3, there are a constant M c and an integer N c such that the map H extends to an M c -qc extension inside P n (c) for all n ≥ N c .

The following lemma can be easily proved by Lemma 7.7.

Lemma 7.10. Given a point c 0 ∈ Crit per (f ) and set

, where l c is the smallest positive integer such that

The following theorem is proved in [PQRTY].

Theorem 7.11. Given a critical point c 0 ∈ Crit p (f )\Crit per (f ). There are two constants S and ∆ 0 > 0, depending on b, δ and µ (see below), and a nested sequence of critical puzzle pieces K n (c 0 ) ⊂⊂ K n-1 (c 0 ), n ≥ 1, with K 0 (c 0 ) to be the critical puzzle piece of depth 0, satisfying (i

Here

(1) for every c ∈ [c 0 ] and every n ≥ 1, H| ∂Kn(c) admits a qc extension inside K n (c) with the maximal dilatation independent of n;

(2

Proof.

(1) We first prove that H| ∂Kn(c 0 ) admits an L ′ -qc extension inside K n (c 0 ) where L ′ is independent of n. This part is similar to the proof of Proposition 3.1 in [PT].

Since H preserves the degree information, the puzzle piece bounded by

Notice that H| ∂K 1 (c 0 ) has a qc extension on a neighborhood of ∂K 1 (c 0 ). It extends thus to an L 1 -qc map K 1 (c 0 ) → K1 (c 0 ), for some L 1 ≥ 1 (see e.g. [CT], Lemma C.1).

(3

For the case that c 0 ∈ Crit r (f ), we apply Lemma 7.7 to

and easily get the conclusion.

Lemma 7.14. Suppose c 0 ∈ Crit e (f ). Then (1) there is a nested sequence of puzzle pieces containing c 0 , denoted by {K n (c 0 )} n≥1 , such that for each n ≥ 1, H| ∂Kn(c 0 ) admits a qc extension inside K n (c 0 ) with the maximal dilatation independent of n,

(2) for every

and in Lemma 7.6 (ii), we distinguish three cases for Forw(c 0 ). In the following, we will discuss the three cases.

In Case 1, i.e., Forw(c 0 )∩(Crit n (f )∪Crit r (f )) = ∅, by using Lemma 7.13, we can get a nested sequence of puzzle pieces containing c 0 , denoted by {K n (c 0 )} n≥1 , and H| ∂Kn(c) admits a qc extension inside K n (c 0 ) whose maximal dilatation independent of n.

We divide Case 2 (Forw(c 0 ) ⊂ Crit p (f )) into two subcases. Subcase 1. There is a critical point c 1 ∈ Forw(c 0 ) ∩ (Crit p (f )\Crit per (f )). Let (K n (c 1 ), K - n (c 1 )) n≥1 be the sequence of pairs of critical puzzle pieces constructed in Theorem 7.11.

For n ≥ 1, set

, by Lemma 7.8, we conclude that for all 1 ≤ i < r n , (f i (K n (c 0 ))\f i (K - n (c 0 ))) ∩ Forw(c 1 ) = ∅. We claim that for every n ≥ 1 and every 1 ≤ i ≤ r n ,

If not, there is some n such that

Hence for every n ≥ 1 and every 0 ≤ i < r n ,

From the equation ( 4) and

Proposition A.2. Suppose g, g are two rational maps and V, Ṽ are two completely invariant Fatou components of g and g respectively satisfying the conditions (1) and

(2) in Theorem A.1. Then there is a qc map from the Riemann sphere C onto itself conjugating g and g on C and this conjugation is conformal on the Fatou set of g.

The other part is to show the Julia set of the model g carries no invariant line fields (see Proposition 1.4 in [CP]).

In this appendix, we will apply Theorem 2.3 (b) to give an another proof of Proposition A.2.

Proof of Proposition A.2. First starting from (g, V ) and (g, Ṽ ), we construct (g : U → V) and (g : Ũ → Ṽ) satisfying the conditions as in the set-up.

We may assume that the fixed points of g and g in V and Ṽ are the infinity point. By Koenigs Linearization Theorem and Böttcher Theorem, we can take a disk

where Crit(g) denotes the critical point set of g.

Let D n be the connected component of g -n (D 0 ) containing D 0 for each n ≥ 1. Then D n ⊂⊂ D n+1 and V = ∞ n=0 D n . There is an integer N 0 satisfying that for any n ≥ 0, g -n (D N 0 ) has only one component, the set Crit(g) is contained in g -n (D N 0 ) and every component of C\D N 0 contains at most one component of C\V having critical points.

By Theorem A.1 (1), there is a conformal map H : V → Ṽ with g • H = H • g on V . Set V := C\D N 0 , Ṽ := C\H(D N 0 ) and U := g -1 (V) and Ũ := g-1 ( Ṽ). One can check that (g : U → V) and (g : Ũ → Ṽ) satisfying the conditions as in the set-up.

Clearly, K g = C\V and K g = C\ Ṽ . Since H : V → Ṽ is a qc conjugacy from g to g and V := C\D N 0 , Ṽ := C\H(D N 0 ), we know that H : V\K g → Ṽ\K g is a qc conjugacy off K g . Let E be a periodic critical component of K g which is mapped to a periodic critical component of K g under some forward iterate of g. According to Theorem A.1 (2), E is a quasi-circle which bounds an eventually superattracting Fatou component containing a critical point c. In the proof of Proposition 4.4 in [CP], a qc map h E is constructed. That map is defined on a puzzle piece P n E (c) containing E. From the definition of that map, one can easily check that h E | ∂Pn E +n (c) = H for all n ≥ 0. Set N c := n E and let M c be the maximal dilatation of the map h E . Then by Theorem 2.3 (2), H extends to a qc conjugacy off intK g . Notice that every component of intK g is a bounded eventually superattracting Fatou component and vice versa. So H can extend to a conformal map in every component of intK g which is again a conjugacy (refer to the proof of Claim 4.1). Hence H extends to a qc conjugacy on V, that is, H extends to a qc conjugacy on C which is conformal on the Fatou set of g.