A Blokh

X Buff

A Ch Éritat

L Oversteegen

THE SOLAR JULIA SETS OF BASIC QUADRATIC CREMER POLYNOMIALS

Keywords: Mathematics Subject Classification. Primary 37F10, Secondary 37B45, 37C25 Complex dynamics, Julia set, Cremer fixed point

In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.

Introduction

A long outstanding problem of whether polynomial Julia sets of positive area exist has recently been solved in [START_REF] Buff | Quadratic Julia sets with positive area[END_REF]. In that paper the authors implemented a program initiated by A. Douady and showed that there exist quadratic polynomials of various types all of which have Julia sets of positive area. The proofs are based on McMullen's results [START_REF] Mcmullen | Self-similarity of Siegel disks and Hausdorff dimension of Julia sets[END_REF] regarding the measurable density of the filled-in Julia set near the boundary of a Siegel disk with bounded type rotation number, Chéritat's techniques of parabolic explosion [C01] and Yoccoz's renormalization techniques [START_REF] Yoccoz | Petits diviseurs en dimension 1[END_REF] to control the shape of Siegel disks, and Inou and Shishikura's results [IS] to control the post-critical sets of perturbations of polynomials having an indifferent fixed point.

In particular, there exist quadratic polynomials with a fixed Cremer point whose Julia set has positive area. The topological structure of Julia sets containing a Cremer point (including the above mentioned ones of positive area) has remained elusive. In the present paper we want to shed some light upon this problem.

Let us first discuss the topological structure of connected polynomial Julia sets in general. The best case scenario is when the Julia set J is locally connected. Then J is homeomorphic to the quotient space S 1 / ∼= J ∼ of the unit circle with respect to a specific equivalence relation ∼ called an invariant lamination. Moreover, the factor map in this case conjugates the polynomial to the map f ∼ induced on J ∼ by the map z d : S 1 → S 1 with d being the degree of the polynomial. Spaces like J ∼ are called below topological Julia sets while the induced maps f ∼ on J ∼ are called topological polynomials. Thus, in the locally connected case topological polynomials are good (one-to-one) models for true complex polynomials on their Julia sets.

The picture is more complicated when J is not locally connected. Still even in that case recently developed tools allow one to use topological polynomials as models (albeit not as good as above). Given a polynomial P , denote by J P its Julia set. Call irrational neutral periodic points CS-points; a CS-point p is said to be a Cremer point if the iterate of the map which fixes p is not linearizable in a small neighborhood of p. Suppose that P is a polynomial with connected Julia set and no CS-points. In his fundamental paper [K04] Jan Kiwi introduced for P an invariant lamination ∼ P on S 1 such that P | J P is semi-conjugate to the induced map f ∼ P : J ∼ P → J ∼ P by a monotone map m : J P → J ∼ P (by monotone we mean a continuous map whose point preimages are connected). Then J ∼ P is a locally connected model for J, and P | J is monotonically semiconjugate to f ∼ P . In addition Kiwi proved in [K04] that at all periodic points p of P in J P the set J P is locally connected at p and m -1 • m(p) = {p}. However in some cases the entire approach which uses modeling of the Julia set by means of a monotone map onto a locally connected continuum breaks down. From now on we will consider only quadratic polynomials; the critical point will be always denoted by c.

A quadratic polynomial P with a Cremer fixed point (i.e. with a neutral non-linearizable fixed point p ∈ J such that P (p) = e 2πiα with α irrational) is said to be a basic Cremer polynomial, and its Julia set is called a basic Cremer Julia set (in this case we always denote the Cremer fixed point of P by p). The main result of a recent preprint [START_REF] Blokh | Monotone maps of Cremer Julia sets[END_REF] is that if P is a basic Cremer polynomial then any monotone map m : J P → A with m(J P ) locally connected must collapse all of J P to a point. Hence studying the topology of basic Cremer Julia sets requires new tools (see, e.g., the fundamental papers [START_REF] Perez-Marco | Topology of Julia sets and hedgehogs[END_REF] and [START_REF] Perez-Marco | Fixed points and circle maps[END_REF] by Perez-Marco); some tools here are provided by continuum theory and developed in [START_REF] Grispolakis | Building Blocks for Julia sets[END_REF] (in which results of [START_REF] Yoccoz | Petits diviseurs en dimension 1[END_REF] were used) and further in [START_REF] Blokh | The Julia sets of quadratic Cremer polynomials[END_REF]. Before introducing them let us have an overview of a few known results.

By Sullivan [START_REF] Sullivan | Conformal dynamical systems[END_REF], a basic Cremer Julia set J is not locally connected. Moreover, by Kiwi [K00] the critical point c is not accessible. Still, there are points in J at which rays are landing (e.g., repelling periodic points [START_REF] Douady | Étude dynamique des polynômes complexes I[END_REF]), so it makes sense to study in more detail the pattern in which such landing can occur. In this respect the following important question is due to C. McMullen [START_REF] Mcmullen | Complex dynamics and renormalization[END_REF]: can a basic Cremer Julia set contain any points at which at least two rays are landing (so-called biaccessible points)? This question was partially answered by Schleicher and Zakeri in [SZ99, Theorem 3] (see also [START_REF] Zakeri | Biaccessibility in quadratic Julia sets[END_REF]Theorem 3]) where they show that if a basic Cremer Julia set contains a biaccessible point then this point eventually maps to the Cremer point. In [START_REF] Blokh | The Julia sets of quadratic Cremer polynomials[END_REF] it is shown that if a basic Cremer Julia set has a biaccessible point then it is a solar Julia set as described in Theorem 1.1. However it is still unknown if there exist basic Cremer Julia sets with biaccessible points. Another paper studying the topology of basic Cremer Julia sets is that of Sørensen [START_REF] Sørensen | Describing quadratic Cremer polynomials by parabolic perturbations[END_REF]. In that paper the author constructs basic Cremer polynomials with external rays which accumulate on both the Cremer point and its preimage and thus gives examples of basic Cremer polynomials whose Julia sets have very interesting topological properties.

Now we would like to state the results of [START_REF] Blokh | The Julia sets of quadratic Cremer polynomials[END_REF]. If P is a basic Cremer polynomial then by [START_REF] Mañé | On a theorem of Fatou[END_REF] p ∈ ω(c) (see also [START_REF] Perez-Marco | Fixed points and circle maps[END_REF] and [C05, Theorem 1.3]). Following Kiwi [K04] we say that two angles θ, γ are K-equivalent if there are angles x 0 = θ, . . . , x n = γ such that the impressions of x i-1 and x i are non-disjoint for 1 ≤ i ≤ n; a class of K-equivalence is called a K-class, and an angle whose impression is disjoint from all other impressions is said to be K-separate. A continuum X is connected im kleinen at a point x provided for each open set U containing x there exists a connected set C ⊂ U such that x is in the interior of C (relative to X). A continuum X is locally connected at a point x provided there exists a basis of open connected neighborhoods at x. Observe that sometimes different terminology is used. For example, in Milnor's book [START_REF] Milnor | Dynamics in one complex variable[END_REF]p. 168] the property of local connectivity is called "open local connectivity" while to the property of being connected im kleinen at a point Milnor refers to as the property of being "locally connected at a point". On the other hand, in the textbook by Munkres [Mun00,p. 162] connected im kleinen is called "weakly locally connected". Using our terminology, if a space is locally connected at x, then it is connected im kleinen at x. It is well known that if a continuum is connected im kleinen at each point, then it is locally connected (see, e.g., [START_REF] Munkres | Topology[END_REF]p. 162,Ex. 6]). However, a continuum can be connected im kleinen at a point without being locally connected at this point (as an example one can consider the so-called infinite broom, see [START_REF] Munkres | Topology[END_REF]p. 162,Ex. 7]).

The main result of [START_REF] Blokh | The Julia sets of quadratic Cremer polynomials[END_REF] is the following theorem (by a degenerate impression we mean an impression consisting of one point).

Theorem 1.1. Let P be a basic Cremer polynomial. Then its Julia set J must be one of the following two types.

Solar Julia set: J has the following equivalent properties:

(1) there is an impression not containing the Cremer point;

(2) there is a degenerate impression;

(3) the set Y of all K-separate angles with degenerate impressions contains all angles with dense orbits (Y contains a full Lebesgue measure G δ -set dense in S 1 ) and a dense in S 1 set of periodic angles, and the Julia set J is connected im kleinen at the landing points of these rays; (4) there is a point at which the Julia set is connected im kleinen; (5) not all angles are K-equivalent. Red dwarf Julia set: J has the following equivalent properties:

(1) All impressions are non-degenerate.

(2) The intersection of all impressions is a non-degenerate subcontinuum of J containing the Cremer point and the limit set of the critical point.

(3) J is nowhere connected im kleinen.

The main aim of this paper is to prove the following theorem.

Theorem 1.2. There exist basic Cremer polynomials with solar Julia sets of positive area.

We prove Theorem 1.2 combining results from [START_REF] Buff | Quadratic Julia sets with positive area[END_REF] and [START_REF] Blokh | The Julia sets of quadratic Cremer polynomials[END_REF]. An interesting remaining problem then is that of the existence of red dwarf Julia sets. It is related to a well-known problem concerning the existence of indecomposable Julia sets (a continuum is indecomposable if it cannot be represented as the union of two proper subcontinua). It is known that if the Julia set is indecomposable then every impression coincides with the entire Julia set (see, e.g., [START_REF] Mayer | Indecomposable continua and the Julia sets of polynomials[END_REF] and [START_REF] Childers | Indecomposable continua and the Julia sets of polynomials, II[END_REF]).

Preliminaries

General facts and notation.

In what follows we use standard tools of Carathéodory theory. An unshielded continuum X ⊂ C is a continuum which coincides with the boundary of the infinite complementary component U of X in the complex sphere. Given an unshielded continuum X let φ be the normalized Riemann map from the unit disk D onto U . The external ray R α is the image of the radius of D corresponding to the external angle α. The impression Imp(α) of R α is defined (see, e.g., [START_REF] Ch | Boundary behavior of conformal maps[END_REF]) as the set of all limit points of sequences φ(x i ) taken over all sequences x i → e 2πiα , x i ∈ D (if we do not want to specify the angle we will omit it from the notation). By a component of a set we always mean a connected component of this set, and by a non-separating planar set we mean a set whose complement in the plane is connected. Also, given sets A, B such that A∩B is a singleton, we shall say that A is attached to B.

Given a polynomial f : C → C with non-separating Julia set J(P ) and a continuum K ⊂ J(P ), a set K is said to be a K-pullback

(of order n by f ) if K is a component of f -n (K).
Clearly, any Kpullback K maps onto K as a branched covering map. For us the most interesting is the case of quadratic polynomials with non-separating Julia sets and subcontinua K of the Julia set. Then if K contains f (c), the first K-pullback is unique and maps onto K in a 2-to-1 branched covering fashion while if K does not contain f (c) then there are two first K-pullbacks each of which maps onto K homeomorphically. Given a quadratic polynomial f with the critical point c f = c, we denote by ψ f = ψ the involution which maps any z to the other preimage of f (z) (e.g., ψ(c) = c, and if f (z) = z 2 + v then ψ(z) = -z). Below we often consider forward invariant continua K ⊂ J f which contain c but are such that the first n-segment of the orbit of c avoids ψ(K) = K (i.e., f (c) ∈ K , . . . , f n (c) ∈ K ). In this case we consider K -pullbacks of order at most n (K is considered its own pullback of order 0). Because of the assumptions, all these K -pullbacks map univalently onto K .

Siegel polynomials.

A quadratic polynomial is said to be a basic Siegel polynomial if it has an invariant Siegel disk. Denote by S lc the family of basic Siegel polynomials with locally connected Julia set. We need a few well-known facts concerning their Julia sets J (see, e.g., [START_REF] Grispolakis | Building Blocks for Julia sets[END_REF]). Given a polynomial P ∈ S lc , let ∆ be its closed Siegel disk, S = ∂∆, and ψ(∆) = ∆ . Since J is locally connected, {c} = ∆ ∩ ∆, and so ∆ is attached to ∆ at c and contains no forward images of c. Consider the branch c -1 , c -2 , . . . of the backward orbit of c = c 0 consisting only of points of S (here P (c -n-1 ) = c -n ). At each point c -n the appropriate n-th pullback of ∆ , corresponding to c -n as the pullback of c, is attached to ∆. However all other ∆ -pullbacks are disjoint from ∆. Since all forward images of c avoid ∆ then the picture described in the previous subsection applies to ∆ and we have a family of well-defined univalent ∆ -pullbacks of all orders. The entire

∆ ∆ L R Figure 1.
Example of a locally connected basic Siegel polynomial Julia set: that with the golden mean rotation number. We put the two external rays landing at the critical point and grayed the half-plane R.

set P -n (∆) is a connected union of ∆ and 2 n -1 ∆ -pullbacks, n of which are attached to ∆, while others are disjoint from ∆.

The way ∆ and ∆ -pullbacks intersect (but not their relative location on the plane!) is the same for all polynomials P ∈ S lc . To describe it, observe that two external rays landing at c cut C into two half-planes. Assign 1 to the closed half-plane L ⊃ ∆ and 0 to the closed half-plane R ⊃ ∆ and study the symbolic dynamics of points of J in terms of this partition of the plane. To each point x ∈ J we associate its infinite itinerary i(x) defined in the obvious way. Clearly, the only ambiguity in i(x) arises if the point x is a critical preimage because the only point of J which belongs to both L and R is c (and hence c can be assigned both 0 and 1 as the first entry in its itinerary). This ambiguity is resolved though if instead of points we deal with ∆ -pullbacks. Namely, if Q is ∆ or a ∆ -pullback then we assign as its infinite itinerary i(Q) the itinerary of any non-precritical point of Q (any such itinerary i(Q) from some time on consists of 1's). To simplify the notation, let us denote by 1 i the string of 1's of length i (possibly, i = ∞). Similarly, if a finite

string i = i 0 i 1 . . . i l is given then i k = {i 0 . . . i l } k is the concatenation L R 1 ∞ 01 ∞ 001 ∞ 0001 ∞ 101 ∞ 1101 ∞ 11101 ∞ 01101 ∞ 0101 ∞ 00101 ∞ 1001∞ 11001∞ 01001 ∞ 10101 ∞ 00001 ∞ 10001 ∞ Figure 2.
Example of itineraries of ∆ and of pullbacks of ∆ and of their dynamics for the Golden mean basic Siegel Julia set. Some of the missing arrows can be determined by using the central symmetry: two symmetric components are mapped to the same component.

of k copies of i (here again k can be ∞). Then we have, e.g., that

i(∆) = 1 ∞ , i(∆ ) = 01 ∞ , etc.
For c and its preimages the above ambiguity can be dealt with by considering c as a point of ∆ or as a point of ∆ and assigning different itineraries respectively. That is, if we take a preimage of c then until it maps onto c its itinerary is well-defined but at the time when it maps onto c we either consider it as a point of ∆ and assign 1 to it, or consider it as a point of ∆ and assign 0 to it. For the forthcoming images of c we again have no ambiguity (1 is assigned to all of them). In other words, to each preimage of c exactly two itineraries are assigned as explained above. This implies that two ∆ -pullbacks with itineraries ī = i 0 i 1 . . . and j = j 0 j 1 . . . have a point in common if and only if there exists k such that i 0 = j 0 , . . . , i k-1 = j k-1 , i k = 1, j k = 0 while i r = j r = 1 for any r > k (in particular, no more than two ∆ -pullbacks can intersect at one point). In what follows we denote the ∆ -pullback with itinerary ī as ∆ī. Observe that since ∆ and ∆ have only one point -namely c -in common, then any two ∆ -pullbacks (or ∆) may have at most one point in common and no three ∆ -pullbacks (or ∆) intersect (any common point of two ∆ -pullbacks maps onto c while the two pullbacks in question map onto ∆ and ∆ ).

So far we have considered as examples points of ∆ -pullbacks (or ∆ -pullbacks themselves). To deal with other points of J we need well-known facts about Julia sets of polynomials P ∈ S lc (see, e.g., [START_REF] Grispolakis | Building Blocks for Julia sets[END_REF]); the notation introduced here will be used from now on. Let p be the fixed point of P belonging to ∆, σ be the angle doubling map on S 1 , (γ, γ ) (resp. [γ, γ ]) be the open (resp. closed) counterclockwise circle arc from γ to γ , and let γγ be the chord connecting γ and γ in the disk. If P (p) = e 2πiρ (ρ is irrational) then there exists a special rotational σ-invariant Cantor set F ⊂ S 1 such that σ| F is no more than 2-to-1 semiconjugate to the irrational rotation by the angle 2πρ [START_REF] Bullett | Ordered orbits of the shift, square roots, and the devill's staircase[END_REF]. More precisely, call an arc complementary to F an F -hole. Then the semiconjugacy maps the endpoints of every F -hole into one point and otherwise is one-to-one. The most important F -hole is the longest one which is the half-circle (β, α) with the endpoints denoted below by 0 < α < 1/2 and β = α + 1/2; the chord αβ connecting α and β is called the critical leaf (diameter). Other F -holes are preimages of (β, α).

The limit set F = ω(α) is exactly the set of angles whose entire orbits are contained in [α, β]; also, the angles in F are exactly the angles whose rays land at points of S = ∂∆. The endpoints of an Fhole can be denoted by α -n , β -n (since they are appropriate preimages of α and β). Both rays R α -n and R β -n land at the point c -n , the n-th pullback of c belonging to ∆. We need the following simple fact.

Lemma 2.1. Suppose that two angles α = β are given none of which maps into F by any power of σ. Then there exists m such that σ m (α ) and σ m (β ) are separated by the critical leaf αβ.

Proof. Define d(θ, θ ) as the length of the shortest arc between θ and θ (we normalize the circle so that its length is equal to 1). It is easy to see that d(σ(θ), σ(θ

)) = T (d(θ, θ )) where T : [0, 1/2] → [0, 1/2]
is the appropriate scaling of the full tent map. Since T (x) > x for 0 < x < 1/3, there exists m such that d(σ m (α ), σ m (β )) ≥ 1/3. If d(σ m (α ), σ m (β )) = 1/2 then since α , β are not preimages of σ(α) we see that σ m (α ), σ m (β ) are separated by αβ and we are done. Assume that σ m (α ), σ m (β ) are not separated by αβ and d(σ m (α ), σ m (β )) < 1/2. Since the longest complementary arcs to the set σ -1 (F ) are of length 1/4, we see that σ m (α ), σ m (β ) belong to two distinct complementary arcs of F ∪ F + 1/2 located on one side of αβ. Thus after several steps the σ-image of, say, α will belong to (β, α) while the corresponding image of β will still be inside (α, β). This means that these two images of α , β will be separated by αβ.

Call a point y ∈ J a local cutpoint of J if the point y is a cutpoint of some connected neighborhood U of y in J. The union of boundaries of ∆ and all ∆ -pullbacks forms the set of all local cutpoints of J. The remaining points of J are called the endpoints of J, or, more informally, the "dust". We say that a connected set X connects a connected set A and a connected set B if the union A ∪ B ∪ X is connected (in practice we use this term when A and B are disjoint). Also, by a string of ∆ -pullbacks we mean a countable collection of ∆ -pullbacks concatenated to each other (so that consecutive pullbacks in the string intersect over exactly one point). Lemma 2.2 studies how dust points in J are connected to ∆. Lemma 2.2. Let y ∈ J be a dust point with itinerary ī = (i 0 i 1 . . . ). Then there exists a unique string S y of ∆ -pullbacks which connects ∆ and y. Denote the ∆ -pullbacks in S y as follows: ∆ 1 (y) is the closest to ∆ (in the sense of the spatial order on the string), ∆ 2 (y) is the second one, etc. Then the itinerary of ∆ j (y) is obtained from the itinerary ī = i 0 i 1 . . . of y as follows: choose the j-th appearance of 0 in ī, keep all the entries before that, and replace all other entries in ī by 1.

Proof. Since J is locally connected, it is arcwise connected. Hence there exists an arc (homeomorphic image of the interval [0, 1]) I ⊂ J connecting y and c. Let us show that I ∩ ∆ = I ∩ S is connected (recall, that S = ∂∆). Indeed, suppose that there exists an arc in J whose endpoints belong to S while otherwise the arc is disjoint from S. Then it follows that parts of S are "shielded" from infinity by this arc, i.e. are not accessible from infinity, a contradiction. Hence such arcs in J do not exist which shows that I ∩ S is either an arc or the point c. If I ∩ S = {c} is a point set π(y) = c, otherwise let π(y) be the other endpoint of I ∩ S. Observe that by the above argument such point is unique so that π is well-defined. Denote the arc connecting y and π(y) by I; then I ∩ S = π(y). Intuitively, one can think of the point π(y) as a "projection" of y into S.

Since all points of I except for y are not from the dust, it follows that I \ {y, π(y)} is contained in the union of some ∆ -pullbacks. The union of these ∆ -pullbacks forms the desired string S y . Observe, that the string S y is unique by the same geometric argument as above -otherwise some points of the Julia set are not accessible from infinity because they will be "shielded" from infinity by other connected subsets of J (these subsets will be parts of the two strings which hypothetically connect ∆ and y and will form the boundary of a simply connected domain in the plane containing points of J). A similar geometric argument is often used in the paper, so we explain it here in detail while simply alluding to it in the future.

By the definition we do not include ∆ in S y and begin S y from ∆ 1 (y), the ∆ -pullback closest to ∆ in S y . Thus, ∆ 1 (y) is either ∆ or a pullback of ∆ attached to ∆. Therefore ∆ 1 (y) has itinerary 1 k 1 01 ∞ with k 1 ≥ 0 digits 1 to begin with (if k 1 = 0 then ∆ 1 (y) = ∆ ). The next pullback ∆ 2 (y) in the string S y is attached to ∆ 1 (y), hence its itinerary coincides with that of ∆ 1 (y) until ∆ 1 (y) maps onto ∆. At this moment ∆ 2 (y) becomes the closest to ∆ pullback of ∆ in the appropriate image of S y and the process repeats itself with the only difference that now the image of ∆ 2 (y) needs k 2 steps to move around ∆ until it finally gets mapped onto ∆ and then onto ∆. This argument yields that the "spatially" j-th pullback ∆ j (y) of ∆ in S y has itinerary 1 k 1 01 k 2 0 . . . 1 k j 01 ∞ . It has to coincide with the itinerary of y until the first time ∆ j (y) maps onto ∆. Also, numbers k i may be equal to zero. Hence the itinerary of ∆ j (y) is obtained from the itinerary ī = i 0 i 1 . . . of y as follows: choose the j-th appearance of 0 in ī, keep all the entries before that, and replace all other entries in ī by 1.

As an example consider the periodic point x with itinerary ī = {011} ∞ . Then the following are the pullbacks of ∆ forming the string

S x : ∆ 1 (x) = ∆ 01 ∞ , ∆ 2 (x) = ∆ 01101 ∞ , ∆ 3 (x) = ∆ 01101101 ∞ etc
. By Lemma 2.1 any two distinct periodic points u ∈ J and v ∈ J have distinct itineraries (it is enough to consider rays landing at u and v). Therefore their strings S u and S v may have a certain initial piece in common (perhaps empty), and then will separate. For example, let z be the periodic point of itinerary {0110111} ∞ . Then the string S z consists of the following pullbacks of ∆ : ∆

1 (z) = ∆ 01∞ , ∆ 2 (z) = ∆ 01101∞ , ∆ 3 (z) = ∆ 011011101∞ , . . . . Comparing S x and S z we see that ∆ 1 (x) = ∆ 1 (z), ∆ 2 (x) = ∆ 2 (z), but ∆ 3 (x) = ∆ 3 (z)
. It follows that ∆ 3 (x) and ∆ 3 (z) are disjoint (both are attached to the same ∆ -pullback but at distinct points, hence if they meet then some points of the Julia set will not be accessible from infinity). From this moment on the strings S x and S z go their own ways, converging to x and z respectively.

Consider now the dynamics of the string S x . Since x is of period 3 then S x must cover itself under P 3 while P 3 (S x ) contains ∆. In fact, already the first application of P restricted onto S x maps ∆ 1 (x) onto ∆. Then S x rotates about ∆ for one step and on the next step S x maps over itself. Thus,

P 3 (∆ 2 (x)) = ∆ 1 (x), P 3 (∆ 3 (x)) = ∆ 2 (x) etc.
In other words, P 3 shifts the pullbacks in S x 1 pullback "down" (i.e. closer to ∆). The number 1 is then called the basic length of S x ; the string S x consists of countably many fragments whose length is 1 and who are shifted by P 3 one onto another closer to ∆ except for the first fragment of S x which maps by P 3 onto ∆.

In the case of the string S z the picture is a bit more complicated, however it has essentially the same properties (in our description of the dynamics of S z we skip discussing simple rotations of S z around ∆). It is easy to check that first P 3 shifts ∆ 2 (z) onto ∆ 1 (z) while ∆ 3 (z) = ∆ 011011101 ∞ maps onto ∆ 011101 ∞ . Then P 4 maps ∆ 011101 ∞ onto ∆ = ∆ 1 (z) and the entire string S z finally covers itself. In other words, since the periodic fragment of the itinerary of z contains two zeros, then P 7 shifts the pullbacks in S z down (closer to ∆) by 2 pullbacks. In this case the basic length of S z is 2; the string S z consists of countably many fragments whose length is 2 and who are shifted by P 7 one onto another closer to ∆ except for the first fragment of S z which maps by P 7 onto ∆. Clearly, P 7 maps the first fragment of S z onto ∆ as a continuous map (there have to be critical points), but otherwise the map P 7 shifts fragments in S z homeomorphically.

In general, given a periodic orbit y of period k whose itinerary has the minimal periodic fragment with l zeros we see that the map P k shifts the ∆ -pullbacks in S y down (closer to ∆) by l pullbacks. In this case the first fragment of S y is ∆ 1 (y)∪• • •∪∆ l (y), the second fragment of S y is the union ∆ l+1 (y) ∪ • • • ∪ ∆ 2l (y), etc. Observe that all the fragments in S y are in fact the pullbacks of the first fragment by the appropriate branch of the inverse function to P l , and from the second fragment on all the fragments in S y are disjoint from ∆. Moreover, since in this case the critical limit set ω P (c) coincides with ∆, we see that by well-known shrinking properties of pullbacks under polynomial/rational maps (see, e.g., Shrinking Lemma [START_REF] Lyubich | Laminations in holomorphic dynamics[END_REF]) the diameters of the pullbacks in S y converge to 0. If diameters of the sets in a sequence converge to 0 then the sets are said to form a null sequence; it follows that in this case the fragments in S y described above form a null sequence. Of course, in the case at hand any string S ζ converges to the point ζ defining it -after all, the Julia set J is locally connected. However even in the case of basic Cremer Julia sets (which are not locally connected) these ideas, with some modifications, still apply.

Let us go back to the example we have already partially considered before, i.e. to the periodic points x, z and their strings S x , S z . As we have seen, S x and S z converge to the periodic points x and z respectively. On the other hand, we have seen that the strings S x and S z ∆ ∆ separate after the pullback ∆ 2 (x) = ∆ 2 (z) = ∆ 01101∞ . Let R γ , R γ be the external rays landing at x and z respectively. Denote by I the open arc between γ and γ contained in the 0-semicircle. Denote by Ŝx the closure of the "tail" of S x taken from ∆ 01101∞ to the point x and by Ŝz the closure of the "tail" of S z taken from ∆ 01101∞ to the point z (so that Ŝx ∩ Ŝz = ∆ 01101∞ ). Then the union of Ŝx , Ŝz , R γ and R γ is a "fork" which encloses a wedge containing all rays R θ , θ ∈ I. The impression of θ ∈ I must be contained in the closure of this wedge and therefore is disjoint from ∆. Again, for the polynomials in S lc this fact can be shown in a much easier way. However the argument is valuable because it applies in more general situations. Let σ 2 be the one-sided shift on the space of all sequences with symbols 1 and 0, and let A n = σ -n 2 (1 ∞ ). Notice that A n includes all σ 2 -preimages of 1 ∞ of order less than or equal to n. Consider elements of A n as vertices of a graph connected by edges if and only if the ∆pullbacks with these itineraries intersect. Clearly, this makes A n into a tree (if there is a loop in A n then as before there must be points of J shielded from the infinity, a conradiction). From now on we always consider A n endowed with the tree structure. Suppose that for each s ∈ A n a set M s is given and g : ∪M t → ∪M t is a function such that 1) g(M s ) ⊂ M σ 2 (s) , and 2) M s ∩ M t = ∅ if and only if ∆ s ∩ ∆ t = ∅. Then we say that (g, {M s , s ∈ A n }) has the same (dynamical) intersection pattern as {∆ s , s ∈ A n }. It follows from the results of [START_REF] Buff | Quadratic Julia sets with positive area[END_REF] (see below) that for certain basic Cremer polynomials f there is an invariant continuum M which plays the role of the Siegel disk ∆ above in the sense that the intersection pattern of M and all appropriately defined M -pullbacks of order n and the intersection pattern of ∆ and all ∆pullbacks of order n are the same.

∆ 2 (x)=∆ 2 (z) x R γ z R γ P 3 P 7
2.3. Basic Cremer polynomials. Let us introduce terminology and notation. Let P α (z) = e 2πiα z + z 2 . Clearly, the critical point of P α is c α = (-.5)e 2πiα ; set M α = ω(c α ) and call it the critical limit set of P α . It is well-known (see, e.g., [START_REF] Mañé | On a theorem of Fatou[END_REF]) that if P α is a basic Cremer polynomial then its Cremer fixed point p α belongs to M α . The next lemma is obtained in [C05].

Lemma 2.3 (Childers). If P α is a basic Cremer polynomial then the set M α is a continuum.

The construction in the next section aims at proving that certain Julia sets of positive area are solar. We need the following definition: an angle θ is said to be of bounded type if there exists K such that in the continued fraction expansion [a 1 , a 2 , . . . ] of θ we have a i < K for all i. Consider the polynomial P θ (z) assuming that θ is of bounded type. Then P θ has an invariant closed Siegel disk ∆ θ , ∂∆ θ = M θ is a simple closed curve and J P θ is locally connected [START_REF] Petersen | Local connectivity of some Julia sets containing a circle with an irrational rotation[END_REF]. For every N , define SN as the set of all θ's with a i ≥ N, i = 1, 2, . . . . Given two sets A, B let us denote by ∂[A, B] the number sup{dist(a, B) :

a ∈ A} (thus, if ∂[A, B] = ε then the set A is contained in the closed ε-neighborhood of B). Theorem 2.4 is proven in [BC06].
The key tool in the proof is the use of Inou-Shishikura's results on near parabolic renormalization [IS]. In what follows we will use the notation introduced above.

Main theorem

The aim of this section is to prove Theorem 1.2. From now on we fix N as in Theorem 2.4. For a polynomial P α , we denote by ψ α the involution which maps any z to the other preimage of P α (z). Set ψ α (M α ) = M α . The following lemma uses self-explanatory notation.

Lemma 3.1. Given n, θ ∈ SN of bounded type, and ε > 0 there exists a neighborhood U of θ such that for any α ∈ SN ∩ U we have P i α (c α ) ∈ M α , i ≤ n, and there exists a collection of well-defined M αpullbacks M α,j , j ∈ A n by P α such that ∂[M α,j , ∆ θ,j ] < ε for any j ∈ A n and the M α -pullbacks {M α,j , j ∈ A n } have the same intersection pattern as ∆ θ -pullbacks {∆ θ,j , j ∈ A n }.

Proof. The proof is based upon Theorem 2.4 and continuity arguments. By Theorem 2.4, M α is very close to ∆ θ for α ∈ SN sufficiently close to θ.

Since P i θ (c θ ) ∈ ∆ θ , it follows then that P i α (c α ) ∈ M α , i ≤ n for α ∈ SN sufficiently close to θ (so that an i-th pullback of M α is mapped 1-to-1 if i ≤ n).
Moreover, by continuity and by Theorem 2.4 given an itinerary j ∈ A n we can correctly define M α -pullbacks M α,j by P α corresponding to ∆ θ -pullbacks ∆ θ,j by P θ and guarantee that 1) ∂[M α,j , ∆ θ,j ] < ε for any j ∈ A n , and hence 2) for two itineraries

s ∈ A n , t ∈ A n if ∆ θ,s ∩ ∆ θ,t = ∅ then M α,s ∩ M α,t = ∅. Let us show that if ∆ θ,s ∩ ∆ θ,t = ∅ then M α,s ∩ M α,t = ∅.
Observe that since A n is a tree then if we remove the edge in A n connecting s and t then A n falls into two trees, A s and A t . Now, it is easy to see that the union of all pullbacks M α,r , r ∈ A n is the set P -n α (M α ) which is connected because the entire orbit of c α is contained in M α (c α is recurrent for P α and M α = ω(c α )). On the other hand suppose that M α,s ∩ M α,t = ∅. Then the unions M α,A s = ∪ i∈A s M α,i and M α,A t = ∪ i∈A t M α,i are disjoint because they could only intersect over M α,s ∩M α,t which is empty. This contradiction shows that M α,s ∩ M α,t = ∅ as desired.

We are now ready to prove Theorem 1.2. However it will be convenient to first prove a simple technical lemma. Lemma 3.2. Let P be a basic Cremer polynomial and K ⊂ J be a continuum such that for some k > 0 we have that P -k (K) ∩ K = ∅ and P i (c) ∈ P k (K) for any i ≥ 0. Then there exists a unique sequence K(0) = K, K(-1), . . . of K pullbacks by P k such that K(-i) ∩ K(-i -1) = ∅ and P k (K(-i -1)) = K(-i) for all i ≥ 0. Moreover, if P 2k (K) is disjoint from K then in the sequence of sets P 2k (K), P k (K), K, . . . non-empty intersections are only possible between two consecutive sets.

Proof. Since J is a non-separating one-dimensional continuum, J is a tree-like continuum (i.e., for each ε > 0 there exists a map φ : J → T where T is a finite tree such that for each t ∈ T the diameter of φ -1 (t) < ε). Since P i (c) ∈ K then it is clear that for every i the family of K-pullbacks by P k consists of 2 ki pairwise disjoint continua contained in J. Since P -k (K) ∩ K = 0, there exists a K-pullback K(-1) of order 1 by P k non-disjoint from K. Such pullback is unique. Indeed, otherwise there exists another K-pullback K of order 1 by P k which is not disjoint from K. Set K(-1) ∪ K ∪ K = Y . Then P k | Y is not 1-to-1 which implies by [H96] that P k | Y has a critical point y ∈ Y . This means that for some j, 0 ≤ j ≤ k -1 we have P j (y) = c. If y ∈ K(-1)∪K then this implies that P 2k-j (c) ∈ P k (K), a contradiction to the assumptions. On the other hand, if y ∈ K then P k-j (c) ∈ P k (K), again a contradiction to the assumptions. Hence such point y ∈ Y does not exist, and K(-1) is unique. Since K(-1) clearly satisfies the same assumptions as K itself we see that the desired sequence of pullbacks of K exists and is unique.

Suppose that P 2k (K) ∩ K = ∅. Then P k (K) ∩ K(-1) = ∅ and for any i ≥ 0 we have K(-i) ∩ K(-i -2) = ∅ (otherwise we apply the appropriate power of P k to get a contradiction). Let us show that then K(-j) ∩ P 2k (K) = ∅ for any j ≥ 0. Indeed, otherwise choose the minimal such j that K(-j) ∩ P 2k (K) = ∅. Then j ≥ 1 and by the choice of j the only non-empty intersections among sets P 2k (K), P k (K), K, . . . , K(-j) are intersections among consecutive pullbacks and the intersection K(-j) ∩ P 2k (K) = ∅. Set P 2k (K) ∪ P k (K) ∪ • • • ∪ K(-j + 1) = E. Then E ∩ K(-j) = [P 2k (K) ∩ K(-j)] ∪ [K(-j + 1) ∩ K(-j)] is disconnected as the union of two disjoint non-empty continua. However all continua are contained in a non-separating continuum J with empty interior. Hence the intersection of any two sub-continua of J must be connected. It follows that K(-j) ∩ P k (K) = ∅ for any j ≥ 1 as desired.

To prove Theorem 1.2 we need the following construction. Choose two P θ -periodic points u and v of periods k and l respectively. Then depending on their itineraries the strings S u , S v of ∆ θ -pullbacks will have a few common pullbacks and then, starting at the last common pullback, will consist of two pairwise disjoint sequences of pullbacks.

Figure 3 .

 3 Figure 3. Illustration, for the golden mean basic Siegel Julia set, of the examples of periodic points x and z considered in the text, together with the associated strings of ∆-pullbacks.

  Theorem 2.4 (Buff-Chéritat). There exists N such that for every θ ∈ SN of bounded type and every ε > 0 if α ∈ SN is sufficiently close to θ then ∂[M α , ∆ θ ] < ε and M α = J Pα . Moreover, there exist basic Cremer polynomials P α with α ∈ SN arbitrarily close to θ and with positive area Julia sets.

The first author was partially supported by NSF grant DMS-0456748. The fourth author was partially supported by NSF grant DMS-0405774.

1

Clearly, the points u and v can be chosen so that the strings S u and S v have at least two common ∆ θ -pullbacks. For example, if u = x has itinerary {011} ∞ and v = z has itinerary {0110111} ∞ then the strings S x and S z have two pullbacks ∆ θ = ∆ θ,01 ∞ and ∆ θ,01101 ∞ in common, yet from the third pullback on the strings S x and S z are disjoint. In any case, and this is important for what follows, the last common pullback of the strings S x and S z is ∆ θ,01101 ∞ , and it is disjoint from ∆ θ .

Let us assume that the basic length of S u is w, the basic length of S v is q, and the initial finite string F , common to both S u and S v , consists of m ≥ 2 ∆ θ -pullbacks. For simplicity and without loss of generality assume that m < min(w, q). Denote the last common ∆ θ -pullback in F by L. Consider w ∆ θ -pullbacks in S u immediately following F and denote their union by Fu ; also, denote the union of w ∆ θ -pullbacks in S u immediately following Fu by F u . Similarly, consider q ∆ θ -pullbacks in S v immediately following F and denote their union by Fv ; also, denote the union of q ∆ θ -pullbacks in S v immediately following Fv by F v . By Lemma 3.2 there exists a string of F u -pullbacks by

. Set n = m + 3k + 3l. We choose α ∈ SN very close to θ so that a few conditions are satisfied. By Theorem 2.4 we may assume that the area of J(P α ) is positive and that P α is a basic Cremer polynomial. We use Lemma 3.1 to guarantee that ∂[M α , ∆ θ ] is so small that the first n iterates of c α avoid M α (so that M α -pullbacks by P α of order at most n map to M α univalently), all M α -pullbacks by P α of order n are very close to the corresponding ∆ θ -pullbacks by P θ and the intersection patterns of ∆ θ,i , i ∈ A n and M α,i , i ∈ A n are the same. Hence we may talk about the strings

for any i ≥ 0. On the other hand, by the choice of n we have that

and M α united with M α itself). By Lemma 3.2 we conclude that the set

is a chain of "concatenated" continua such that intersections among them are only possible between F α u and F α u = F α u (0) and between two consecutive pullbacks of F α u . Analogous claims can be proven for F α v and F α v . For the corresponding pullbacks of F α v we use similar notation and get the set

a chain of "concatenated" continua such that intersections among them are only possible between F α v and F α v = F α v (0) and between two consecutive pullbacks of F α v . It is easy to see that the sets Q α u and Q α v are disjoint. Indeed, suppose otherwise. Then we can choose minimal r, s such that

have a disconnected intersection (it consists of L α and a non-empty compact set F α u (-r) ∩ F α v (-l) disjoint from L α ) despite the fact that they both are contained in the Julia set J(P α ) (recall that the Julia sets of basic Cremer polynomials are non-separating continua with empty interior).

Since M α ∩ ( F α u ∪ F α v ) = ∅ by the choice of α then by the Shrinking Lemma [START_REF] Lyubich | Laminations in holomorphic dynamics[END_REF] we know that diam(F α u (-i)) → 0 as i → ∞ (i.e., F α u (-i), i = 0, 1, . . . is a null-sequence). Hence by continuity any limit point a of this sequence of sets is P k α -fixed. Since there are finitely many P k α -fixed points while the set of all limit points of the sequence of sets F α u (-i), i = 0, 1, . . . is connected (recall, that this sequence of sets is a chain of "concatenated" continua) we conclude that the sequence F α u (-i), i = 0, 1, . . . converges to a P k α -fixed point which we will denote by u . Similarly, the sequence F α v (-i), i = 0, 1, . . . converges to a P l αfixed point which we will denote by v .

Let us study possible intersections between some of these sets. Set

Indeed, since u is a P k -fixed point then u ∈ Q α u because of the way sets F α u (-i), i ≥ 0 intersect. Suppose that u ∈ Z and consider two continua,

notice that by the above u , v ∈ M α . Let R γ be the external ray for P α landing at u and R β be the external ray for P α landing at v . The above implies that the union

into two open half-planes W and H, one of which (say, W ) contains M α (and therefore the Cremer point p α of P α ). Choose any external angle τ whose external ray R τ is contained in H. Then it follows that the impression of R τ is contained in L α ∪ Q α u ∪ Q α v and hence does not contain p α . By Theorem 1.1 this implies that J(P α ) is a solar Julia set which completes the proof of Theorem 1.2.


Xavier Buff

Arnaud Chéritat
email: cheritat@picard.ups-tlse.fr

How regular can the boundary of a quadratic Siegel disk be?

may come

How regular can the boundary of a quadratic Siegel disk be?

Xavier Buff, Arnaud

Introduction

In 1995, R. Pérez-Marco [START_REF] Pérez-Marco | Siegel disks with smooth boundary[END_REF] constructed the first known examples of Siegel disks with C ∞ boundaries. The authors [BC] discovered later an independent proof of the existence of Siegel disks with C ∞ boundaries in the family of quadratic polynomials P θ (z) = exp(i2πθ)z + z 2 , θ ∈ R/Z.

A. Avila formalized, simplified and extended these results in [A] (these two preprints have now been published as [ABC]). L. Geyer further simplified it in [G]. These smooth Siegel disks were provided by an elementary construction based on a perturbation lemma (recalled as lemma 2 below), which was unknown at the time. Here, we describe a procedure which introduces irregularities on the boundaries of Siegel disks. For this, we use an old result of Herman (see lemma 1 below). Combining it with our previous work, it enables us to prove much more: there exist quadratic Siegel disks for which the regularity of the boundary is anywhere in the full spectrum between analytic and continuous. Since the possibilities are numerous, we will formulate our main theorem in a general and abstract form. As a particular case, we will obtain that for all n ∈ N, there exist quadratic Siegel disks whose boundaries are Jordan curves which are C n but not C n+1 embedded curves.

To state the main theorem, let us introduce a few notations. We will denote by U the set of complex numbers of norm 1. If P θ is linearizable in a neighborhood of its indifferent fixed point z = 0, we denote by ∆ θ its Siegel disk (maximal domain of linearization) and by φ θ : r θ D → ∆ θ the unique conformal map with φ θ (0) = 0 and φ θ (0) = 1. This φ θ is also the linearizing map, and thus satisfies where R θ (z) = exp(i2πθ)z. If P θ is not linearizable, we set ∆ θ = ∅ and r θ = 0. Let C 0 be the space of holomorphic functions from D to C having a continuous extension to D. This is a Banach space for the supremum norm. Let C ω be the space of functions from D to C having a holomorphic extension to a neighborhood of D.

φ θ • R θ = P θ • φ θ , 2000 Mathematics Subject Classification. 37F50, 37F10, 46B50.
Definition 1. Let X and Y be topological spaces and X ⊂ Y . We write X ⊂ 0 Y if the canonical injection X → Y is continuous. If moreover X is a normed vector space and Y a Fréchet or a Banach space, we write X ⊂ c Y if every bounded set in X has compact closure in Y .

Theorem 1. Let F be any Fréchet space such that C ω ⊂ F ⊂ 0 C 0 , and let B ⊂ c F be a Banach space. Then, there exists a Bruno number θ such that

• the map z → φ θ (r θ z) belongs to F but not to B

• the boundary of ∆ θ is a Jordan curve that does not contain the critical point of P θ . Equivalently, one can replace the Banach space B ⊂ c F by a countable union of Banach spaces B n ⊂ c F or by a countable union of compact sets K n ⊂ F .

There are many Fréchet and Banach spaces or compact subsets that can be considered here. To give a sample of what can be obtained from theorem 1, we will restrict ourselves to the following C α conditions. Let f :

D → C be holomorphic. If α = n is an integer, we say that f ∈ C n if f (k) ∈ C 0 for 0 ≤ k ≤ n. If α = ∞, we say that f ∈ C ∞ if f ∈ C n for every integer n. If α = n + ε for some integer n and some ε ∈ ]0, 1[, we say that f ∈ C n+ε if f ∈ C n and f (n) is ε-Hölder on D.
For α ∈ [0, +∞[, C α is a Banach space 1 and for α < β, the injection of C β in C α is compact (the proof is similar to the proof of Ascoli's theorem).

Geometry vs Analysis: we control the regularity of the boundaries of Siegel disks via the regularity of the linearizing maps. We will need a few results on the boundary behavior of conformal maps for which we refer the reader to [Po]. For α ≥ 1, there is a notion of C α Jordan curve (see [Po], section 3.3): these are curves that have a parameterization that is C α with non-vanishing first derivative.

Theorem 2 (Kellog-Warschawski). Assume α ∈ ]1, +∞] is not an integer and Γ ⊂ C is a Jordan curve bounding a simply connected domain U containing 0. Then, Γ is a C α Jordan curve if and only if the conformal map φ : (D, 0) → (U, 0) belongs to C α with non-vanishing derivative on D.

The situation is less simple when α = n is a positive integer. If the conformal map of a Jordan curve J is in C n and if the first derivative does not vanish, then obviously J is a C n Jordan curve. However, there are C n Jordan curves whose conformal maps are not in the space C n .

1 for φ ∈ C n+ε and ε ∈]0, 1[, we set

φ C n+ε = n k=0 φ (k)
sup + min λ > 0 λx ε is a modulus of continuity for φ (n) .

In the quadratic family, if the conformal map of a Siegel disk is at least C 1 , then its derivative cannot vanish on D (this is an easy exercise). Theorem 2 has the following consequence: for a non-integer α ∈ ]1, +∞],

z → φ θ (r θ z) ∈ C α ⇐⇒ ∂∆ θ is a C α Jordan curve.
Let us now state some corollaries of theorem 1.

Corollary 1. There exist quadratic Siegel disks with C ∞ boundaries.

Proof. Take F = C ∞ and K n = ∅.
In [PZ], C. Petersen and S. Zakeri proved that for almost every θ ∈ R/Z, the quadratic polynomial P θ has a Siegel disk whose boundary contains the critical point, is a Jordan curve but is not a quasicircle. Theorem 1 implies:

Corollary 2. There exist quadratic Siegel disks whose boundaries are Jordan curves that do not contain the critical point, and are not quasicircles.

Proof. The conformal map of a quasidisk is always Hölder (see [Po]). So, choose

F = C 0 and B n = C 1/n (n ≥ 2).
Corollary 3. For every n ∈ N, there exist quadratic Siegel disks whose boundaries are Jordan curves that do not contain the critical point, and are C n but not C n+1 Jordan curves.

Proof. Take F = C n , B = C n+1/2 . Then, theorem 1 gives Siegel disks whose boundaries are C n but not C n+1/2 Jordan curves. (Note that taking B = C n+1 does not work; see the discussion Geometry vs Analysis above).

If P θ has a Siegel disk, set

I θ = α ∈ [0, +∞[ φ θ (r θ z) ∈ C α .
This is an interval, either empty or with left bound = 0. It contains 0 if and only if the boundary of ∆ θ is a Jordan curve (see [M] or [Ro]). However, it is yet unknown if ∂∆ θ is always a Jordan curve. So, it is unknown if I θ can be empty, but the following proves that all other cases happen.

Corollary 4. For every α ≥ 0, there exists a Bruno number θ such that I θ = [0, α] and for every β > 0, there exists a Bruno number θ such that

I θ = [0, β[. Proof. For the first case, choose F = C α and B n = C α+1/n . For the second case, choose F = α∈[0,β[ C α , B = C β .
Remark. For n ∈ N, let C n+Lip be the set of holomorphic functions on D whose n-th derivative is Lipschitz. The inclusion of B = C n+1 in F = C n+Lip is not compact (it is an isometry). Thus we cannot apply theorem 1. We may wonder whether there exist (fixed quadratic) Siegel disks whose conformal maps belong to C n+Lip but not C n+1 . In fact, for n > 0, this is impossible by a theorem of Gottschalk and Hedlund (see [H1] proposition 4.2 in part IV).

Proof of theorem 1.

It is equivalent to prove theorem 1 for a Banach space B ⊂ c F , for a countable union of Banach spaces B n ⊂ c F or for a countable union of compact sets K n ⊂ F , as shown in the following Proposition 1. If F is a Fréchet space, then

(1) the image of any Banach space B under a compact injection is contained in a countable union of compact subsets of F ;

(2) any countable union of compact subsets K n of F is contained in the image of a Banach space by a compact injection.

Proof. For (1), let K n be the closure in F of the ball of radius n in the B-norm.

Obviously, B ⊂ n∈N K n . Moreover, by the very definition of a compact injection, K n is compact. For (2), let us reduce the problem to the case of a single compact set. Let λ n > 0 be a sequence so that λ n K n -→ 0. Let K = {0} ∪ n∈N λ n K n . Then K is compact, and any vector subspace of F containing K will contain n∈N K n . Next, we replace K by DK = λf |λ| ≤ 1, f ∈ K . Then we can replace K by the closure of the convex hull of K, which is compact by a well known property of compact sets in Fréchet spaces (see [Ru], Theorem 3.20). Now K is both convex and invariant under multiplication by elements of D. Now B = t∈R+ tK equipped with the norm

f B = inf{t ∈ R + | f ∈ tK} is a Banach space that is compactly injected in F .
We will prove theorem 1 for a countable union of compact sets K n ⊂ F . In fact, we will prove the following stronger version of theorem 1.

Theorem 3. Let F be any Fréchet space such that

C ω ⊂ F ⊂ 0 C 0 ,
and let (K n ) n∈N be any sequence of compact subsets of F . Then, for every Bruno number θ, every r < r θ and every ε > 0, there exists a Bruno number θ such that

(1) |θ -θ| < ε (2) r < r θ < r + ε (3) the map z → φ θ (r θ z) belongs to F but to no K n (4) the distance in F between z → φ θ (r θ z) and z → φ θ (rz) is < ε. Moreover, the continuous extension ψ of φ θ to r θ D, which exists by (3), satisfies

(5) ψ(r θ z) -φ θ (rz) ∞ < ε on D (6) ψ(r θ U) = ∂∆ θ is a Jordan curve that does not contain the critical point.

It is enough to prove (1)-( 4) only. Indeed, (3) implies the existence of the extension since F ⊂ C 0 . This extension must be injective (see [M], lemma 18.7), which shows that the boundary of the Siegel disk is a Jordan curve. Then, (5) follows from (4) (with a smaller ε), by continuity of the inclusion F ⊂ C 0 . And (5) implies that for ε small enough, the boundary of the Siegel disk does not contain the critical point.

2.1. Tools. The proof is based on a perturbation lemma 9, that will be stated in section 2.4. There are two results that make the perturbation lemma possible: lemma 1 provides irregularity and lemma 2 provides regularity.

Let D 2 denote the set of bounded type irrational numbers. For a proof of the following lemma, see for example [H2].

Lemma 1 (Herman). Let θ ∈ D 2 , U be a connected open set containing 0 and f : U → C be a holomorphic function which fixes 0 with derivative e 2πiθ . Let ∆ be the Siegel disk of f at 0 (which exists by a theorem of Siegel). If U is simply connected and f is univalent, then ∆ cannot have a compact closure in U .

Remark. In fact, Herman's theorem is stronger: U needs not be simply connected, and the condition on θ can be weakened to the so-called Herman condition. See [H2] for a reference. In the case of quadratic polynomials, lemma 1 is a consequence of the following fact: a bounded type quadratic Siegel disk contains the critical point on its boundary.

For a proof of the following lemma, see [BC] or [ABC].

Lemma 2. For every Bruno number θ and every r < r θ , there exists a sequence θ n ∈ D 2 such that θ n -→ θ and r θn -→ r.

The references [A] and [G] contain similar statements without requiring θ n ∈ D 2 . It is in fact easy to adapt the proof in [A] and get D 2 numbers. We do not know if [G] can produce D 2 numbers.

Elementary properties of the linearizations.

Let us state well known properties concerning the dependance of φ θ on θ.

Lemma 3. Let θ n -→ θ be real numbers and assume that r = lim inf r θn > 0. Then r θ ≥ r and φ θn tends to φ θ uniformly on compact subsets of rD.

Proof. The univalent maps φ θn all have derivative 1 at 0, hence form a normal family. Let ψ be the limit of a convergent subsequence (for uniform convergence on compact subsets of rD). Then ψ (0) = 1, so in particular ψ is not constant. Moreover, by passing to the limit in the equation P θn • φ θn = φ θn • R θn , we have

P θ • ψ = ψ • R θ .
Therefore, uniqueness of the linearization implies r θ ≥ r and ψ = φ θ on rD.

Lemma 4. Assume r n > 0, r n -→ r > 0, θ ∈ R and θ n -→ θ is a sequence of real numbers such that r θn > r n , and the maps z → φ θn (r n z) converge uniformly on U to some ψ. Then r θ ≥ r, and z → ψ(z) extends continuously φ θ to the closure of B(0, r) (if r θ > r, this just means φ θ (rz) = ψ(z)). Moreover, ψ is injective. In the case r θ = r, we have ∂∆ θ = ψ(U).

Preparatory work.

Note that C ω is not a Fréchet space. It is the union over ε > 0 of spaces C ω ε of holomorphic functions on (1 + ε)D. Each C ω ε is endowed with the topology of uniform convergence on compact sets (in fact it is a Fréchet space; if you prefer Banach spaces, take the space of bounded holomorphic functions on (1 + ε)D and the discussion below will be the same). We do not put a topology on C ω . Lemma 5. Let F ⊂ 0 C 0 be a Fréchet space and K be a compact subset of F . Assume f n ∈ K tend to some map f : D → C uniformly on compact subsets of D.

Then f ∈ K and a) d F (f n , f ) -→ 0 where d F is the distance function of the Fréchet space F , b) f n -f ∞ -→ 0.
Proof. Part b) is a corollary of part a) and the continuity of the injection F ⊂ C 0 . Since f n lies in the compact set K, it has a subsequence

f n k such that d F (f n k , h) -→ 0 for some h ∈ K. The injection F ⊂ C 0 being continuous, f n k -h ∞ -→ 0.
Since f n tends to f uniformly on compact subsets of D, we must have h = f . This proves f ∈ K, and that f is the only limit of convergent subsequences of f n in the d F -metric. Since f n lies in a compact set, this gives part a).

Lemma 6. Assume F is a Fréchet space with C ω ⊂ F ⊂ 0 C 0 . Then, for all ε, the injection C ω ε ⊂ F is continuous. Proof. This is a corollary of the closed graph theorem, since the injection of C ω ε in C 0 is continuous. Lemma 7. There exist subsets D n of C ω with C ω = D n such that for every Fréchet space F with C ω ⊂ F ⊂ 0 C 0 , D n is compact in F .
Proof. D n be the set of holomorphic functions in (1 + 1 n+1 )D whose absolute value is bounded by n. Each D n is compact in C ω 1 n+1 (Montel's theorem). According to lemma 6, D n is also compact in F .

We will consider the following property of a subset A of C 0 : (H)

For every ε > 0, A contains a neighborhood of 0 in C ω ε .

Lemma 8. For every Fréchet space F with C ω ⊂ F ⊂ 0 C 0 , there exists a compact subset L of F with property H.

Proof. Consider the D n from lemma 7. Since D n is compact in F , λD n tends to 0 when λ -→ 0. Choose λ n > 0 such that λ n D n is included in the ball of F of center 0 and radius 1/(n + 1). Then the set L = λ n D n is compact (note that 0 ∈ L) and has the property H since D n contains a neighborhood of 0 in C ω 1 n .

Main lemma.

Lemma 9 (perturbation). Assume that F is a Fréchet space with C ω ⊂ F ⊂ 0 C 0 , and K is a compact subset of F . For every Bruno number θ, every ρ 1 < ρ 2 < r θ , and every ε > 0 there exists a Bruno number (in fact bounded type) θ and ρ > 0 such that

(1) |θ -θ| < ε (2) ρ 1 < ρ < ρ 2 (3) r θ > ρ (4) d F (φ θ (ρ z), φ θ (ρ z)) < ε (5) φ θ (ρ z) does not belong to K.
Proof. Let ρ 3 = ρ1+ρ2 2 . If, for some ρ ∈]ρ 1 , ρ 3 [, φ θ (ρ z) satisfies ( 5), then we are done with θ = θ. Otherwise, use lemma 2 to find θ n -→ θ such that r θn -→ ρ 3 . By lemma 3, φ θn tends to φ θ uniformly on compact subsets of ρ 3 D. According to lemma 6, the injection C ω ε ⊂ F is continuous for all ε > 0. On the other hand, there is some

ε > 0 such that the function [ρ 1 , ρ 3 ] → C ω ε defined by ρ → φ θ (ρz) is continuous. Therefore, its image K 0 is compact in C ω ε , hence compact in F .
Assume L is a compact subset of F with property H (see lemma 8). Set K = (K+L)∪(K 0 +L). We claim that for large enough n, there exists a ρ ∈ [ρ 1 , r θn [ such that φ θn (ρz) does not belong to K . Otherwise, by lemma 5 φ θn (r θn z) would belong to K . Lemma 5 b) would imply that φ θn (r θn z) -φ θ (ρ 3 z) ∞ -→ 0. Thus eventually, ∂∆ θn would be contained in ∆ θ , contradicting lemma 1 since θ n ∈ D 2 .

Let then ρ n be the infimum of the set of ρ ∈ [ρ 1 , r θn [ such that φ θn (ρz) does not belong to K . Note that φ θn (ρ n z) does not belong to K. Otherwise, since L has property H, φ θn (ρz) would belong to K + L ⊂ K for all ρ > ρ n sufficiently close to ρ n , which contradicts the definition of ρ n . This gives (5) with ρ = ρ n for n large enough.

Let us prove that ρ n -→ ρ 3 . Otherwise, for a subsequence, we would have ρ n -→ r ∈ [ρ 1 , ρ 3 [. With the definition of ρ n , this would yield a sequence ρ n -→ r such that φ θn (ρ n z) does not belong to K . The sequence of holomorphic functions z → φ θn (ρ n z) would converge uniformly on compact subsets of ρ3 r D to φ θ (rz). Because of property H, the function (φ θn -φ θ )(ρ n z) would eventually belong to L, so φ θn (ρ n z) would belong to L + K 0 ⊂ K , which is a contradiction.

As soon as ρ n > ρ 1 , φ θn (ρ n z) is in K : indeed, for all ρ ∈ [ρ 1 , ρ n [, φ θn (ρz) is in K , and the claim follows from lemma 5.

Lemma 5 a) now implies that d F (φ θn (ρ n z), φ θ (ρ 3 z)) -→ 0 when n -→ +∞. And ρ n -→ ρ 3 implies d F (φ θ (ρ 3 z), φ θ (ρ n z)) -→ 0. This gives (4).

Proof of theorem 3.

Let D n be given by lemma 7, and L by lemma 8.

We are going to define by induction on n ≥ -1 a sequence θ n of parameters, an increasing sequence ρ n ≥ r, and real numbers ε n > 0.

The induction hypothesis will be H n :

• r θn > ρ n

• for all k with 0 ≤ k ≤ n, the F -distance between φ θn (ρ n z) and the set

K k ∪ D k ∪ L is > ε k
Let θ -1 = θ, ρ -1 = r, ε -1 = 1 (ε -1 will not be used).

Given n ≥ 0, assume that θ k , ρ k , ε k are defined for -1 ≤ k < n, and that H n-1 holds. There exists an η > 0 such that for all f ∈ F and all k with 0 ≤ k < n, the condition d F (f, φ θn-1 (ρ n-1 z)) < η implies the F -distance between f and K k ∪D k ∪L remains > ε k . Let ρ 1 = ρ n-1 and ρ 2 such that ρ 1 < ρ 2 < r θn-1 , close enough to ρ 1 so that for every ρ ∈ [ρ 1 , ρ 2 ],

( * ) d F (φ θn-1 (ρ z), φ θn-1 (ρ 1 z)) < 1 2 min(η, ε/2 n+1 ) (this is possible since the injection of C ω ε ⊂ F is continuous for all ε > 0). Let θ n and ρ n be provided by lemma 9 such that

• |θ n -θ n-1 | < ε/2 n+1 • ρ 1 < ρ n < ρ 2 • r θn > ρ n
• d F (φ θn (ρ n z), φ θn-1 (ρ n z)) < 1 2 min(η, ε/2 n+1 ) • φ θn (ρ n z) does not belong to K n ∪ D n ∪ L

  1

We then define ε n = 1 2 d F (K n ∪ D n ∪ L, φ θn (ρ n z)). It is easy to check that H n holds. Now that the sequences θ n , ρ n , ε n have been defined, let θ be the limit of the Cauchy sequence θ n , and ρ the limit of the increasing sequence ρ n (which is bounded from above by 4). Observe that for all n, r θn > ρ n , and that by equation ( * ) the sequence of maps φ θn (ρ n z) (restricted to D) is a Cauchy sequence for d F , thus converges in F to some ψ (we use here the completeness of Fréchet spaces). Convergence in F implies convergence in C 0 , thus we can apply lemma 4 to conclude that ψ(z) = φ θ (ρ z) on D. Also, d F (K n ∪ D n ∪ L, ψ) ≥ ε n , so ψ does not belong to any K n nor to any D n . Since D n = C ω , this implies ψ does not extend holomorphically to a neighborhood of D, thus r θ = ρ .

Acknowledgments

We would like to thank Adrien Douady for useful discussions. We would like to thank the referee for his constructive comments and his work in examine our paper.

Concluding remark

All of the above discussion extends to any analytic family for which an analogue of lemma 2 can be proved.


Arnaud Chéritat

Dierk Schleicher

The hunt for Julia sets with positive measure

The hunt for Julia

Introduction

The title of the talk I gave at Hubbard's 60th birthday conference was "Are there Julia sets with positive measure?". Four months later I gave a talk in Denmark where the first two words were inverted and the question mark became a period.

I do not know when Adrien Douady set up his plan for finding a Julia set with positive measure, but in 1998 he offered me to study it for my PhD thesis. He added that his opinion about whether or not such Julia sets existed depended on his mood. If the plan worked it would solve an open conjecture. But in case of failure, it would not say anything on the measure of the objects he was looking at (quadratic Julia sets with a Cremer point). 1When I finished my thesis in 2001, I had convinced Adrien that his plan would work, by proving it would follow from a pair of reasonable conjectures, supported by analogy with known results for Julia sets with a bounded type Siegel disk, also supported by computer pictures.

Adrien told me cylinder renormalization could help in proving these conjectures and had me and Xavier Buff meet and begin a very productive collaboration on quadratic Julia sets. I also went to Toronto for six months where I met Yampolsky and had many discussions with him. However cylinder renormalization turned out to be difficult. We lacked first an invariant class, second an argument to turn this into a proof of my conjectures.

An invariant class was found by Inou and Shishikura in 2003. To turn it into a proof still required some time and effort. I remember three key periods. First, I was asked to write a review for the French Mathematical Society on the book "The Mandelbrot Set, Theme and Variations" edited by Tan Lei [T]. In the process, I learnt important methods in holomorphic dynamics, and this knowledge proved very useful. Second, this is in the favorable atmosphere of Hubbard's 60th birthday conference that I saw the end of the tunnel, i.e. how things may fit together to provide a proof. The third period was when Xavier and I worked hard during the following summer holiday to make it work.

Crash-course

This section introduces some terminology for the non-experts or readers in the distant future. See for instance [M] for a more thorough introduction.

In this article, it is understood that we work with complex polynomials, i.e. with some P ∈ C[X]. A quadratic polynomial is just any degree 2 polynomial. We are interested in the iteration of P , i.e. in the behavior of sequences z n defined by z n+1 = P (z n ), and how this behavior depends on z 0 and on P . Here P n will refer to the composition P • • • • • P , and not to the product.

The orbit of a point is the set P n (z) n ∈ N . A periodic point is a point z 0 such that z k = z 0 for some k > 0. Its period is the least value of k such that this hold. A cycle is the orbit of a periodic point. A period one periodic point is also called a fixed point. Its multiplier is the derivative λ = (P k ) ′ (z 0 ). The periodic point is

repelling if |λ| > 1, indifferent if |λ| = 1, attracting if |λ| < 1 and superattracting if λ = 0.
Superattracting is considered as a subcase of attracting. For instance, ∞ is a superattracting fixed point of any polynomial of degree > 1.

The periodic point z 0 is linearizable when there exists a homeomorphism φ from a neighborhood of z 0 to a neighborhood of 0 such that φ • P k • φ -1 (z) = λz holds in a neighborhood of z 0 . For dimension one analytic maps, the existence of a continuous conjugacy to the linear part is known to imply the existence of an analytic one.

If |λ| = 0 and |λ| = 1, then the periodic point is automatically linearizable. If |λ| = 0 then it can be conjugated to z → z d in a neighborhood of 0, for some integer d ≥ 2. If |λ| = 1, then write λ = e 2iπθ for some θ ∈ R. This θ is called the rotation number. In the non-linearizable case, z 0 is called a Cremer point, and the dynamics becomes very complicated.

If θ ∈ Q ( ⇐⇒ λ is a root of unity), then z 0 is a parabolic point. It is most of times non-linearizable. If θ / ∈ Q, then z 0 is
The basin of an attracting cycle is the set of points whose orbit tends to the cycle (i.e. end up staying in every neighborhood of the cycle). The filled-in Julia set K(P ) of a polynomial P of degree ≥ 2 is the complement of the basin of ∞. It is a non-empty compact subset of C. The Julia set J(P ) has three equivalent and standard definitions. It is the boundary of K(P ) (which is the same as the boundary of the basin of ∞). It is the closure of the set of repelling periodic points. It is the complement of the Fatou set F , where z ∈ F ⇐⇒ there exists a neighborhood V of z on which the family {P n } n∈N is normal.

A degree 2 polynomial has a connected Julia set J(P ) ⇐⇒ K(P ) is connected ⇐⇒ the (unique) critical point of P belongs to K(P ). The Mandelbrot set M is the set of c ∈ C such that the quadratic polynomial P c (z) = z2 +c has a connected Julia set. It is thus also the set of c such that the critical point z = 0 of P c has a bounded orbit. The latter definition is used for making computer pictures of M .

A hyperbolic polynomial P is one which has an iterate P n , n > 0, whose derivative is expanding (for the spherical metric) on all its Julia set.

Status of the hunt at the time of the conference

This section chiefly presents the content of the talk I gave for Hamal's 60 th birthday conference, in a more detailed way and completed by references.

Motivation

The Julia set is the locus of chaos. To have a Julia set of positive measure means that a randomly chosen point in the Riemann sphere has a non zero probability of being in the locus of chaos. It is interesting in itself to know whether or not this may happen for polynomials.

Yet, the question has strong links with a set of still open conjectures:

Conjecture (Fatou) In the set of degree 2 polynomials, hyperbolic ones are dense. 2

Conjecture (MLC) The Mandelbrot set is locally connected.

Conjecture (NILF)

The Julia set of a quadratic polynomial carries no invariant line field. 3

It was also conjectured for higher degree polynomials, and also for rational maps whose Julia set is not the whole Riemann sphere (there exist rational maps whose Julia set is the whole Riemann sphere).

We have the following known relations between them (as of 2005):

Fatou

Mañe, Sad, Sullivan

⇐⇒ NILF = ⇒ MLC Douady Hubbard = ⇒ immediate Zero
For MLC =⇒ Fatou, see [DH] and [Sc]. For Fatou ⇐⇒ NILF, see [MSS] and [Mc].

The Zero conjecture thus fitted well in the picture, all the more so as its analog for Kleinian group, Ahlfors' conjecture, was proved in 2004.

Yet, the Zero conjecture fails [BC].

3.2 Known results (where not to look at) for quadratic polynomials

In the rest of this article P denotes a polynomial of degree ≥ 2 (and most of times = 2), J(P ) its Julia set, Leb the Lebesgue measure, dim H the Hausdorff dimension, M the Mandelbrot set, and Pol 2 the set of polynomials of degree 2, also called quadratic polynomials. All degree 2 polynomials are C-affine conjugated to a unique polynomial of the form

Q c = z 2 + c.
Recall that a hyperbolic polynomial or rational fraction is one which has an iterate whose derivative is expanding (for the spherical metric) on all its Julia set. This is equivalent to all the critical points belonging to attracting or superattracting basins. The Lebesgue measure of such a Julia set is then equal to 0 (see [DH]), and even better (see [Su]):

Theorem 1.1 (Sullivan) A hyperbolic rational fraction has a Julia set J of Hausdorff dimension < 2 (thus zero Lebesgue measure):

dim H J < 2. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐

Status of the hunt at the time of the conference 5

Let's recall a basic fact (yet not obvious):

Theorem 1.2 (Fatou, Julia, Douady, Hubbard) A degree 2 polynomial has at most one non repelling cycle.

The proof of this theorem can be found in [D2] and [Bl]. Next, by [L]:

Theorem 1.3 (Lyubich, Shishikura) If P has no indifferent periodic points and is not infinitely renormalizable, then

Leb J(P ) = 0.

So, to find positive measure in Pol 2 , we must look either at infinitely renormalizable polynomials at polynomials with an indifferent cycle These two conditions are mutually exclusive, and prevent P from being hyperbolic.

The three theorems mentioned above show that there are plenty of "small" Julia sets, with different notions of what is "small". Let us cite a theorem by [Sh] providing plenty, in some sense, of Julia sets that are "big", in some sense: Remark 1.5 This may sound like an encouragement, but in case (S1), for a Baire generic set of values of c ∈ ∂M, P has no indifferent cycle, and is not renormalizable, thus by a previously mentioned theorem: Leb J(P ) = 0.

Let us focus on quadratic polynomials with an indifferent cycle. According to the Douady-Hubbard theory of tuning (quadratic-like renormalization of quadratic maps, [START_REF] Douady | On the dynamics of polynomiallike mappings[END_REF]) it is enough to study the case where the cycle has period 1. Then, P is C-affine conjugated to

P θ : z → e 2iπθ z + z 2
for a unique θ ∈ R. In the rational case we have [ADU]:

✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 6
1. The hunt for Julia sets with positive measure 2 . The basin of infinity is in white, the Julia set in black and the rest in gray. The latter consists in the Siegel disk and its iterated preimages. We also drew a few invariant curves within the Siegel disk and marked its center z = 0 with a cross.

Theorem 1.6 (Aaronson, Denker, Urbanski) If θ ∈ Q then dim H J(P θ ) < 2.
Recall that the set of bounded type irrationals, i.e. of numbers θ = a 0 + 1 a 1 + . . . with a n bounded, has measure equal to 0. From [START_REF] Mcmullen | Self-similarity of Siegel disks and Hausdorff dimension of Julia sets[END_REF]:

Theorem 1.7 (McMullen) If θ is a bounded type irrational then dim H J(P θ ) < 2.
Let PZ be the set of irrational θ = a 0 + 1

a 1 + . . . such that ln a n = O( √ n).
This is a set of full measure in R. By [PZ]:

✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 3.
Status of the hunt at the time of the conference 7

Theorem 1.8 (Petersen, Zakeri) If θ ∈ PZ, then z = 0 is linearizable and Leb J(P ) = 0.

Douady's plan

It takes place in the family

P θ (z) = e 2iπθ z + z 2
with θ ∈ R. We recall that z = 0 is an indifferent fixed point and that all other periodic points are repelling (by Theorem 1.2).

Conjecture (Douady) There exists a value of θ ∈ R such that z = 0 is a Cremer point of P θ and such that Leb J(P θ ) > 0.

In short, the idea is to look at the filled-in Julia set K(P θ ) when its interior is not empty and perturb θ so as to lose much interior but not too much measure. Let us detail this.

Let us list a few properties concerning J = J(P ) and K = K(P ) where P is a polynomial of degree at least 2:

(Fatou, Julia) J = ∂K, (Fatou, Julia, Sullivan) int(K) = ∅ ⇐⇒ K has a cycle which is either attracting, parabolic or Siegel, (Douady [D]) the map P → Leb(K) is upper semi-continuous.

The idea is then to construct a sequence of θ n ∈ R such that (D1) θ n -→ θ ∈ R, (D2) each P θn is Siegel or parabolic at 0 (then int K(P θn ) = ∅), (D3) P θ is Cremer at z = 0, (D4) Leb int(K(P θn )) is bounded below, i.e. we do not lose too much measure between two successive θ n . Then, by (D1), (D4) and the semi-continuity mentioned above, Leb K(P θ ) > 0. By (D3), int(K(P θ )) = ∅, i.e. J(P θ ) = K(P θ ):

Leb J(P θ ) > 0.
To build such a sequence, θ n+1 is defined as a perturbation of θ n . The main problem is to ensure that the measure loss for the interior of the filled-in Julia set is small, so as to have (D4) hold. Adrien proposed to do this in two steps:

✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 1.
The hunt for Julia sets with positive measure Start from a θ n ∈ R \ Q with P θn having a Siegel disk, and take θ n+1 ∈ Q to be one of its convergents.

Start from θ n+1 ∈ Q and take θ n+2 ∈ R \ Q close to θ n+1 to be one of its special irrational perturbations, as described below. Such a P θn+2 has a Siegel disk.

It is easy to get (D3) if the perturbations can be taken arbitrarily small. The special sequence of irrational perturbations (α k ) k∈N of a rational p/q is defined as follows: write one of the two (necessarily finite) continued fraction expansions p/q = [a 0 , . . . , a m ] = a 0 + 1

a 1 + 1 . . . + 1 a m then let 4 α k = [a 0 , a 1 , . . . , a m , k, 1, 1, . . .].
So here is a Wish list he asked me to check as a PhD thesis subject:

Wish 1 Let α be such that P α has a Siegel disk (with possible additional conditions, like "α has bounded type"). Let p n /q n be the continued fraction convergents of α. Then

lim inf n→+∞ Leb • K(P pn/qn ) ≥ Leb • K(P α ),
in other words, the loss of measure of interiors is tame.

This allows to do Adrien's first step: replacing some θ n for which P θn has a Siegel disk by one of its convergent without losing too much measure, and calling this convergent θ n+1 .

Yet, when one perturbs a rational, the theory of parabolic implosion [D] easily implies that there is a definite loss of measure, which necessarily complicates the second step:

lim sup α-→p/q Leb K(P α ) < Leb K(P p/q ) = Leb • K(P p/q ).
To miminize this loss, Adrien proposed to take the special sequence α k -→ p/q described above. Let loss(p/q) be defined by:

lim inf k→+∞ Leb • K(P α k ) Leb K(P p/q ) = 1 -loss(p/q),
which is sort of a measure of the area loss for this special sequence. One has 0 < loss(p/q) < 1 (the lower bound follows from the definite loss aforementioned, the upper bound from the fact that the Siegel disk of cannot shrink too much in terms of its area, which is proved using the the theory of parabolic implosion and a lower bound on the size of Siegel disks with a fixed rotation numer (diophantine in our case), due to Siegel. See also my thesis [C] completed with [C2]).

Wish 2 With possible additional conditions, for big values of q, loss(p/q) is arbitrarily close to 0.

Then in the second step of Adrien's plan (defining the irrational θ n+2 once the rational θ n+1 has been fixed), to keep the loss of measure small so that (D4) can hold, it is necessary that in the previous step (deciding which approximant of the irrational θ n the rational θ n+1 shall be), we took the approximant deep enough, so that its denominator is big, so that the quantity loss(θ n+1 ) is small.

Analysis of the plan, reduction

Based on the preparatory work of Jellouli [J], I gave a positive answer to Wish 1 in my thesis [C]:

Theorem 1.9 (Chéritat) Wish 1 is granted, without any additional condition on α.

To prove this theorem, it was enough to prove that the set U n of points which under iteration of P pn/qn never leave the Siegel disk ∆ of the unperturbed map, takes most of the area of ∆, in the sense that Leb(∆\U n ) -→ 0 as n -→ +∞. This is done by comparing the dynamics to a vector field. For the expert reader, let us mention that the comparison is made possible thanks to a control on parabolic explosion, allowed by Yoccoz's inequality on the size of the limbs of the Mandelbrot set. See [C] for details.

Towards Wish 2, I gave in my thesis an upper bound on loss(p/q) in terms of the image of the basin of ∞ in Fatou coordinates (see Figure 3.5 for an illustration): let us put the natural Euclidean metrics on C/Z inherited from C, and do the same with the Lebesgue measure. For p/q ∈ Q, the basin of ∞ for P p/q is an open set, and its image in a repelling cylinder of the parabolic point is a bounded non empty open set U = U (p/q). Let

✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Figure 3.2.
On this picture, we superposed (semi-transparent) the Julia set of Pα for the approximant α = 13/8 of the golden mean over the Julia set for α = the golden mean. Adrien's Wish 1 was based on that kind of pictures. In particular, notice how thin is the intersection between the unperturbed Siegel disk and the basin of infinity for the perturbed Julia set.

A(p/q) be its Lebesgue measure, and h(p/q) its height: h(p/q) = sup

(A ′ , h ′ ) of A ′ > 0 and h ′ > 0 s.t. 1. ∀p/q ∈ Q, if A(p/q) ≤ A ′ and h(p/q) ≤ h ′ then loss(p/q) ≤ m(A ′ , h ′ ), 2. for all fixed h ′ > 0, m(A ′ , h ′ ) -→ 0 as A ′ -→ 0.
See [C]. Hence if U (p/q) has small area and not too big height, then the loss is small, allowing Adrien's plan to work: Corollary 1.11 If the next conjecture holds, then ∃θ such that Leb J(P θ ) > 0.

To state this conjecture, let us say that an irrational α is gilded if its continued fraction expansion has its entries eventually all equal to 1: α = [a 0 , . . . , a k , 1, 1, . . .].

Conjecture (Chéritat) For all gilded irrational α, denoting p n /q n its convergents, we have

(C1) A(p n /q n ) -→ 0, (C2) h(p n /q n ) is bounded.
Backing this conjecture, Figure 3.6 is the kind of experiment that allowed me to convince Adrien that his method worked.

Proof (of Corollary 1.11):

The above conjecture with Theorem 1.10 imply that for all gilded irrational α, loss(p n /q n ) -→ 0 where p n /q n denote the convergents of α. Now we want to construct a sequence θ n satisfying conditions (D1) to (D4) of Douady's plan. Let θ 0 be any gilded irrational, for instance the golden mean. Let θ n be defined by induction according to the two steps of Douady's plan, which are reminded here: θ 2p is irrational, θ 2p+1 is one of its convergents and is thus rational, θ 2p+2 is a special gilded irrational perturbation of the rational number θ 2p+1 . We claim we can choose the sequence (θ n ) so that for all n > 0: Given θ 2p , the existence of a convergent θ 2p+1 satisfying (3.1) with n = 2p + 1 is granted by Theorem 1.9. We moreover choose θ 2p+1 so that loss(θ 2p+1 ) < 2 -2p-2 . Therefore, there exists a special perturbation θ 2p+2 of θ 2p+1 so that (3.1) holds for n = 2p + 2. The sequence (θ n ) n∈N thus satisfies ( D2) and (D4). To have it also satisfy (D1) it is enough to require at each step that |θ n+1 -θ n | < 2 -n , which is possible since in the construction above, there is a whole sequence of valid choices of θ n+1 , sequence which tends to θ n . So now the sequence θ n tends to some θ. To also satisfy (D3), it is enough to have a θ that satisfies Cremer's Liouvillian sufficient condition (see [M]) for non-linearizability, which can be achieved for instance by requiring that max

Leb • K(P θn ) > 1 -2 -n Leb • K(P θn-1 ). ( 3
|θ 2p+1 -θ 2p |, |θ 2p -θ 2p-1 | < 1/ exp(p • 2 q )
where q is the denominator of θ 2p-1 .

Analogy with McMullen's results

This subsection is rather informal, because first the details are a bit complicated, and second at that time we did not know how to make it work.

The results we are referring to will be found in [START_REF] Mcmullen | Self-similarity of Siegel disks and Hausdorff dimension of Julia sets[END_REF].

McMullen proved that the critical point is a Lebesgue density point of the interior of the filled-in Julia set of P α for any bounded type α. Now, the pictures of the parabolic filled-in Julia sets K(P pn/qn ) in Fatou coordinates bear some kind of resemblance to that of zooms on the critical point of the Siegel disk bearing filled-in Julia set K(P α ). This is illustrated in Figure 3.7.

Could McMullen's density results be adapted to our setting, i.e. to the parabolic filled-in Julia sets seen in Fatou coordinates, and yield my conjecture, henceforth positive measure?

McMullen proved more: for instance, when the entries of the continued fraction of α are eventually periodic of period p, then there exist a map λ : z → τ z if p is even and τ z if p is odd, with τ ∈ C, 0 < |τ | < 1 and analytic functions g k such that the sequence

λ -n • P q n+kp α • λ n -→ n→+∞ g k .
We conjecture5 the existence of a similar tower of limits g k λ each frame, the scale has been set so that the distance between the critical point (at the center of the frame) and its q n -th iterate, is constant (about a 16th of the frame width). The convergence of these Julia sets echoes the aforementioned conjectured convergence of the rescaled q n -th iterate.

-n • P q n+kp p n+kp /q n+kp • λ n -→ n→+∞ g k .

What remained to be done

Together with Xavier Buff, we were addressing my conjecture (on the areas A(p n /q n ), and heights h(p n /q n )) via the study of cylinder renormalization (Buff, Epstein, Shishikura, Yampolsky, . . . ). 6 Let us denote R this renormalization.

H. Inou and M. Shishikura had recently proved the existence of a stable class for R. This was, and still is, done by perturbation of the parabolic case thus requires the rotation number to be close enough to 0. In particular, there is an integer N > 0 such that this class contains the quadratic polynomials P α for which α has all its continued fraction entries ≥ N .

In the discussion above, we used gilded numbers, i.e. numbers whose continued fraction entries are eventually constant equal to 1. The whole discussion is valid if instead we look at numbers whose continued fraction entries are eventually constant equal to N . With this modification, Inou and Shishikura's result proved part (C2) of my conjecture (i.e. the height h(p n /q n ) is bounded).

We then had a framework to attack part (C1) (which claims that the area A(p n /q n ) should tend to 0).

After the conference 4.1 Present results

Xavier Buff and I finished the proof during the summer following the conference (2005). To bound the area A(p n /q n ), we first used Inou-Shishikura's results to prove that the post-critical set of P pn/qn stays arbitrarily close to the Siegel disk ∆ of P α :

6 Roughly, it consists in the following: draw a well chosen curve C between the fixed point 0 and the other fixed point of Pα. Among other conditions, we want that Pα be injective on C and send it to a curve disjoint from C apart from its endpoints. Consider the bounded region D bounded by C and Pα (C). Consider the first return map from D to D. Now glue the boundary of D by identifying z ∈ C with Pα(z). Remove the two endpoints from this. One gets a Riemann surface isomorphic to a cylinder, or if one prefers to C * . The first return map is conjugated by this isomorphism to a map which is the cylinder renormalization of Pα. The fixed point 0 of Pα corresponds to one end of the cylinder, and drawing another curve from this end to a fixed point of the first return map allows to renormalize the renormalized map, and so on. . . Theorem 1.12 Let α be any irrational with every continued fraction entry ≥ N and with moreover bounded type, i.e. the entries form a bounded sequence. Then ∀ε > 0, ∃n 0 such that ∀n ≥ n 0 , every point in the postcritical set of P pn/qn is at distance < ε from ∆ (or in ∆).

Then, using Wish 1 (which I proved earlier in my thesis), McMullen's theorem of Lebesgue density of int K(P α ) near ∆, a pull-back argument, a lemma à la Vitali, and the bound on h(p n /q n ), it is easy to conclude and prove part (C1) of my conjecture, thus finishing the proof of the existence of a Julia set with positive Lebesgue measure.

We also extended the method to yield examples of Julia sets with positive measure for quadratic polynomials with a Siegel disk, and also for infinitely renormalizable quadratic polynomials.

The preprint [IS] presenting the near-parabolic renormalization of Inou and Shishikura has been just been submitted; our preprint [BC] giving the full details of our positive measure result too.

Prospects

There are still open questions: 7 Does there exist a polynomial or a rational map, of degree ≥ 2, with a Cremer point, but whose Julia set has Lebesgue measure = 0? Quadratic Julia sets bearing a Cremer point, or a fixed Siegel disk not containing the critical point in its boundary, are non-locally connected. There is also a construction by Dan Sørensen of some infinitely renormalizable quadratic polynomials with a non-locally connected Julia set [So]. Could it happen that for quadratic polynomials, J is non-locally connected ⇐⇒ J has positive Lebesgue measure? Similarly, is it equivalent for a quadratic Julia set with a fixed Siegel disk ∆, to have a positive measure Julia set and to have ∆ not containing the critical point? Is it possible to characterize the set of irrational numbers α such that Leb J(P α ) > 0?

Can we produce accurate computer pictures quadratic Julia sets with positive measure?

7 Thanks to the people who formulated some of them. Would one be able to see anything on an accurate picture? The problem being that around the critical point, the Julia set could be so dense that it would not be possible to see the difference between J and a neighborhood of the Cremer point, unless one draws an astronomical number of pixels.

  called an irrationally indifferent periodic point. Several subcases occur. In the linearizable case, the Siegel disk refers to the of the hunt at the time of the conference maximal domain on which a conjugacy to a rotation on a disk is possible.

  For a Baire generic set of values of c ∈ ∂M, P = z 2 + c has a Hausdorff dimension 2 Julia set. (S2) For a Baire generic set of values of θ ∈ R, P = e 2iπθ z + z 2 has a Hausdorff dimension 2 Julia set.

Figure 3 . 1 .

 31 Figure 3.1. The Julia set of P θ for θ = the golden mean = 1+ √ 5

  of the hunt at the time of the conference

z∈U

  Im z -inf z∈U Im z. The Figure 3.5 shows an example. Theorem 1.10 (Chéritat) There exists a function m

  Figure 3.3. The fat Douady rabbit (α = 1/3 = [0, 3] = [0, 2, 1]), and one of its perturbations (α = [0, 2, 1, k, 1+ √ 5 2 ], for k = 150).

  Figure 3.4. The Julia set for α = 89/55 (9-th approximant to the golden mean). Compare with Figure 3.1.

Figure 3

 3 Figure 3.5. Left: closeup at the parabolic fixed point on the Julia set of Pα for α = 1/3. We sketched a fundamental domain. By P 3 1/3 , its inner boundary is mapped to the outer, and the quotient is a Riemann surface isomorphic to C/Z, called the Ecalle-Voronin cylinder. The isomorphism is called the Fatou coordinate. Right: image of the Julia set by the Fatou coordinate. The cylinder C/Z has been unwrapped: the picture is invariant by z → z+1 and vertical lines mark a fundamental domain of the cylinder C/Z. We hatched its intersection with the basin of ∞. The quantities h and A are the euclidean height and area of the hatched set.

Figure 3 .

 3 Figure3.8 shows closeups on the Julia sets for the convergents p n /q n of the golden mean (whose period p = 1), with n ranging from 5 to 13. In

Figure 3

 3 Figure 3.7. The image on the left shows the Julia set of P pn/qn in Fatou coordinate for some approximant pn/qn of the golden mean. The image in the middle shows a closeup on the same Julia set, but in the initial coordinates, centered on the critical point. The image on the right shows a closeup at the same scale, also centered on the critical point, on the Julia set of Pα for α = the golden mean.There is some kind of resemblance, especially when one looks at the thick part of the basin of infinity.

  Is P ergodic on J in our examples? Moreover, let us mention the following open conjecture, which if it were true, would give another proof of the existence of Cremer point bearing quadratic Julia sets with positive measure: Conjecture (Hubbard) inf |λ|<1 Leb K(λz + z 2 ) > 0

We now have examples with positive measure, but it is still unknown whether or not some of them have measure 0.

Fatou also conjectured that this holds in the set of degree d polynomials and in the set of degree d rational maps.

It is chosen so that in the cylinder renormalization associated to the parabolic implosion at θ n+1 , the virtual multiplier = the golden mean.

This conjecture is still unproven today. We found another way to transfer Mc-Mullen's Lebesgue density result to parabolic filled-in Julia sets seen in cylinders.