Rigidity and non local connectivity of Julia sets of some quadratic

arXiv:0710.4406v1 [math.DS] 24 Oct 2007

Rigidity and non local connectivity of Julia sets of some quadratic polynomials Genadi Levin Inst. of Math., Hebrew University, Jerusalem 91904, Israel

Abstract For an infinitely renormalizable quadratic map fc : z 7→ z 2 +c with the sequence of renormalization periods {nm } and the rotation numbers {tm = pm /qm }, we prove that if lim sup n−1 m log |pm | > 0, then the Mandelbrot set is locally connected at c. We prove also that if lim sup |tm+1 |1/qm < 1 and qm → ∞, then the Julia set of fc is not locally connected provided c is the limit of corresponding cascade of successive bifurcations. This quantifies a construction of A. Douady and J. Hubbard, and strengthens a condition proposed by J. Milnor.

1

Introduction

Theorem 1 Suppose that for some quadratic polynomial f (z) = z 2 + c there is an increasing sequence nm → ∞ of integers, such that f nm is simply renormalizable, and log |pm | > 0, lim m→∞ nm where pm /qm ∈ (−1/2, 1/2] denotes the rotation number of the separating fixed point of the renormalization f nm . Then c lies in the boundary of the Mandelbrot set M and M is locally connected at c. The following problems are central in holomorphic dynamics: MLC conjecture: “The Mandelbrot set is locally connected”, and its dynamical counterpart: “For which c is the Julia set Jc of fc locally connected?”. The MLC conjecture is equivalent to the following rigidity conjecture: if two quadratic polynomials with connected Julia sets and all periodic points repelling are combinatorially equivalent, then they are conformally conjugate. The MLC implies that hyperbolic dynamics is dense in the space of complex quadratic polynomials [6] (see

1

also [39]). Yoccoz (see [14]) solved the above problems for finitely renormalizable quadratic polynomials as follows: for such a non hyperbolic map fc , the Julia set Jc of fc is locally connected (provided fc has no neutral periodic points), and the Mandelbrot set M is locally connected at c. (For the (sub)hyperbolic maps and maps with a parabolic point the problem about the local connectivity of the Julia set had been sattled before in the works by Fatou, Douady and Hubbard and others, see e.g. [3].) At the same time, infinitely renormalizable dynamical systems have been studied intensively, see e.g. [27], [28], [11] and references therein. In his work on renormalization conjectures, Sullivan [41] (see also [30]) introduced and proved so-called “complex bounds” for real infinitely renormalizable maps with bounded combinatorics. Roughly speaking, complex bounds mean that a sequence of renormalizations is precompact. This property became in the focus of research. It has played a basic role in recent breakthroughs in the problems of local connectivity of the Julia set and rigidity for real and complex polynomials, see [12], [22], [23], [38], [16],[17], [21], [1], [18]. On the other hand, there are maps without complex bounds. Indeed, Douady and Hubbard showed the existence of an infinitely renormalizable fc , such that its small Julia sets do not shrink (and thus fc has no complex bounds) but still such that the Mandelbrot set is locally connected at this c, see [31], [40]. Theorem 1 provides a first class of combinatorics of infinitely renormalizable maps fc without (in general) complex bounds, for which the Mandelbrot set is locally connected at c. Obviously, previous methods in proving the local connectivity of M do not work in this case. Our method is based on an extension result for the multiplier of a periodic orbit beyond the domain where it is attracting, see next Section. That the maps with the combinatorics described in Theorem 1 in general do not have complex bounds and can have non locally connected Julia sets follow from the second result, Theorem 2, which makes explicit Douady-Hubbard’s construction. It is based on the phenomenon known as cascade of successive bifurcations, and is the following. Let W be a hyperbolic component of the Mandelbrot set, so that fc has an attracting periodic orbit of period n for c ∈ W . Given a sequence of rational numbers tm = pm /qm 6= 0 in (−1/2, 1/2], choose a sequence of hyperbolic components W m as follows: W0 = W , and, for m ≥ 0, the hyperbolic component W m+1 touches the hyperbolic component W m at the point cm with the internal argument tm = pm /qm (see next Sect.). When the parameter c crosses cm from W m to W m+1 , the periodic orbit of period nm = nq0 ...qm−1 which is attracting for c ∈ W m “gives rise” another periodic orbit of period nm+1 = nm qm which becomes attracting for c ∈ W m+1 . Thus when the parameter c moves to a limit parameter through the hyperbolic components W m , the dynamics undergoes a sequence of bifurcations precisely at the parameters cm , m = 0, 1, .... In the case when n = 1 and tm = 1/2 for all m ≥ 0, the parameters cm are real, and we get the famous period-doubling cascade on the real line known since

2

1960’s [34]. The corresponding limit parameter lim cm = cF = −1.4.... The Julia set JcF is locally connected [15], [23]. Douady-Hubbard’s construction shows that if the sequence {tm } tends to zero fast enough, then periodic orbits generated by the cascade of successive bifurcations at the parameters cm , m > 0, stay away from the origin. This implies that the Julia sets of the renormalizations of fc∗ for c∗ = lim cm do not shrink, and Jc∗ is not locally connected at zero. Their construction is by continuity, and it does not give any particular sequence of tm . Milnor suggests in [32], p. 21, that the convergence of the series: ∞ X

|tm |1/qm−1 < ∞

(1)

m=1

could be a criterion that the periodic orbits generated by the cascade after all bifurcations stay away from the origin. Theorem 2 shows that the condition (1) is indeed sufficient for this, but not the optimal one (cf. [24]. We give a weaker sufficient condition and thus prove: Theorem 2 For every a ∈ (0, 1) there exists Q as follows. Let t0 , t1 ,...,tm ,... be a sequence of rational numbers tm = pm /qm ∈ (−1/2, 1/2] \ {0}, such that lim supm→∞ qm > Q and lim sup |tm |1/qm−1 < a. (2) m→∞

Let W be a hyperbolic component of period n, and the sequence of hyperbolic components W m , m = 0, 1, 2, ..., is built as above, so that W m+1 touches W m at the point cm with the internal argument tm . Then: (1) the sequence {cm }∞ m=0 converges to a limit parameter c∗ , (2) the map fc∗ is infinitely renormalizable with non locally connected Julia set. If, for instance, |tm+1 |1/qm ≤ 1/2 for big indexes m, then qm+1 ≥ 2qm → ∞, and Theorem 2 applies. Let us make some further comments. As it is mentioned above, we prove (2) by showing that the conditions of Theorem 2 imply that the cascade of bifurcated periodic orbits of fc∗ stay away from the origin. By Theorem 1, if, for the sequence tm = pm /qm , lim sup m→∞

log |pm | > 0, q0 ...qm−1

then the components W m tend to a single point, and M is locally connected at this point.

3

The conclusion (2) of Theorem 2 breaks down if not all its conditions are valid. Indeed, if tm = 1/2 for all m, then |tm+1 |1/qm = 1/21/2 < 1 At the same time, as it is mentioned above, the limit Julia set JcF is locally connected. Theorem 2 is a consequence of a more general Theorem 5, see Section 4. Note in conclusion that there are similarities between Theorems 2, 5 and the celebrated Bruno-Yoccoz criterion of the (non-)linearizability of quadratic map near its irrational fixed point. Throughout the paper, B(a, r) = {z : |z − a| < r}. Acknowledgments. The author is indebted to Alex Eremenko for discussions, and especially for answering author’s question about the function H and for finding the reference [35] (see Section 4).

2 2.1

Multipliers Hyperbolic components

A component W of the interior of M is called an n-hyperbolic if fc , c ∈ W , has an attracting periodic orbit O(c) of period n. Denote by ρW (c) the multiplier of O(c). By the Douady-Hubbard-Sullivan theorem [5], [29], [3], ρW is a analytic isomorphism of W onto the unit disk, and it extends homeomorphically to the boundaries. Given a number t ∈ (−1/2, 1/2], denote by c(W, t) the unique point in ∂W with the internal argument t, i.e. ρW at this points is equal to exp(2πt). Root of W is the point cW = c(W, 0) with the internal argument zero. If t = p/q is a rational number, we will always assume that p, q are co-primes. For any rational t 6= 0, denote by L(W, t) the connected component of M \{c(W, t)} which is disjoint with W . It is called the t-limb of W . Denote also by W (t) a nq-hyperbolic component with the root point c(W, t); it touches W at this point. The limb L(W, t) contains W (t). Root of W is the point cW = c(W, 0) with the internal argument zero. The root cW of W is the landing point of precisely two parameter rays, i.e. external rays of the Mandelbrot set [6]. In what follows, it will be important the notion of the wake of a hyperbolic component W [14]: it is the only component W ∗ of the plane cut by two external rays to the root of W that contains W . The points of periodic orbit O(c) as well as its multiplier ρW extend as analytic functions to the wake W ∗ . Moreover, |ρW | > 1 in W ∗ \ W .

4

2.2

Analytic extension of the multiplier

Given C > 1, consider an open set Ω of points in the punctured ρ-plane defined by the inequality |ρ − 1| > C log |ρ| (3) It obviously contains the set D∗ = {ρ : 0 < |ρ| ≤ 1, ρ 6= 1} and is disjoint with an interval 1 < ρ < 1 + ǫ. Denote by Ω(C) the connected component of Ω which contains the set D∗ completed by 0. Denote also by Ωlog (C) the set of points L = log ρ = x+iy, ρ ∈ Ω(C), |y| ≤ π. Ω(C) is simply-connected. More precisely, the intersection of Ωlog (C) with any vertical line with x = x0 > 0 is either empty or equal to two (mirror symmetric) intervals. If C > 4, then x < 2/(C − 2) for all L = x + iy in Ωlog (C). If C is large enough, Ωlog (C) contains two (mirror symmetric) domains bounded by the lines y = ±(C/2)x (x > 0) and y = ±π Let W be an n-hyperbolic component of M . The map ρW from W onto the unit disk c 7→ ρW (c) has an inverse, which we denote by c = ψW (ρ). It is defined so far in the unit disk. In [24] we prove the following. Theorem 3 (a) There exists B0 as follows. Let O(c) be a repelling periodic orbit of fc of exact period n, and the multiplier of O(c) is equal to ρ. Assume that |ρ| < e. Then |ρ′ (c)| 4n (1 + o(1))} (4) |ρ − 1| ≤ B0 {log |ρ(c)| + n |ρ(c)| as n → ∞. (b) Denote Ωn = Ω(n−1 4n B0 ) and log −1 n Ωlog n = Ω (n 4 B0 ) = {L = x + iy : exp(L) ∈ Ωn , |y| ≤ π}

Then the function ψ = ψW extends to a holomorphic function in the domain ˜ n of Ωn defined by its log-projection Ωn . Moreover, ψ is univalent in a subset Ω log ˜ n = {log ρ : ρ ∈ Ω ˜ n } as follows: Ω log ˜ log Ω n = Ωn \ {L : |L − Rn | < Rn }

where Rn depends on n only and has an asymptotics Rn = (2 + O(2−n ))n log 2 as n → ∞. ˜ n by ψ is contained in the wake W ∗ . Finally, the image of Ω Next inequalities can be checked easily.

5

Proposition 1 There exists K > 0, such that, for every n > 0 and t ∈ (−1/2, 1/2], ˜ log the Euclidean distance between the point 2πit and the boundary of Ω n , ˜ log K −1 P (t, n) < dist(2πit, ∂ Ω n ) < KP (t, n) where P (t, n) = min{

2.3

(5)

n|t| t2 , }. 4n n

Limbs

Let W be an n-hyperbolic component. For every t = p/q 6= 0, consider the limb L(W, t). Then, for every c ∈ L(W, t), a branch of log ρW (c) is contained in the following round disk (Yoccoz’s circle): Yn (t) = {L : |L − (2πit +

n log 2 n log 2 )| < }, q q

(6)

see [14], [37] and references therein. In [24] we derive from Theorem 3 and the latter result the following bound. Theorem 4 There exists A > 0, such that, for every n-hyperbolic component W and every t = p/q ∈ [−1/2, 1/2], the diameter of the limb L(W, t) is bounded by: diamL(W, t) ≤ A

3 3.1

4n p

(7)

Rigidity Simple renormalization

We follow some terminology as in [27]. Let f be a quadratic polynomial. The map f n is called renormalizable if there are open disks U and V such that f n : U → V is a polynomial-like map with a single critical point at 0 and with connected Julia set Jn . The map f n : Jn → Jn has two fixed points: β (non-separating) and α. Denote them by βn and αn . The renormalization is simple if any two small Julia sets f i (Jn ), i = 0, 1, ..., n − 1 cannot cross each other, i.e. they can meet only at some iterate of βn . The fixed point αn assuming it is repelling separates the Julia set of the renormalization f n : U → V . Moreover, it has a well-defined nonzero rational rotation number p/q ∈ (−1/2, 1/2], which can be defined as follows: f n : U → V is hybrid equivalent to a quadratic polynomial which lies in the p/q-limb of the main cardioid.

6

3.2

Demonstration of Theorem 1

We split the proof into few steps. A. Let f n : U → V be a simple renormalization of f , βn its β-fixed point, n−1 and On = {f i (βn )}i=0 the periodic orbit contaning βn . We use some notions and results from [6], [33]. The characteristic arc I(On ) = (τ− (On ), τ+ (On )) of On is the shortest arc (measured in S 1 ) between the external arguments of the rays landing at the points of On . Then τ± (On ) are the arguments of two dynamical rays that land at the point βn′ = f (βn ) of On , and c lies in the sector bounded by these rays and disjoint with 0. Furthermore, two parameter rays of the same arguments τ± (On ) land at a single parameter c(On ). The point c(On ) is the root of a hyperbolic component denoted by W (On ), and the above parameter rays completed by c(On ) bound the wake W (On )∗ of this component. Denote by L(On ) = W (On )∗ ∩ M the corresponding limb. Thus, (A1) c0 ∈ L(On ), (A2) moreover, the rotation number of the α-fixed point αn of the renormalization is p/q if and only if c0 ∈ L(W (On ), p/q) - the limb of W (On ) which is attached at the point of ∂W (On ) with the internal argument p/q. B. Let m < m′ . By [27], since all renormalizations f nm are simple, nm divides nm′ . Also, the small Julia set Jnm′ is a proper subset of the small Julia set Jnm . Therefore, (B1). τ− (Onm ) < τ− (Onm′ ) < τ+ (Onm′ ) < τ+ (Onm ), thus the sets L(Onm ) form a decreasing sequence of compact sets containing c0 . Therefore, for the local connectivity of M at c0 it is enough to prove that the set S = ∩m L(Onm ) consists of a single point. (B2). It is easy to see that τ+ (Onm ) − τ− (Onm ) → 0. This follows from a well known inequality τ+ (Onm ) − τ− (Onm ) < (2nm − 1)2 /(2nm qm − 1) (see [2], [26]), and qm > 2. Thus, there exists lim τ± (Onm ) = τ0 . (8) Note that τ0 is not periodic under the uniformization map p(t) = 2τ (mod1). C. Let c be any point from S. (C1) All periodic points of fc are repelling. Indeed, obviously, fc cannot have an attracting cycle. If fc has an irrational neutral periodic orbit then c lies in the boundary of a hyperbolic component contained in S, a contradiction with (8). If fc has a neutral parabolic periodic orbit, then c is the landing point of precisely two parameter rays with periodic arguments, again the same contradiction. Thus, all cycles are repelling. Consider the so-called real lamination λ(c) of fc [19]. It is a minimal closed equivalence relation on S 1 that identifies two points whenever their prime end impressions intersect. For every m, τ± (Onm ) ⊂ λ(c). Since λ(c)

7

− , τ + }, such that is closed, τ0 ∈ λ(c). Moreover, for every m, there is a pair {τm m + − ± p(τm ) ⊂ τ± (Onm ), and {τm , τm } is contained in the class of λ(c) corresponding to the point βnm . Passing to the limit, we get that the pair {τ0 /2, τ0 /2 + 1/2} is the critical class of λ(c). The following statements are known after Thurston and Douady and Hubbard and proved in a much more general form in [20], Proposition 4.10: if the critical class {τ0 /2, τ0 /2 + 1/2} is contained in a class of λ(c), then λ(c) is determined by (the itinerary of) τ0 . Since τ0 is the same for all c ∈ S, we conclude that the real laminations of all fc , c ∈ S, coincide. In particular, for every c ∈ S, fcnm is simply renormalizable because by [27] the simple renormalizations are detected by the lamination. The renormalization fcnm is hybrid equivalent k is simply to some fT (c,m) [7]. Denote n ˆ k = nm+k /nm , k > 0. Then fTnˆ(c,m) renormalizable, and the rotation number of its α-fixed point is pm+k /qm+k because it is a topological invariant. (C2) By (A2), T (c, m) is contained in the intersection of a decreasing sequence of limbs Lm,k , k = 1, 2, ..., such that Lm,k is attached to a hyperbolic component of period n ˆ k at the internal argument pm+k /qm+k . By Theorem 4, the diameter of Lm,k , nm+ k

diamLm,k

4 nm 4nˆ k =A . ≤A pm+k pm+k

Since lim(log pm )/nm > 0, one can find and fix m in such a way, that nm+ k

4 nm = 0. lim k→∞ pm+k It means, that, for the chosen m, the limbs Lm,k (k → ∞) shrink to a point cˆ = T (c, m), so that cˆ depends on m but is independent of c ∈ S. Thus fcnm is quasi-conformally conjugate to fcˆ, for all c ∈ S. D. Assume that the compact S has at least two different points. Then S contains a point c1 ∈ ∂M other than c0 . By (C1)-(C2), the renormalizations fcn0m of fc0 and fcn1m of fc1 are quasi-conformally conjugate, all periodic points of fc0 , fc1 are repelling, and λ(c0 ) = λ(c1 ). We are in a position to apply Sullivan’s pullback argument (see [30]). Using a quasi-conformal conjugacy near small Julia sets (rather than on the postcritical set) and an appropriate puzzle structure, we arrive at a quasi-conformal conjugacy between fc0 , fc1 . Since c1 ∈ ∂M , then according to [6], c0 = c1 .

8

4

Non locally connected Julia sets

4.1

Statement

Theorem 5 Let t0 , t1 ,...,tm ,... be a sequence of rational numbers tm = pm /qm ∈ (−1/2, 1/2]. Assume that the following conditions are satisfied. (E0) 1 log |tm+1 | > 0. (9) lim inf m→∞ q0 ...qm−1 (Y0) sup |tm |q0 ...qm−1 < ∞.

(10)

m≥0

(S0) for some k ≥ 0, ∞ X m=k

uk,m H(uk,m+1 ) < ∞, qm (1 − uk,m )

(11)

where the smooth strictly increasing function H : [0, 1) → [0, ∞) is defined below, and 4qk ...qm−1 1/q uk,m = θk,mm , θk,m = C|tm+1 | max{qk ...qm , }, (12) qk ...qm where C > 0 is an absolute constant. Let now W be a hyperbolic component of some period n ≥ 1, and the sequence W m of hyperbolic component is built as above, in other words, the hyperbolic component W m+1 touches the hyperbolic component W m at a point cm with internal argument tm . Then: (1) the sequence of parameters cm , m=0,1,2,..., converges to a limit parameter c∗ , (2) the map fc∗ is infinitely renormalizable with non locally connected Julia set. The function H used above is equal to H(u) = 16uΠ∞ k=1 and is bounded by H(u) ≤

(1 + u2k )8 , (1 − u2k−1 )8

1 exp (−π 2 / log u). 16

Theorem 2 announced in the Introduction is a simple corollary of Theorem 5. Indeed, assume there exists a ∈ (0, 1), such that |tm |1/qm−1 < a

9

for all large m. If now some qk is big enough, then qm → ∞ rapidly enough, and it is not difficult to check that the conditions (E0), (Y0), and (S0) are satisfied. The rest of the Section occupies the proof of Theorem 5.

4.2

Bifurcations

Let W be a n-hyperbolic component, and let c0 ∈ ∂W have an internal argument t0 = p/q 6= 0. Consider a periodic orbit O(c) = {bj (c)}nj=1 of fc which is attracting when c ∈ W . Then all bj (c) as well as the mulptiplier ρ(c) of O(c) are holomorphic in W and extend to holomorphic functions in c in the whole wake W ∗ of W . As we know, the multiplier ρ(c) is injective near c0 . Denote the inverse function by ψ. It ˜ n , which includes the unit disk and a neighborhood is well defined in the domain Ω of the point ρ0 = exp(2πit0 ), so that ψ(ρ0 ) = c0 . Remind that W (t0 ) denotes a hyperbolic component touching W at the point c0 . Lemma 4.1 (cf. [24], [4]) There exists Q > 0 as follows. Given n, q, denote ˜ log r(t0 , n) = 1/2 min{1/(2nq 3 ), dist(2πit0 , ∂ Ω n )}, Let q > Q, n > 0. (1) There exist q function Fk as follows. For each k = 1, ...q, Fk is a holomorphic function in the disk S = {|s| < r(t0 , n)1/q }, such that Fk (0) = 0, Fk′ (0) 6= 0; for every s ∈ S and corresponding ρ = ρ0 + sq , the points bk (c0 ) + Fk (s exp(2πj/q)), k = 1, ..., n, j = 0, ..., q − 1, form a periodic orbit Oq (c) of fc of period nq, where c = ψ(ρ). (2) The disk Bt0 = B(ρ0 , r(t0 , n)) ˜ n , so that ψ is univalent in Bt . The multiplier ρW (t ) of the is contained in Ω 0 0 nq-periodic orbit Oq has a well-defined analytic extension from the hyperbolic component W (t0 ) to the union of the wake W ∗ and the domain ψ(Bt0 ). The function λ = ρW (t0 ) ◦ ψ is defined and holomorphic in Bt0 . For every real T ∈ (−1/2, 1/2], the equation λ(ρ) = exp(2πiT ) has at most one solution ρ in Bt0 , and for such a solution the corresponding c = ψ(ρ) lies on the boundary of the nq-hyperbolic component W (t0 ). A covering property holds: for every r ≤ r(t0 , n), the image of the disk B(ρ0 , r) under the map λ covers the disk B(1, q 2 r/16).

10

Proof. It is known (see e.g. [6]) that the periodic orbit Oq (c) exists locally, for c close enough to c0 , and this is the only periodic orbit other than O(c) in its neighborhood. Therefore, by the Implicit Function theorem, (1) holds locally, in a neighborhood of the point s = 0. Let us show that the functions Fk don’t have singularities in the disk S. Indeed, if the periodic orbit Oq of the period nq has a singularity at some c, then c lies in a limb L(W, p′ /q ′ ) other than L(W, p/q). By a calculation in the external arguments [24] (cf. [4]), then q ≥ q ′ − 1. On the other hand, a simple estimate shows that for all n, q, the ball B(2πit0 , 1/(2nd3 )) is disjoint with every Yoccoz’s circle that touches the vertical line at a point 2πip′ /q ′ with some q ′ ≤ q + 1, p′ /q ′ 6= p/q. Moreover, for q large enough, and every n ≥ 1, the log-projection of the latter ball covers Bt0 . Therefore, ψ is univalent in Bt0 and, furthermore, V = ψ(Bt0 ) is disjoint with any limb L(W, p′ /q ′ ), with q ′ ≤ q + 1, p′ /q ′ 6= p/q. Now (1) follows from the Monodromy Theorem. We have shown also that λ is holomorphic in Bt0 . Let λ(ρ1 ) = exp(2πiT1 ) for some real T1 . Then, for c1 = ψ(ρ1 ), the map fc1 has nq-periodic orbit Oq (c1 ) with the multiplier ρ1 = exp(2πiT1 ). Now, V is disjoint with any p′ /q ′ -limb of W other than L(W, t0 ), where q ′ ≤ q + 1, therefore, c1 ∈ L(W, t) and, moreover, lies in the boundary of an nq-hyperbolic component. This hyperbolic component belongs to some limb L(W, t′ ), t′ = p′ /q ′ , where, by the above, q ′ > q + 1. If t′ 6= t0 , then L(W, t′ ) contains a parameter c˜ on the boundary of this component, such that fc˜ has a nq-periodic orbit with the multiplier 1. But then q ≥ q ′ − 1, in contradiction with the previous condition q ′ > q + 1 or t′ = t0 . Thus c1 is in the limb L(W, t0 ). Hence, it lies in the boundary of some nq-hyperbolic component belonging this time to L(W, t0 ). On the other hand, the multiplier of the periodic orbit Oq (c) is bigger on modulus than 1 off the closure of the component W (t0 ). Thus the hyperbolic component containing c1 in its boundary is just W (t0 ). Since the corresponding multiplier is injective on the boundary, it means that ρ1 is the only solution of the equation λ(ρ) = exp(2πiT1 ). To prove the covering property, we start with the formula q2 dλ (ρ0 ) = − dρ ρ0 (see [10]). Given r ≤ r(t0 , n), consider a function m(w) = (q 2 r)−1 (λ(ρ0 + rw) − 1). It is holomorphic in the unit disk, m(0) = 0, |m′ (0)| = 1 and, by the proved part of the statement, m(w) = 0 if and only if w = 0. Therefore, by a classical result (Caratheodory-Fekete, see e.g. [9] or [36]), the disk B(0, 1/16) is covered by the image of the unit disk under the map m. This is equivalent to the covering property. 

11

4.3

The function H

Definition 4.1 There exists a real strictly increasing smooth function H : [0, 1) → [0, ∞), H(0) = 0 as follows. Let G be the set of all holomorphic functions g : B(0, 1) → C \ {1}, such that g(w) = 0 if and only if w = 0. Then H(u) is defined by: H(u) = sup{|g(w)| : |w| ≤ u, g ∈ G}. There is an explicit expression for H. It is obtained as follows,. Let J(w) be a holomorphic function in B(0, 1), such that J(0) = 0, J ′ (0) > 0, and J : B(0, 1) \ {0} → C \ {0, 1} is an infinite unbranched cover. Such function is investigated in [35], see also [8], [13], [36]. By the Schwarz lemma, H(u) = max |J(w)|. |w|=u

On the other hand, by [35], J(w) = 16wΠ∞ k=1

(1 + w2k )8 . (1 + w2k−1 )8

(Apparently, J is equal to the squere of the so-called elliptic modulus, see e.g.[36].) Thus, (1 + u2k )8 H(u) = −J(−u) = 16uΠ∞ . k=1 (1 − u2k−1 )8 As it is shown in [35]: H(u) ≤

4.4

2 1 exp−π / log u . 16

Main Lemma

Lemma 4.2 There exist Q > 0 and L > 0 as follows. Let t0 , t1 ,...,tm ,... be a sequence of rational numbers tm = pm /qm ∈ (−1/2, 1/2). Denote n0 = 1, nm = nq0 q1 ...qm−1 , m > 0. Denote also d˜m = r(tm , nm ) = 1/2 min{

1 3 2nm qm

12

˜ log , dist(2πitm , ∂ Ω nm )},

and, finally, dm =

320 |t |, 2 m+1 βqm

where β = 1/32000. Assume that, for every m ≥ 0, the following conditions are satisfied. (0) qm > Q, ˜ (Y) dm < d2m . (S) ∞ X um−1 |t0 |H(u0 ) + H(um ) < L, (13) qm−1 (1 − um−1 ) m=1

where um , m = 0, 1, 2, ... is defined by um = θm θm =

1/qm

with

200βdm , m ≥ 0. d˜m

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Let now W0 be the main cardioid. Denote W m = W m−1 (tm−1 ), m = 1, 2, ..., in other words, the hyperbolic component W m touches the hyperbolic component W m−1 at the point cm−1 := c(W m−1 , tm−1 ) with the internal argument tm−1 . Then: (1) the sequence of parameters cm , m=0,1,2,..., converges to a limit parameter c∗ , (2) the map fc∗ is infinitely renormalizable with non locally connected Julia set. Proof. By the definition of d˜m , for every m = 0, 1, 2, ..., the disk Btm := B(exp(2πitm ), d˜m ) ˜ nm , and the conclusions of Lemma 4.1 holds. lies in Ω Introduce more notations. Let ψW m be a function, which is inverse to the multiplier function ρW m , and ψm (w) = ψW m (exp(w)), m = 0, 1, .... ˜ log ψm is holomorphic in Ωlog nm and univalent in its subset Ωnm . In particular, the function ψm extends in a univalent fashion to the disk B(2πitm , d˜m ) Note that cm = ψm (2πitm ). Denote further ′ Rm = ψm (B(2πitm , dm )), Dm = ψm (B(2πitm , βdm )), Dm = ψm (B(2πitm , 100βdm ))

We use classical distortion bounds for univalent maps, see e.g. [9]: if g is univalent in a disk B(0, R) and r < R, then B(g(0), α(r/R)−1 r|g′ (0)|) ⊂ g(B(0, r)) ⊂ B(g(0), α(r/R)r|g′ (0)|)

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where α(x) = (1 − x)−2 . In particular, if r ≤ d˜m /2, then ψm (B((2πitm , r)) is “roughly” a disk: ′ ′ B(cm , 4−1 r|ψm (2πitm )|) ⊂ ψm (B((2πitm , r)) ⊂ B(cm , 4r|ψm (2πitm )|) ′ are “roughly” disks (around c ), and Hence, Rm , Dm and Dm m ′ Dm ⊂ Dm ⊂ Rm , m = 0, 1, ...,

and, moreover, the diameter of each previous set is at least twice smaller than the diameter of the next one. Fact. For m = 0, 1, 2, ..., the following holds. (i) cm+1 ∈ Dm , ′ . (ii) Rm+1 ⊂ Dm For the proof, we use Lemma 4.1. Consider the function λ = ρW m+1 ◦ ψW m . Then λ is holomorphic in Btm . If dm is small enough, then B(exp(2πitm ), βdm /2) ⊂ exp(B(2πitm , βdm )). By the covering property of Lemma 4.1, λ(B(exp(2πitm ), βdm /2)) covers the disk 2 βd /32). But B(1, qm m 2 exp(2πitm+1 ) ∈ B(1, qm βdm /32), 2 βd /320 < q 2 βd /(64π). Therefore, by the part (2) of because |tm+1 | = qm m m m Lemma 4.1, the point

cm+1 = ψm+1 (2πitm+1 ) ∈ ψm (B(2πitm , βdm )) = Dm . ′ , let us notice that by part (b) of To prove that Rm+1 is contained in Dm Theorem 3, the root cm of W m+1 is outside of Rm+1 , and Rm+1 is “roughly” a disk around cm+1 . Hence, Rm+1 ⊂ B(cm+1 , 4|cm+1 − cm |) and then Rm+1 ⊂ B(cm , 17|cm+1 − cm |) Let us estimate |cm+1 − cm |. Since cm+1 ∈ Dm , then |cm+1 − cm | < βα(β)rm ′ (2πit )|d and α(x) = (1−x)−2 . Thus R where rm = |ψm m m m+1 ⊂ B(cm , 17βα(β)rm ). ′ On the other hand, Dm = ψm (B(2πitm , 100βdm ) contains the disk B(cm , 100β(α(100β))−1 rm ). By the choice of β, we check that 17α(β) < 100/α(β), i.e. indeed ′ Rm+1 ⊂ B(cm , 17βα(β)rm ) ⊂ B(cm , 100β(α(100β))−1 rm ) ⊂ Dm .

The Fact is proved. ′ and It implies that for all m, cm+1 ∈ Dm ′ ′ Dm+1 ⊂ Rm+1 ⊂ Dm .

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′ tend to zero, the sequence c tends to the point Since the diameters of Dm m ∞ ′ {c∗ } = ∩∞ m=0 Rm = ∩m=0 Dm .

It proves the conclusion (1). Now we pass to the proof of the conclusion (2). Let us denote by Ok (c) the nk -periodic orbit of fc , which is attracting if c ∈ W k , k = 0, 1, 2, .... As we know, Ok extends holomorphically for c in the wake (W k )∗ of W k . Given m ≥ 1, consider any point b(c) of a periodic orbit Om (c), for c ∈ (W m )∗ . Lemma 4.1 tells us that bm extends also to a neighborhood of cm−1 in the following sense. There is a holomorphic function Z of a local parameter s in the disc Sm−1 := {|s| < vm−1 },

1/qm−1 vm−1 = d˜m−1 ,

such that it matches b(c), i.e., b(c) = Z(s) for c = cm−1 (s) := ψW m−1 (exp(2πitm−1 )+ sqm−1 ). Moreover, the points Z(sejqm−1 ), where j = 1, ..., nm−1 −1 and eq denotes a primitive q-root of unity, also belong to Om . We denote by b+ the point Z(seqm−1 ) of Om , which is uniquely defined for s ∈ Sm−1 . Let us estimate the distance between b = Z(s) and b+ = Z(seqm−1 ). Since |Z(s)| < 3 for s ∈ Sm−1 , we have: |Z(seqm−1 ) − Z(s)|
0, such that, for all m large enough, |tm |q0 ...qm−1 < M.

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(S1) for some k ≥ 0, ∞ X m=k

uk,m H(uk,m+1 ) < ∞, qm (1 − uk,m )

where 1/q

uk,m = θk,mm , θk,m = C|tm+1 | max{qk ...qm ,

4qk ...qm−1 }, qk ...qm−1 qm

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for some universal C > 0. It is enough to find a tail {tk0 , ..., tm , ...} of the sequence {tm }∞ m=0 , which satisfies all the conditions of Lemma 4.2 with n = 1. Indeed, assume this is the case. It means the following. Let’s start with the 1-hyperbolic component W0 (the main cardioid) and the tail {tk0 , tk0 +1 ...} in place of W and {t0 , t1 , ...}. Then we get a sequence of hyperbolic components W0k0 ,m , m ≥ k0 , where W0k0 ,k0 = W0 and, for m > k0 , the component W0k0 ,m touches W0k0 ,m−1 at the point ck0 ,m−1 with internal argument tm−1 . Then by Lemma 4.2 the sequence of parameters ck0 ,m converges, as m → ∞, to some ck0 ,∗ , and the Julia set Jck0 ,∗ is not locally connected. By a

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well known straightening procedure, see [7] and the proof of Theorem 1, it implies the conclusions (1)-(2) of Theorem 5. Now we find k0 . By (Y1), tm → 0, in particular, qm → ∞. Given k ≥ 0, denote nk,k = 1, nk,m = qk ...qm−1 for m > k, the relative period. Denote, for m ≥ k, d˜k,m = r(tm , nk,m ) = 1/2 min{ and, as before, dm =

1 3 2nm qm

˜ log , dist(2πitm , ∂ Ω nk,m )},

320 |t |. 2 m+1 βqm

Using Proposition 1 with the constant K, we check that for all k large enough and m ≤ θk,m , where m ≥ k, θk,m := 200βd d˜ k,m

θk,m = C|tm+1 | max{qk ...qm ,

4qk ...qm−1 }, qk ...qm−1 qm

and C = 6400 max{2, K} is the absolute constant. By (E1)-(Y1), θk,m → 0 as k → ∞ uniformly in m ≥ k. In particular, conditions (0) and (Y) of Lemma 4.2 are satisfied, for the tail {tk , ..., tm , ...} for every k large enough. Assume that the condition (S1) also holds, for some fixed k large enough. Then we show that this implies that the last condition (S) of Lemma 4.2 is satisfied, too, for the tail {tk0 , ..., tm , ...}, for some k0 > k. 1/q Indeed, uk,m = θk,mm is decreasing in k. Therefore, by (S1), for every k0 large enough, ∞ X uk0 ,m H(uk0 ,m+1 ) < L/2. qm (1 − uk0 ,m ) m=k0

Thus to satisfy the condition (S) of Theorem 4.2, it is enough to check that |tk0 |H(uk0 ,k0 ) tends to zero as k0 tends to ∞. Note that uk,m decreases as k increases. Hence, it follows from (S1), that uk,k0−1 uk0 −1,k0 −1 H(uk0 ,k0 ) ≤ H(uk,k0 ) → 0 qk0 −1 (1 − uk0 −1,k0 −1 ) qk0 −1 (1 − uk,k0 −1 ) as k0 → ∞. On the other hand, using (Y1) it is easy to see that |tk0 | uk0 −1,k0 −1 qk0 −1 (1−uk0 −1,k0 −1 )