Asymptotically holomorphic methods for infinitely renormalizable $C^r$ unimodal maps

Abstract

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

1 Introduction

Over the last decades, many remarkable results were obtained for rational maps of the Riemann sphere and, somewhat surprisingly, it turned out that quite a few of these have an analogue in the case of smooth interval maps. For example, the celebrated Julia–Fatou–Sullivan structure theorem for rational maps establishes the absence of wandering domains, showing that each component of the Fatou set is eventually periodic, and moreover gives a simple classification of the possible dynamics on a periodic component of the Fatou set, see [60]. For smooth interval maps, analogous results were obtained, starting with Denjoy’s results for $C^2$ circle diffeomorphisms dating back to 1932. We now know that $C^2$ interval or circle maps cannot have wandering intervals provided all their critical points are non-flat, proved in increasing generality in [7, 18, 19, 29, 47, 50, 62]. Interestingly, although the statements for the Julia–Fatou–Sullivan structure theorem for rational maps and the generalized Denjoy theorems for interval and circle maps are analogous, the proofs use entirely different ideas. In the former case, they rely on the measurable Riemann mapping theorem (MRMT) while in the latter case, the proofs rely on real bounds coming from $C^2$ distortion estimates together with arguments relating to the order structure of the real line.

However, overall, not only the results but also the techniques used in the fields of holomorphic dynamics and interval dynamics have become increasingly intertwined over the last decades. Indeed, within the literature of real one-dimensional dynamics, a growing number of results are obtained under the additional assumption that the maps are real analytic rather than smooth. The reason for this is that a real analytic map (obviously) has a complex extension to a small neighbourhood in $\mathbb C$ of the dynamical interval, and therefore many tools from complex analysis can be applied to such a real map. For instance, many results in the theory of renormalization of interval maps are either not known in the smooth category or were only obtained with a significant amount of additional effort. Specifically, the Feigenbaum–Coullet–Tresser conjectures were first obtained using computer supported proofs, e.g. [40], and later using conceptual proofs for real analytic unimodal interval maps in [5, 45, 51, 61], for real analytic circle homeomorphisms with critical points in [13, 14, 28, 36, 63] and for certain multimodal maps in [55–58]. All these later results heavily use complex analytic machinery and, in particular, rely on the complex analytic extensions of interval maps.

Within the literature on holomorphic dynamics, one sees a similar development: many conjectures about iterations of general polynomials are only solved in the context of polynomials with real coefficients. An example of such a conjecture is density of hyperbolicity which is unsolved in the general case, but was proved for real quadratic maps independently by Lyubich and Graczyk-Swiatek, and in the general case by Kozlovski, Shen and van Strien, see [26, 37, 38, 47]. These results heavily rely on the existence of so-called real and complex bounds, see [12, 25, 42, 46, 54], but such complex bounds do not hold for general non-real polynomials or rational maps. Indeed, they hold for non-renormalizable polynomial maps, as in [30, 39, 64], but, in general, not for non-real infinitely renormalizable quadratic maps, see for example [52, 59].

Of course there are plenty of results on renormalization and towards density of hyperbolicity in the setting of non-real polynomials, see [10, 31–34, 44] and similarly there are plenty of impressive results on interval maps which do not use complex tools, on for example invariant measures, thermodynamic formalism and stochastic stability. Nevertheless, it is fair to say that a growing number of results within the field of real one-dimensional dynamics crucially rely on complex analytic tools, and vice versa many results about polynomial maps are only known when these preserve the real line.

When studying real one-dimensional maps, it is unnatural to restrict attention to maps which are real analytic. Indeed, in certain cases, renormalization results for real analytic interval maps can be extended to $C^3$ or $C^4$ maps. This was done using a functional analytic approach in [16] for unimodal interval maps and by heavily exploiting what is known for real analytic circle homeomorphisms in [27, 28]. A purely real approach, which gives existence of periodic points of the renormalization operator for unimodal maps of the form $g(|x|^\ell )$ , $\ell>1$ , was obtained by Martens in [49].

The purpose of this paper is to initiate a theory for $C^{3+}$ interval maps showing that these have extensions to the complex plane with properties analogous to those of real polynomial maps. Thus, the eventual aim of this theory is to show that $C^{3+}$ maps can be treated with techniques which are very similar to the complex analytic techniques which were so fruitful in the case of polynomial and real-analytic maps.

In this paper, we will establish the first cornerstone of this theory by showing that one has a Julia–Fatou–Sullivan-type description for such maps in a very important situation, namely for infinitely renormalizable maps of bounded type.

Let us be more precise and consider a $C^r$ map $f\colon I\to \mathbb R$ . Such a map f has an extension to a $C^{r}$ map $F\colon \mathbb C \to \mathbb C$ which is asymptotically holomorphic of order r, that is, $({\partial }/{\partial \bar {z}})F(z)=0$ when $\mathrm {Im}\, z=0$ and $({\partial }/{\partial \bar {z}})F(z)=O(| \mathrm {Im}\,{z} |^{r-1})$ uniformly, see [24]. The notion of asymptotically holomorphic maps goes back at least to [9]. In dynamics, this notion was used in [4, 11, 12, 27, 43, 61] (see also [22, 23] for related material on the more restrictive notion of a uniformly asymptotically conformal (UAC) map).

Note that F is not conformal outside the real line and so, in principle, periodic points can be of saddle type. Even if a periodic point is repelling, in general, the linearization at such a point will not be conformal. It follows that F cannot be quasiconformally conjugate to a polynomial-like map (the pullbacks of a small circle in a small neighbourhood of a non-conformal repelling point become badly distorted, but this is not the case in a small neighbourhood of a conformal repelling point). This means that our maps are not uniformly quasiconformal, that is, there exists no K so that all iterates are K-quasiconformal. So we cannot apply the rich theory developed for such maps, see for example [8, 35].

Nevertheless, we are able to provide a surprisingly detailed description for maps from our class. For example, by new estimates on the Jacobian of a quasiconformal mapping in terms of some Poincaré metric, we obtain an analogue of the Julia–Fatou–Sulivan theorem and the absence of wandering domains. We expect that these estimates will be useful more widely.

Main Theorem. Let $f\in C^{3+\alpha }$ ( $\alpha>0$ ) be a unimodal, infinitely renormalizable interval map of bounded type whose critical point has criticality given by an even integer d. Then every asymptotically holomorphic $C^{3+\alpha }$ extension F of f (of order $3+\alpha $ ) to a map defined on a neighbourhood of the interval in the complex plane is such that there exists a sequence of domains $U_n\subset V_n\subset \mathbb {C}$ containing the critical point of f and iterates $q_n$ with the following properties.

1. (1) The map $G:=F^{q_n}\colon U_n\to V_n$ is a degree d, quasi-regular polynomial-like map.

2. (2) For large enough n, each periodic point in the filled Julia set $\mathcal K_{G}:=\{z\in U_n; G^i(z)\in U_n\,\, \text { for all } i\ge 0\}$ is repelling.

3. (3) The Julia $\mathcal J_G:= \partial \mathcal K_G$ and filled-in Julia set of G coincide, that is, $\mathcal J_G=\mathcal K_G$ .

4. (4) The map G is topologically conjugate to a polynomial mapping in a neighbourhood of its Julia set. In particular, G has no wandering domains.

5. (5) The Julia set $\mathcal J_{G}$ is locally connected.

A more precise statement of this theorem can be found in Corollary 6.8 where we use the notion of controlled asymptotically holomorphic polynomial-like (AHPL)-maps, see Definition 5.1. We expect a similar result to hold in much greater generality, for example, for general $C^{3+\alpha }$ real asymptotically holomorphic polynomial-like maps (of order $3+\alpha $ ) with finitely many critical points of integer order. (We believe that a control of the form $({\partial }/{\partial \bar {z}})F(z)\to 0$ as $| \mathrm {Im}\,{z} |\to 0$ is insufficient for the theorem to hold.)

Our plan is to build on the results in this paper to prove the absence of invariant line fields for asymptotically holomorphic maps extending the methods of [51]. In addition, rather than using functional analytic tools, as in [16], we plan to prove renormalization results for $C^r$ maps through the McMullen tower construction directly following the ideas in [51] or more ambitiously following the approach of Avila–Lyubich in [5]. Thus, our ultimate goal is to establish a closer analogy between real and complex one-dimensional dynamics along the lines suggested in Table 1.

Table 1 Establishing a closer analogy between real and complex one-dimensional dynamics.

1.1 Object of study

We shall study the dynamics of certain quasi-regular maps in the complex plane that are generalizations of standard (holomorphic) polynomial-like maps, as defined by Douady–Hubbard in [20]. Such generalized polynomial-like maps arise as deep renormalizations of unimodal interval maps that admit an asymptotically holomorphic extension to a complex neighbourhood of their real domain. Let $\varphi : U\to V$ be a $C^1$ map between two domains in the complex plane, and assume that . We say that $\varphi $ is asymptotically holomorphic of order $r>1$ if $\varphi $ is quasi-regular and its complex dilatation $\mu _\varphi $ satisfies $|\mu _\varphi (z)|\leq C|\mathrm {Im}\,{z}|^{r-1}$ for all $z\in U$ and some constant $C>0$ (in particular, $\mu _\varphi $ vanishes on the real axis, that is, $\varphi $ is conformal there). As mentioned above, every $C^r$ map of the real line admits an extension to a neighbourhood of the real axis which is asymptotically holomorphic of order r. (The notion of asymptotically holomorphic maps can even be defined for maps which are merely quasiconformal on $\mathbb C$ . It can be shown that if such a map is asymptotically holomorphic of order r, then its restriction to the real line is actually $C^r$ , see [2, 21].)

We may now formally define the class of dynamical systems we intend to study. Please note that in what follows, we only consider maps having a unique critical point of finite even order $d\geq 2$ .

Definition 1.1. Let $U,V\subset \mathbb {C}$ be Jordan domains symmetric about the real axis, and suppose U is compactly contained in V. A $C^r$ ( $r\geq 3$ ) map $f:U\to V$ is said to be an asymptotically holomorphic polynomial-like map, or AHPL-map for short, if:

1. (i) f is a degree $d\geq 2$ proper branched covering map of U onto V, branched at a unique critical point $c\in U\cap \mathbb {R}$ of criticality given by d;

2. (ii) f is symmetric about the real axis, that is, for all $z\in U$ ;

3. (iii) f is asymptotically holomorphic of order r.

It follows from the well-known Stoilow factorization theorem (see [3, Corollary 5.5.3]) that an AHPL-map f as above can be written as $f=\phi \circ g$ , where $g:U\to V$ is a (holomorphic) polynomial-like map and $\phi :V\to V$ is a $C^r$ quasiconformal diffeomorphism which is also asymptotically holomorphic of order r.

Just as in the case of standard polynomial-like maps, we define the filled-in Julia set of an AHPL-map $f:U\to V$ to be the closure of the set of points which never escape under iteration, namely,

This is a compact, totally f-invariant subset of U. Its boundary $J_f=\partial \mathcal {K}_f$ is called the Julia set of f. By simple analogy with the case of holomorphic polynomial-like maps, there are natural questions to be asked about AHPL-maps and their Julia sets.

1. (1) Are the (expanding) periodic points dense in $J_f$ ?

2. (2) When is $J_f$ locally connected?

3. (3) What is the classification of stable components of $\mathcal {K}_f\setminus J_f$ ?

4. (4) Can f have non-wandering domains?

5. (5) Is there a (topological) straightening theorem for AHPL-maps?

These questions do not have obvious answers. For instance, in the holomorphic case, the first question has an affirmative answer whose proof is easy thanks to Montel’s theorem—a tool which is not useful here. Likewise, in the holomorphic case, question (4) has a negative answer thanks to Sullivan’s non-wandering domains theorem, whose proof uses quasiconformal deformations of f in a way that is not immediately available here, because, in general, the iterates of an AHPL-map are not uniformly quasiconformal.

Rather than studying very general AHPL-maps, in this paper, we will restrict our attention to those which can be renormalized, in fact infinitely many times. The definition of renormalization in the present context is the same as the one for polynomial-like mappings: an AHPL-map f is renormalizable if there exists a topological disk D containing the critical point of f and an integer $p>1$ so that D is compactly contained in $f^p(D)$ and $f^p:D\to f^p(D)$ is again an AHPL-map. Thanks to a theorem proved in [12], every sufficiently deep renormalization of an asymptotically holomorphic map whose restriction to the real line is an infinitely renormalizable map (in the usual real sense) is an (infinitely renormalizable) AHPL-map with a priori bounds.

One of our goals in the present paper is to provide answers to (some of) the above questions under the assumption that the AHPL-map f is infinitely renormalizable of bounded type. Another goal will be to prove $C^2$ a priori bounds for the renormalizations of such an f, under the same bounded type assumption.

1.2 Summary

Here is a brief description of the contents of this paper. We start by revisiting the real bounds for $C^3$ unimodal maps in §2. In §3, we prove that the successive renormalizations of a $C^3$ infinitely renormalizable AHPL-map of bounded type are uniformly bounded in the $C^2$ topology, and that such bounds are beau in the sense of Sullivan. In proving these bounds, we employ as a tool the matrix form of the chain rule for the second derivative of a composition of maps. This tool does not seem to have been used at all in the literature on low-dimensional dynamics. The key ingredient that allows us to prove our main theorem is a result that, roughly speaking, states that (a deep renormalization of) an AHPL-map is an infinitesimal expansion of the hyperbolic metric on its co-domain minus the real axis. This is the main result in §5.1, namely Theorem 5.4.

In §4, we introduce techniques which are crucial in establishing Theorem 5.4, namely Proposition 4.14 and Theorem 4.15. Specifically, we give a bound for the hyperbolic Jacobian of a $C^2$ quasiconformal map in terms of its local quasiconformal distortion in two situations: for maps with small dilatation and for maps which are asymptotically holomorphic. These bounds are applied to the diffeomorphic part of our AHPL-map, which therefore needs to be at least $C^2$ with good bounds. This is the main reason why we need the $C^2$ bounds developed in §3. This infinitesimal expansion of the hyperbolic metric has several consequences, e.g. the fact that every periodic point of (a sufficiently deep renormalization of) an AHPL-map is expanding—once again, see Theorem 5.4.

Finally, in §6, we go further and construct puzzle pieces for such AHPL-maps, and show with the help of Theorem 5.4 that the puzzle pieces containing any given point of the Julia set of an infinitely renormalizable AHPL-map shrink around that point. This implies that the Julia set of such a map is always locally connected. Even more, as a consequence, such a map is in fact topologically conjugate to an actual (holomorphic) polynomial-like map and therefore does not have wandering domains.

2 Revisiting the real bounds

In this section, we will recall some basic facts about renormalization of real unimodal maps.

2.1 Renormalization of unimodal maps

We need to recall some definitions and a few facts concerning the renormalization theory of interval maps. Let us consider a $C^3$ unimodal map $f:I\to I$ defined on the interval $I=[-1,1]\subset \mathbb {R}$ , with its unique critical point at $0$ and corresponding critical value at $1$ , i.e. with $f'(0)=0$ and $f(0)=1$ . From the viewpoint of renormalization, to be defined below, there is no loss of generality in assuming that f is even, that is, that $f(-x)=f(x)$ for all $x \in I$ . We also assume that the critical point of f has finite even order $d\geq 2$ . Hence, we often refer to f as a d-unimodal map.

We say that such an f is renormalizable if there exist an integer $p=p(f)>1$ and $\unicode{x3bb} =\unicode{x3bb} (f)=f^p(0)$ such that $f^p|[-|\unicode{x3bb} |,|\unicode{x3bb} |]$ is unimodal and maps $[-| \unicode{x3bb} |,|\unicode{x3bb} |]$ into itself. Taking p as the smallest possible, we define the first renormalization of f to be the map $Rf:I\to I$ given by

(2.1) $$ \begin{align} Rf(x) =\frac{1}{\unicode{x3bb}}\,f^p(\unicode{x3bb} x). \end{align} $$

The intervals $\Delta _j=f^j([-|\unicode{x3bb} |,|\unicode{x3bb} |])$ for $0\leq j\leq p-1$ , have pairwise disjoint interiors, and their relative order inside $I_0$ determines a unimodal permutation $\theta $ of $\{0,1,\ldots ,p-1\}$ . Thus, renormalization consists of a first return map to a small neighbourhood of the critical point rescaled to unit size via a linear rescale.

It makes sense to ask whether $Rf$ is also renormalizable, since $Rf$ is certainly a normalized unimodal map. If the answer is yes, then one can define $R^2f=R(Rf)$ , and so on. In particular, it may be the case that the unimodal map f is infinitely renormalizable, in the sense that the entire sequence of renormalizations $f, Rf, R^2f, \ldots , R^nf, \ldots $ is well defined.

We assume from now on that f is infinitely renormalizable. Let us denote by $P(f)\subseteq I$ the closure of the forward orbit of the critical point under f (the post-critical set of f). The set $P(f)$ is a Cantor set with zero Lebesgue measure, see below. It can be shown also that $P(f)$ is the global attractor of f both from the topological and metric points of view.

Note that for each $n\ge 0$ , we can write

where $q_0=1$ , $\unicode{x3bb} _0=1$ , $q_n=\prod _{i=0}^{n-1} p(R^if)$ and $\unicode{x3bb} _n=\prod _{i=0}^{n-1} \unicode{x3bb} (R^if)=f^{q_n}(0)$ . The positive integers $a_i=p(R^if)\geq 2$ are called the renormalization periods of f, and the $q_n$ terms are the closest return times of the orbit of the critical point. Note that $q_{n+1}=a_nq_n=\prod _{i=0}^{i=n}a_i \geq 2^{n+1}$ ; in particular, the sequence $q_n$ goes to infinity at least exponentially fast.

It will be important to consider the renormalization intervals of f at level n, namely $\Delta _{0,n}=[-|\unicode{x3bb} _n|,|\unicode{x3bb} _n|] \subset I_0$ and $\Delta _{i,n}=f^i(\Delta _{0,n})$ for $i=0,1, \ldots , q_n-1$ . The collection ${\mathcal C}_n=\{\Delta _{0,n}, \ldots , \Delta _{q_n-1,n}\}$ consists of pairwise disjoint intervals. Moreover, $\bigcup \{\Delta :\Delta \in {\mathcal C}_{n+1}\} \subseteq \bigcup \{\Delta :\Delta \in {\mathcal C}_n\}$ for all $n \geq 0$ and we have

$$ \begin{align*} P(f)= \bigcap _{n=0}^{\infty} \bigcup_{i=0}^{q_n-1} \Delta_{i,n}. \end{align*} $$

Once we know that $\max _{0\leq i\leq q_n-1} |\Delta _{i,n}|\to 0$ as $n\to \infty $ , it follows that $P(f)$ is, indeed, a Cantor set. This (and much more) follows from the so-called real a priori bounds proved by Sullivan in [61]. The following form of the real bounds is not the most general, but it will be quite sufficient for our purposes. We say that an infinitely renormalizable map f as above has combinatorial type bounded by N if its renormalization periods are bounded by N, that is, $a_n\leq N$ for all $n\in \mathbb {N}$ .

Theorem 2.1. (Real bounds)

Let $f:I\to I$ be a $C^3$ unimodal map as above, and suppose that f is infinitely renormalizable with combinatorial type bounded by $N>1$ . Then there exist constants $K_f>0$ and $0<\alpha _f< \beta _f<1$ such that the following hold for all $n\in \mathbb {N}$ .

1. (i) If $\Delta \in \mathcal {C}_{n+1}$ , $\Delta ^*\in \mathcal {C}_n$ and $\Delta \subset \Delta ^*$ , then $\alpha _f|\Delta ^*|\leq |\Delta | \leq \beta _f|\Delta ^*|$ .

2. (ii) For all $1\leq i<j\leq q_n-1$ and each $x\in \Delta _{i,n}$ , we have

   $$ \begin{align*} \frac{1}{K_f}\frac{|\Delta_{j,n}|}{|\Delta_{i,n}|} \leq |(f^{j-i})'(x)| \leq K_f \frac{|\Delta_{j,n}|}{|\Delta_{i,n}|}. \end{align*} $$

3. (iii) We have $\|R^nf\|_{C^1(I)} \leq K_f$ .

Moreover, there exist positive constants $K=K(N)$ , $\alpha =\alpha (N)$ , $\beta =\beta (N)$ , with $0<\alpha < \beta <1$ , and $n_0=n_0(f)\in \mathbb {N}$ such that for all $n\geq n_0$ , the constants $K_f$ , $\alpha _f$ and $\beta _f$ in items (i), (ii) and (iii) above can be replaced by K, $\alpha $ and $\beta $ , respectively.

For a complete proof of this theorem, see [19]. In informal terms, the theorem states three things. First, that the post-critical set $P(f)$ of an infinitely renormalizable d-unimodal map with bounded combinatorics is a Cantor set with bounded geometry. Second, that the successive renormalizations of such a map are uniformly bounded in the $C^1$ topology. Third, that the bounds on the geometry of the Cantor set and on the $C^1$ norms of the renormalizations become universal at sufficiently deep levels (such bounds are called beau by Sullivan in [61]—see also [19]).

Further analysis of the nonlinearity of renormalizations yields the following consequence of the real bounds.

Corollary 2.2. ( $C^2$ real bounds)

Under the assumptions of Theorem 2.1, the successive renormalizations of f are uniformly bounded in the $C^2$ topology, and the bound is beau in the sense of Sullivan.

The following consequence of the real bounds, namely Lemma 2.3 below, is adapted from [13, Lemma A.5, pp. 379], and also from [17, §2.1].

Let $f:I\to I$ be a $C^3$ unimodal map as defined above, and suppose f is infinitely renormalizable with renormalization periods bounded by N. For each $n\geq 1$ , let $\mathcal {C}_n=\{\Delta _{i,n}:\, 0\leq i\leq q_n-1\}$ denote the collection of renormalization intervals of f at level n. For each $n \geq 1$ , we define

$$ \begin{align*}S_n=\sum_{\mathcal{C}_n\ni \Delta\neq \Delta_{0,n}}\frac{|\Delta|}{d(c,\Delta)}\,,\end{align*} $$

where $d(c,\Delta )$ denotes the Euclidean distance between $\Delta \subset I$ and the critical point $c=0$ . Roughly speaking, the result states that for each infinitely renormalizable unimodal map of bounded type, the sequence $\{S_n\}_{n \geq 1}$ is bounded and the bound is beau in the sense of Sullivan.

Lemma 2.3. There exists a constant $B_1=B_1(N)>0$ with the following property. For each infinitely renormalizable unimodal map f of combinatorial type bounded by N, there exists $n_1=n_1(f)\in \mathbb {N}$ such that for all $n\geq n_1$ , we have $S_n\leq B_1$ .

Proof. The desired bound can be proved by a recursive estimate. Note that we can write

(2.2) $$ \begin{align} S_{n+1} = \sum_{\mathcal{C}_{n+1}\ni J\subset \Delta_{0,n}\setminus \Delta_{0,n+1}} \frac{|J|}{d(c,J)} + \sum_{\mathcal{C}_n\ni \Delta\neq \Delta_{0,n}}\bigg( \sum_{\mathcal{C}_{n+1}\ni J\subset \Delta} \frac{|J|}{d(c,J)} \bigg). \end{align} $$

Now, since $d(c,J)> \tfrac 12|\Delta _{0,n+1}|$ for each $J\in \mathcal {C}_{n+1}$ , we certainly have

(2.3) $$ \begin{align} \sum_{\mathcal{C}_{n+1}\ni J\subset \Delta_{0,n}\setminus \Delta_{0,n+1}} \frac{|J|}{d(c,J)} \leq 2\frac{|\Delta_{0,n}|}{|\Delta_{0,n+1}|}. \end{align} $$

From the real bounds, Theorem 2.1, we know that there exists a constant $0<\alpha =\alpha (N)<1$ such that $|\Delta _{0,n}|\leq \alpha ^{-1}|\Delta _{0,n+1}|$ for all sufficiently large n. For each $\Delta \in \mathcal {C}_n$ , let $J_1,J_2,\ldots , J_{a_n}\in \mathcal {C}_{n+1}$ be all the intervals at level $n+1$ which are contained in $\Delta $ . Then, again from the real bounds, we have $\sum _{i=1}^{a_n}|J_i|\leq \beta |\Delta |$ , where $0<\beta =\beta (N)<1$ , provided the renormalization level n is sufficiently large. Moreover, $d(c,J_i)\geq d(c,\Delta )$ for all i. Hence, we have for all n sufficiently large,

(2.4) $$ \begin{align} \sum_{\mathcal{C}_n\ni \Delta\neq \Delta_{0,n}}\bigg( \sum_{\mathcal{C}_{n+1}\ni J\subset \Delta} \frac{|J|}{d(c,J)} \bigg) &\leq \sum_{\mathcal{C}_n\ni \Delta\neq \Delta_{0,n}}\bigg(\frac{\sum_{\mathcal{C}_{n+1}\ni J\subset \Delta}|J|}{d(c,\Delta)}\bigg) \nonumber\\ &\leq \beta \sum_{\mathcal{C}_n\ni \Delta\neq \Delta_{0,n}}\frac{|\Delta|}{d(c,\Delta)} = \beta S_n. \end{align} $$

Putting equations (2.3) and (2.4) back into equation (2.2), we deduce that there exists $n_0=n_0(f)$ such that $S_{n+1}\leq \beta S_n +2\alpha ^{-1}$ for all $n\geq n_0$ . By induction, it follows that $S_{n_0+k}\leq \beta ^k S_{n_0} + 2\alpha ^{-1}(1+\beta +\cdots + \beta ^{k-1})$ for all $k\geq 0$ . Since $\beta <1$ , this shows that the sequence $(S_n)_{n\geq 1}$ is bounded, and eventually universally so.

What we will need is in fact a consequence of this lemma. Given f as in Lemma 2.3, write for all $n\geq 1$ ,

(2.5) $$ \begin{align} S_n^* = \sum_{i=1}^{q_n-1} \frac{|\Delta_{i,n}|^2}{|\Delta_{i+1,n}|} [d(c,\Delta_{i,n})]^{d-2}\, , \end{align} $$

where d is the order of f at the critical point c.

Lemma 2.4. There exists a constant $B_2=B_2(N)>0$ with the following property. For each infinitely renormalizable unimodal map f of combinatorial type bounded by N, there exists $n_2=n_2(f)\in \mathbb {N}$ such that for all $n\geq n_2$ , we have $S_n^*\leq B_2$ .

Proof. Since f has a critical point of order d at c, we have $|f'(x)|\geq C_0|x-c|^{d-1}$ for all $x\in I$ , for some $C_0=C_0(f)>0$ . Replacing, if necessary, f by $R^kf$ for sufficiently large k, we can assume that $C_0$ depends in fact only on N. Now, for each i, we can write $|\Delta _{i+1,n}|/|\Delta _{i,n}|=|f'(x_{i,n})|$ for some $x_{i,n}\in \Delta _{i,n}$ by the mean-value theorem. Hence, using that $|x_{i,n}-c|\geq d(c,\Delta _{i,n})$ , we have

$$ \begin{align*} &\frac{|\Delta_{i,n}|^2}{|\Delta_{i+1,n}|} [d(c,\Delta_{i,n})]^{d-2} = \frac{|\Delta_{i,n}|}{|f'(x_{i,n})|} [d(c,\Delta_{i,n})]^{d-2}\\ &\quad\leq C_0^{-1} \frac{|\Delta_{i,n}|}{|x_{i,n}-c|} \leq C_0^{-1} \frac{|\Delta_{i,n}|}{d(c,\Delta_{i,n})}. \end{align*} $$

This shows that $S_n^*\leq C_0^{-1}S_n$ for all (sufficiently large) n, and the desired result follows from Lemma 2.3.

3 The $C^2$ bounds for AHPL-maps

In this section, we prove that the successive renormalizations of an infinitely renormalizable AHPL-map of bounded combinatorial type are uniformly bounded in the $C^2$ topology, and the bounds are beau. Such bounds will be required when we study the diffeomorphic part of an AHPL-map.

The main result of this section can be stated more precisely as follows.

Theorem 3.1. Let $f:U\to V$ be an infinitely renormalizable, $C^3,$ AHPL-map of combinatorial type bounded by $N\in \mathbb {N}$ , and let $R^n(f):U_n\to V_n$ , $n\geq 1$ , be the sequence of renormalizations of f. There exists a constant $C_f>0$ such that $\|R^n(f)\|_{C^2(U_n)}\leq C_f$ . Moreover, there exist $C=C(N)>0$ and $m=m(f)\in \mathbb {N}$ such that $\|R^n(f)\|_{C^2(U_n)}\leq C$ for all $n\geq m$ .

The proof will use the real bounds as formulated in §2.1, Lemma 2.4 as well as the complex bounds established in [12], in the form stated in §3.1 below. In fact, the complex bounds are essential even to make sure that the renormalizations $R^nf$ appearing in Theorem 3.1 are well-defined AHPL-maps (see Remark 3.3 below).

3.1 The complex bounds

We conform with the notation introduced earlier when dealing with infinitely renormalizable interval maps and with AHPL-maps.

Theorem 3.2. (Complex bounds)

Let $f{\kern-1.2pt}:{\kern-1pt} U{\kern-1.5pt}\to{\kern-1.2pt} V$ be an AHPL-map with $I{\kern-1.2pt}=[-1,{\kern-1.2pt}1]{\kern-1.2pt}\subset{\kern-1.2pt} U$ , and suppose that $f|_I:I\to I$ is an infinitely renormalizable quadratic unimodal map with combinatorial type bounded by N. There exist $C=C(N)>1$ and $n_3=n_3(f)\in \mathbb {N}$ such that the following statements hold true for all $n\geq n_3$ .

1. (i) For each $0\leq i\leq q_n-1$ , there exist Jordan domains $U_{i,n}, V_{i,n}$ , with piecewise smooth boundaries and symmetry about the real axis, such that $\Delta _{i,n}\subset U_{i,n}\subset V_{i,n}$ , the $V_{i,n}$ are pairwise disjoint and we have the sequence of surjections

   $$ \begin{align*} U_{0,n}\xrightarrow{\;f\,} U_{1,n}\xrightarrow{\;f\,}\cdots \xrightarrow{\;f\,} U_{q_n-1,n}\xrightarrow{\;f\,} V_{0,n} \xrightarrow{\;f\,} V_{1,n}\xrightarrow{\;f\,} \cdots\xrightarrow{\;f\,} V_{q_n-1,n}. \end{align*} $$

2. (ii) For each $0\leq i\leq q_n-1$ , $f_{i,n}=f^{q_n}|_{U_{i,n}}:U_{i,n}\to V_{i,n}$ is a well-defined AHPL-map with critical point at $f^i(c)$ .

3. (iii) We have $\mod{\kern-1.2pt} (V_{i,n}{\kern-1pt}\setminus{\kern-1pt} U_{i,n})\geq C^{-1}$ and $\mathrm {diam}(V_{i,n})\leq C |\Delta _{i,n}|$ for all $0\leq i{\kern-1.2pt}\leq q_n-1$ .

4. (iv) The map $f_{i,n}:U_{i,n}\to V_{i,n}$ has a Stoilow decomposition $f_{i,n}=\phi _{i,n}\circ g_{i,n}$ such that $K(\phi _{i,n})\leq 1+C|\Delta _{0,n}|$ for each $0\leq i\leq q_n-1$ .

This theorem is a straightforward consequence of (a special case of) the complex bounds proved in [12].

Remark 3.3. For each $n\geq 1$ , consider the linear map $\Lambda _n(z)=|\Delta _{0,n}|z$ , and consider the Jordan domains $U_n=\Lambda _n^{-1}(U_{0,n})\subset \mathbb {C}$ and $V_n=\Lambda _n^{-1}(V_{0,n})\subset \mathbb {C}$ . Note that $I\subset U_n\subset V_n$ . We define $R^nf:U_n\to V_n$ by $R^nf=\Lambda _n^{-1}\circ f_{0,n}\circ \Lambda _n$ . This is the nth renormalization of f that appears in the statement of Theorem 3.1. Note that the complex bounds given by this theorem guarantee that $\mathrm {diam}(V_n)\asymp |I|$ ; in particular, the $C^0$ norms $\|R^nf\|_{C^0(U_n)}$ are uniformly bounded (by a beau constant).

3.2 Digression on the chain rule

Let $\phi :U\to \mathbb {R}^n$ be a $C^2$ map defined on an open set $U\subset \mathbb {R}^n$ . In matrix form, the second derivative $D^2\phi $ of $\phi $ is an $n\times n^2$ matrix obtained by the juxtaposition of the Hessian matrices of each of the n scalar components of $\phi $ . For instance, in dimension $n=2$ , the second derivative of a map $\phi =u+iv$ is given by the $2\times 4$ matrix $D^2\phi = [\begin {smallmatrix} u_{xx} & u_{xy} & v_{xx} & v_{xy}\\u_{yx} & u_{yy} & v_{yx} & v_{yy} \end {smallmatrix}]$ obtained by adjoining the Hessian matrices of the two components of $\phi $ .

Now, if $U,V, W\subseteq \mathbb {R}^n$ are open sets with $V\subseteq W$ , and if $\psi :U\to V$ and $\phi :W\to \mathbb {R}^n$ are both $C^2$ , then the composition $\phi \circ \psi $ is $C^2$ , and

(3.1)

This is the chain rule for the second derivative of a composition in matrix form. Here, we denote by the tensor (or Kronecker) product of two square matrices $A,B$ of the same size; thus, in our case, is a square $n^2\times n^2$ matrix. For a proof of this formula, see [48].

We will need in fact a formula for the second derivative of an (arbitrarily high) iterate of a given map. We formulate it as a lemma. (We use the abbreviation (m times).)

Lemma 3.4. Let $\phi :U\to \mathbb {R}^n$ , $U\subseteq \mathbb {R}^n$ open, be a $C^2$ map. Then for each $k\geq 0$ , we have

wherever the kth iterate $\phi ^k$ is defined.

Proof. This easily established from equation (3.1) by induction (write $\phi ^{k+1}=\phi \circ \phi ^k$ for the induction step).

Of course, in this paper, we will only need these formulae in dimension $n=2$ .

3.3 Proof of Theorem 3.1

Here we prove our first main result, namely Theorem 3.1. It is natural to divide the proof into two steps: in the first step, we bound the $C^1$ norms of renormalizations, and in the second step, we bound the $C^2$ norms. Throughout the proof, we shall successively denote by $C_0,C_1,C_2,\ldots $ positive constants that are either absolute or depend only on the constants given by the real and complex bounds. Also, in the estimates to follow, we use the operator norm on matrices; to wit, we define $\|A\|=\sup _{|v|=1}|Av|$ (here, $|v|$ denotes the Euclidean norm of the vector v). This norm has the advantage of being sub-multiplicative, which is to say that $\|AB\|\leq \|A\|\cdot \|B\|$ whenever the product $AB$ is well defined. It also satisfies .

3.3.1 Bounding the $C^1$ norms

First we prove that the successive renormalizations of f are uniformly bounded in the $C^1$ topology with beau bounds. We will prove a bit more than what is required. Let us fix $n\in \mathbb {N}$ so large that the real and complex bounds given by Theorems 2.1 and 3.2 hold true for $R^nf$ . We divide our argument into a series of steps.

1. (i) Replacing f by a sufficiently high renormalization, we may assume, using Corollary 2.2, that the $C^2$ norm of $f|_{I}$ is bounded by a beau constant (that depends only on N). In particular, there exists an open complex neighbourhood $\mathcal {O}$ of the dynamical interval $I\subset \mathbb {R}$ , with $\mathcal {O}\subseteq U$ , such that $\|f\|_{C^2(\mathcal {O})}\leq C_0$ . Additionally, because the critical point c has order d, we may also assume that $\|Df(y)\|\leq C_0|y-c|^{d-1}$ and $\|D^2f(y)\|\leq C_0|y-c|^{d-2}$ for all $y\in \mathcal {O}$ .

2. (ii) We may assume that n is so large that $V_{i,n}\subset \mathcal {O}$ for all i. This is possible because, by the complex bounds (Theorem 3.2), $\mathrm {diam}(V_{i,n})\asymp |\Delta _{i,n}|$ , and therefore the $V_{i,n}$ shrink exponentially fast as $n\to \infty $ , by the real bounds.

3. (iii) Let $j,k$ be positive integers such that $1\leq j<j+k\leq q_n$ . Then for each $x\in \Delta _{j,n}$ , we have, by Theorem 2.1,

   (3.2) $$ \begin{align} C_1^{-1}\frac{|\Delta_{j+k,n}|}{|\Delta_{j,n}|} \leq \|Df^k(x)\|=|(f^k)'(x)| \leq C_1\frac{|\Delta_{j+k,n}|}{|\Delta_{j,n}|}. \end{align} $$

4. (iv) Given $x\in \Delta _{j,n}$ and $y\in U_{j,n}$ , let us write $x_i=f^i(x)$ , $y_i=f^i(y)$ for all $i=0, 1,\ldots ,k$ . By step (i) and since f has a critical point at c of order d, we have

   (3.3) $$ \begin{align} \frac{ \|Df(x_i) - Df(y_i)\|}{ [d(c,\Delta_{i+j,n})]^{d-2}} \leq C_2 |x_i-y_i| \leq C_3|\Delta_{i+j,n}| \end{align} $$
   for $i=0,1,\ldots , k-1$ . From equation (3.3), we obviously have
   (3.4) $$ \begin{align} \|Df(y_i)\| \leq \|Df(x_i)\|+ C_3|\Delta_{i+j,n}| \cdot [d(c,\Delta_{i+j,n})]^{d-2} \end{align} $$
   for $i=0,1,\ldots , k-1$ .

5. (v) By the chain rule for first derivatives, we have

   (3.5) $$ \begin{align} \|Df^k(y)\| \leq \bigg\|\prod_{i=0}^{k-1} Df(y_i)\bigg\| \leq \prod_{i=0}^{k-1} \|Df(y_i)\|. \end{align} $$

6. (vi) Using equations (3.4) and (3.5), we get

   (3.6) $$ \begin{align} \|Df^k(y)\|&\leq \prod_{i=0}^{k-1} ( \|Df(x_i)\|+ C_3|\Delta_{i+j,n}| \cdot [d(c,\Delta_{i+j,n})]^{d-2} ) \nonumber\\ &\leq \prod_{i=0}^{k-1} \|Df(x_i)\| \cdot \prod_{i=0}^{k-1} \bigg(1+ C_3\frac{|\Delta_{i+j,n}|}{\|Df(x_i)\|} [d(c,\Delta_{i+j,n})]^{d-2} \bigg). \end{align} $$

7. (vii) However, since $x_i$ is real (and f preserves the real line), we have

   (3.7) $$ \begin{align} \prod_{i=0}^{k-1} \|Df(x_i)\| = \bigg|\prod_{i=0}^{k-1} f'(x_i)\bigg| = \|Df^k(x)\|. \end{align} $$
   Moreover, for each $i=0,1,\ldots , k$ , we have
   (3.8) $$ \begin{align} \|Df(x_i)\| =|f'(x_i)| \asymp \frac{|\Delta_{i+j+1,n}|}{|\Delta_{i+j,n}|}. \end{align} $$

8. (viii) Putting equations (3.7) and (3.8) back into equation (3.6), we get

   (3.9) $$ \begin{align} \|Df^k(y)\| \leq \|Df^k(x)\|\cdot \prod_{i=0}^{k-1} \bigg(1+ C_4\frac{|\Delta_{i+j,n}|^2}{|\Delta_{i+j+1,n}|} [d(c,\Delta_{i+j,n})]^{d-2} \bigg). \end{align} $$
   However now, using Lemma 2.4, we see that the product in the right-hand side of equation (3.9) is uniformly bounded, because
   (3.10) $$ \begin{align} \prod_{i=0}^{k-1} &\bigg(1+ C_4\frac{|\Delta_{i+j,n}|^2}{|\Delta_{i+j+1,n}|} [d(c,\Delta_{i+j,n})]^{d-2} \bigg) \nonumber \\[-4pt] & \leq \exp\bigg\{C_4\sum_{i=0}^{k-1} \frac{|\Delta_{i+j,n}|^2}{|\Delta_{i+j+1,n}|} [d(c,\Delta_{i+j,n})]^{d-2}\bigg\} \nonumber \\ &\leq\exp\bigg\{C_4 \sum_{i=1}^{q_n-1} \frac{|\Delta_{i,n}|^2}{|\Delta_{i+1,n}|} [d(c,\Delta_{i,n})]^{d-2}\bigg\} \nonumber \\ &=\exp\{C_4S_n^*\} \leq \exp\{B_2C_4\}. \end{align} $$

9. (ix) Hence, we have proved that $\|Df^k(y)\|\leq C_5\|Df^k(x)\|$ for all $y\in U_{j,n}$ and all $x\in \Delta _{j,n}$ . From equation (3.2), it follows that

   (3.11) $$ \begin{align} \|Df^k(y)\|\leq C_6 \frac{|\Delta_{j+k,n}|}{|\Delta_{j,n}|} \quad \textrm{for all } y\in U_{j,n}. \end{align} $$
   In particular, taking $j=1$ and $k=q_n-1$ , we see that the first derivative of the map $f^{q_n-1}|_{U_{1,n}}:U_{1,n}\to V_{0,n}$ satisfies (recall that $\Delta _{q_n,n}=\Delta _{0,n}$ )
   (3.12) $$ \begin{align} \|Df^{q_n-1}(y)\| \leq C_6 \frac{|\Delta_{0,n}|}{|\Delta_{1,n}|} \quad \textrm{for all } y\in U_{1,n}. \end{align} $$

10. (x) However, since f has a critical point of order d at $c=0$ , the restriction $f|_{U_{0,n}}:U_{0,n}\to U_{1,n}$ satisfies $\|Df(y)\|\leq C_7|y|^{d-1}\leq C_8|\Delta _{0,n}|^{d-1}$ for all $y\in U_{0,n}$ (we are implicitly using step (i) here). Combining this fact with step (ix), equation (3.12) and using the chain rule, we see that the first derivative of the map

    $$ \begin{align*}f_{0,n}=f^{q_n}|_{U_{0,n}}= f^{q_n-1}|_{U_{1,n}}\circ f|_{U_{0,n}}:U_{0,n}\to V_{0,n}\end{align*} $$
    satisfies
    (3.13) $$ \begin{align} \|Df^{q_n}(y)\| \leq C_9 \frac{|\Delta_{0,n}|^d}{|\Delta_{1,n}|} \quad \textrm{for all } y\in U_{0,n}. \end{align} $$
    However, again using that the critical point has order d, we have $|\Delta _{1,n}|\asymp |\Delta _{0,n}|^d$ . Putting this information back in equation (3.13), we deduce that
    $$ \begin{align*}\|Df_{0,n}\|_{C^0(U_{0,n})}=\|Df^{q_n}\|_{C^0(U_{0,n})}\leq C_{10}.\end{align*} $$
    Therefore, $\|DR^nf\|_{C^0(U_n)}\leq C_{10}$ also, since $R^nf$ is simply a linearly rescaled copy of $f_{0,n}$ . This shows that the successive renormalizations of f around the critical point are indeed uniformly bounded in the $C^1$ topology and the bounds are beau.

3.3.2 Bounding the $C^2$ norms

We now move to the task of bounding the second derivatives of the renormalizations of f. Here we use the chain rule for the second derivative of a (long) composition, as given by Lemma 3.4. Once again, we break the proof into a series of (short) steps.

1. (xi) Since $R^nf=\Lambda _n^{-1}\circ f_{0,n}\circ \Lambda _n$ , with $\Lambda _n(z)=|\Delta _{0,n}|z$ , we have

   (3.14) $$ \begin{align} \|D^2R^nf\|_{C^0(U_n)} \leq |\Delta_{0,n}|\cdot\|D^2f_{0,n}\|_{C^0(U_{0,n})}. \end{align} $$
   We need to bound the norm on the right-hand side of equation (3.14).

2. (xii) Recall from step (x) the decomposition $f_{0,n}= f^{q_n-1}|_{U_{1,n}}\circ f|_{U_{0,n}}$ . By the chain rule for second derivatives, for each $y\in U_{0,n}$ , we have

   (3.15)

   
   Note from step (i) that $\|D^2f(y)\|\leq C_0|y-c|^{d-2} \leq C_{11}|\Delta _{0,n}|^{d-2}$ . Moreover, applying equation (3.12) with y replaced by $f(y)$ , we have
   (3.16) $$ \begin{align} \|Df^{q_n-1}(f(y))\| \leq C_6 \frac{|\Delta_{0,n}|}{|\Delta_{1,n}|}.\\[-12pt] \nonumber \end{align} $$
   These two estimates combined yield an upper bound for the matrix norm of the second summand in the right-hand side of equation (3.15), namely
   (3.17) $$ \begin{align} \|Df^{q_n-1}(f(y))D^2f(y)\| \leq C_{12} \frac{|\Delta_{0,n}|^{d-1}}{|\Delta_{1,n}|}, \end{align} $$
   where $C_{12}=C_6C_{11}$ .

3. (xiii) It remains to bound the matrix norm of the first summand in the right-hand side of equation (3.15). Applying Lemma 3.4 with $\phi =f$ and $k=q_n-1$ to any point $z\in U_{1,n}$ , we have

   (3.18)

   
   Note that $\|D^2f(f^{q_n-2}(z))\|\leq C_0$ , by step (i). Since $f^{j-1}(z)\in U_{j,n}\subset \mathcal {O}$ , it also follows from step (i) that
   $$ \begin{align*} \|D^2f(f^{j-1}(z))\|\leq C_0|f^{j-1}(z)-c|^{d-2} \leq C_{13}[d(c,\Delta_{j,n})]^{d-2}\\[-12pt] \nonumber \end{align*} $$
   for all $j\leq q_n$ . Using this information in equation (3.18), we get
   (3.19) $$ \begin{align} &\|D^2f^{q_n-1}(z)\|\leq C_0 \|Df^{q_n-2}(z)\|^2 \nonumber\\ &\quad + C_{13}\sum_{j=1}^{q_n-2} \|Df^{q_n-j-1}(f^j(z))\|\, \|Df^{j-1}(z)\|^2 [d(c,\Delta_{j,n})]^{d-2}. \end{align} $$

4. (xiv) We now need to bound the norms on the right-hand side of equation (3.19). Using the estimate of equation (3.11) given in step (ix), we have

   (3.20) $$ \begin{align} \|Df^{q_n-2}(z)\| \leq C_6\frac{|\Delta_{q_n-1,n}|}{|\Delta_{1,n}|}, \end{align} $$
   as well as
   (3.21) $$ \begin{align} \|Df^{q_n-j-1}(f^j(z))\| \leq C_6\frac{|\Delta_{q_n-1,n}|}{|\Delta_{j+1,n}|} \end{align} $$
   and
   (3.22) $$ \begin{align} \|Df^{j-1}(z)\| \leq C_6\frac{|\Delta_{j,n}|}{|\Delta_{1,n}|} \\[-14pt] \nonumber \end{align} $$
   for all $j\leq q_n-1$ . Putting equations (3.20), (3.21) and (3.22) back in equation (3.19), we get
   (3.23) $$ \begin{align} \|D^2f^{q_n-1}(z)\|\leq C_{14}\bigg[ \frac{|\Delta_{q_n-1,n}|^2}{|\Delta_{1,n}|^2} + \sum_{j=1}^{q_n-2} \frac{|\Delta_{q_n-1,n}|}{|\Delta_{j+1,n}|}\frac{|\Delta_{j,n}|^2}{|\Delta_{1,n}|^2}[d(c,\Delta_{j,n})]^{d-2}\bigg]. \end{align} $$

5. (xv) Now we note that $|\Delta _{q_n-1,n}|\asymp |\Delta _{0,n}|$ , by the real bounds. (We have $|\Delta _{0,n}|=|f'(\xi )||\Delta _{q_n-1,n}|$ for some $\xi \in \Delta _{q_n-1,n}$ , by the mean value theorem, so $|\Delta _{0,n}|\leq C_0|\Delta _{q_n-1,n}|$ (where $C_0$ is the constant of step (i)). An inequality in the opposite direction follows from the fact, due to Guckenheimer (and using [18, Theorem IV.B] if f is not symmetric), that when $f|_{I}$ has negative Schwarzian derivative, the renormalization interval containing the critical point is the largest among all renormalization intervals at its level. Here we have not assumed the negative Schwarzian property for f, but it can be proved that $R^nf|_{I}$ has this property for all sufficiently large n. For details, see [16, pp. 760].) Using this information in equation (3.23), we deduce that

   (3.24) $$ \begin{align} \|D^2f^{q_n-1}(z)\| \leq C_{15}\frac{|\Delta_{0,n}|}{|\Delta_{1,n}|^2}\bigg[ |\Delta_{0,n}| + \sum_{j=1}^{q_n-2} \frac{|\Delta_{j,n}|^2}{|\Delta_{j+1,n}|}[d(c,\Delta_{j,n})]^{d-2}\bigg].\\[-15pt] \nonumber \end{align} $$
   Applying Lemma 2.4, we see that the sum inside square-brackets in the right-hand side of equation (3.24) is bounded (by a beau constant). Hence, we have established that
   (3.25) $$ \begin{align} \|D^2f^{q_n-1}(z)\| \leq C_{16}\frac{|\Delta_{0,n}|}{|\Delta_{1,n}|^2}.\\[-15pt] \nonumber \end{align} $$

6. (xvi) Carrying the estimates in equations (3.17) and (3.25) back into equation (3.15), we deduce that

   (3.26) $$ \begin{align} \|D^2f_{0,n}(y)\| \leq C_{17}\bigg( \frac{|\Delta_{0,n}|^{2d-1}}{|\Delta_{1,n}|^2} + \frac{|\Delta_{0,n}|^{d-1}}{|\Delta_{1,n}|}\bigg).\\[-15pt] \nonumber \end{align} $$
   This inequality is established for all $y\in U_{0,n}$ .

7. (xvii) Finally, combining equation (3.26) with equation (3.14), we get

   $$ \begin{align*} \|D^2R^nf\|_{C^0(U_n)} \leq C_{18}\bigg( \frac{|\Delta_{0,n}|^{2d}}{|\Delta_{1,n}|^2} + \frac{|\Delta_{0,n}|^{d}}{|\Delta_{1,n}|}\bigg). \end{align*} $$
   Using once again the fact that $|\Delta _{1,n}|\asymp |\Delta _{0,n}|^{d}$ , we deduce at last the inequality $\|D^2R^nf\|_{C^0(U_n)}\leq C_{20}$ . Hence, the successive renormalizations of f are uniformly bounded in the $C^2$ topology, as claimed (and the bounds are beau).

This finishes the proof of Theorem 3.1.

Remark 3.5. If we consider the Stoilow decomposition $R^nf=\phi _n\circ g_n$ coming from Theorem 3.2(iv), where $g_n:U_n\to V_n$ is a d-to- $1$ holomorphic branched covering map and $\phi _n:V_n\to V_n$ is an asymptotically holomorphic diffeomorphism, then it is possible to prove, using similar estimates, that $\|\phi _n\|_{C^2(V_n)}$ , $\|\phi _n^{-1}\|_{C^2(V_n)}$ and $\|g_n\|_{C^2(U_n)}$ are uniformly bounded and the bounds are beau.

4 Controlling the distortion of hyperbolic metrics

This section is a conformal/quasiconformal intermezzo. Here we develop the distortion tools that will be used in the proof of Theorem 5.4 in §5. We believe that these tools—especially those concerning the control of infinitesimal distortion of hyperbolic metric by an asymptotically conformal diffeomorphism, see Proposition 4.14 (for self-maps of the disk) and Theorem 4.15 (for other domains)—are of independent interest, and may find applications in other topics of study, such as Riemann surface theory.

4.1 Comparison of hyperbolic metrics

We view any non-empty open set $Y\subset \mathbb {C}$ whose complement has at least two points as a hyperbolic Riemann surface. As such, Y admits a conformal metric of constant negative curvature equal to $-1$ , the so-called hyperbolic or Poincaré metric of Y. We denote by $\rho _Y(z)|dz|$ this metric; $\rho _Y(z)$ is the Poincaré density at $z\in Y$ . Integrating this metric along a given rectifiable path $\gamma \subset Y$ , we get its hyperbolic length $\ell _Y(\gamma )$ . This gives rise to a distance $d_Y$ in the usual way: for any given pair of points $z,w\in Y$ , we set $d_Y(z,w)=\inf \ell _Y(\gamma )$ , where $\gamma $ ranges over all paths joining z to w (this will be equal to $\infty $ if z and w lie in distinct components of Y). We call $d_Y$ the hyperbolic distance of Y. Accordingly, given $E\subseteq Y$ , we denote by $\mathrm {diam}_Y(E)$ the hyperbolic diameter of E. We also use the following notation: if $z\in Y$ and $v\in T_zY$ is a tangent vector to Y at z, then we write $|v|_Y$ for the hyperbolic length of v (that is, the length of v in the above infinitesimal conformal metric).

Thus, when Y is the upper or lower half-plane, we have $\rho _Y(z)=|\mathrm {Im}\,z|^{-1}$ . When Y is the disk of centre $z_0\in \mathbb {C}$ and radius $R>0$ , we have

(4.1) $$ \begin{align} \rho_Y(z) = \frac{2R}{R^2-|z-z_0|^2}. \end{align} $$

In the case of the unit disk, one can easily compute that

$$ \begin{align*} d_{\mathbb{D}}(0,z) = \log{\frac{1+|z|}{1-|z|}}. \end{align*} $$

This yields the following elementary estimate which will be used in §5.1 (see Remark 5.2).

Lemma 4.1. Let $0\in E\subset \mathbb {D}$ and $0<\delta \leq 1$ . If $z\in \mathbb {D}$ is any point whose distance to the boundary of $\mathbb {D}$ is at least $\delta $ , and if $w\in E$ , then

$$ \begin{align*} d_{\mathbb{D}}(z,w) \leq \mathrm{diam}_{\mathbb{D}}(E) + \log{\frac{1}{\delta}}. \end{align*} $$

The well-known Schwarz lemma states that any holomorphic map $\varphi : X\to Y$ between two hyperbolic Riemann surfaces weakly contracts the underlying hyperbolic metrics. In other words, $|D\varphi (z)v|_Y \leq |v|_X$ for all $z\in X$ and every tangent vector $v\in T_zX$ . If equality holds for some z even at a single non-zero vector $v\in T_zX$ , then $\varphi $ is a local isometry between (a component of) X and (a component of) Y. In particular, if X is connected and $X\subset Y$ is a strict inclusion, and $\varphi : X\to Y$ is the inclusion map, then $\varphi $ is a strict contraction of the hyperbolic metrics. This leads, in the case when X is connected and $X\subset Y\subset \mathbb {C}$ , to the strict monotonicity of Poincaré densities: $\rho _X(z)> \rho _Y(z)$ for all $z\in X$ . The following comparison of Poincaré densities follows from monotonicity and will prove useful later.

Lemma 4.2. Let $Y\subseteq \mathbb {C}\setminus \mathbb {R}$ be an non-empty open set, and let $z,w\in Y$ be such that $\mathrm {Re}\,z=\mathrm {Re}\,w$ and $|\mathrm {Im}\,z|\leq |\mathrm {Im}\,w|$ . If $z\in D(w,|\mathrm {Im}\,w|)\subseteq Y$ , then

(4.2) $$ \begin{align} \frac{1}{|\mathrm{Im}\,z|} \leq \rho_Y(z) \leq \frac{1}{|\mathrm{Im}\,z|}\bigg(1-\frac{1}{2}\frac{|\mathrm{Im}\,z|}{|\mathrm{Im}\,w|}\bigg)^{-1}. \end{align} $$

Proof. Look at the inclusions $D(w,|\mathrm {Im}\,w|)\subseteq Y\subseteq \mathbb {C}\setminus \mathbb {R}$ and use equation (4.1) with $z_0=w$ and $R=|\mathrm {Im}\,w|$ .

4.2 Expansion of hyperbolic metric

It so happens that contraction sometimes leads to expansion. If $\psi : X\to Y$ is a bi-holomorphic map between two hyperbolic Riemann surfaces and $X\subset Y$ , then the inverse $\psi ^{-1}$ , viewed as a map from Y into Y, can be written as a composition of $\psi ^{-1}:Y\to X$ with the inclusion $X\subset Y$ . The first map in the composition is an isometry between the underlying hyperbolic metrics, whereas the second map is a contraction. Therefore, $\psi $ expands the hyperbolic metric of Y. In the present paper, we shall need a more quantitative version of this fact. This is given by the following lemma due to McMullen (see [51]).

Lemma 4.3. Let $X, Y$ be hyperbolic Riemann surfaces with $X\subset Y$ , and let $\psi :X\to Y$ be holomorphic univalent and onto. Then for all $x\in X$ and each tangent vector $v\in T_xX$ , we have

(4.3) $$ \begin{align} |D\psi(x)v|_Y \geq \Phi(s_{X,Y}(x))^{-1}|v|_X, \end{align} $$

where $s_{X,Y}(x)=d_Y(x,Y\setminus X)$ and $\Phi (\cdot )$ is the universal function given by (in [51], McMullen gives $\Phi (s)=2{|t\log t|}/({1-t^2})$ , where $0\leq t<1$ is such that $s=d_{\mathbb {D}}(0,t)$ . Eliminating t yields equation (4.4))

(4.4) $$ \begin{align} \Phi(s) =\sinh{(s)} \log{\bigg( \frac{1+ e^{-s}}{1- e^{-s}} \bigg)}. \end{align} $$

We remark that $\Phi (s)$ is a continuous monotone increasing function with $\Phi (0)=0$ and $\Phi (\infty )=1$ . Instead of equation (4.4), we shall need merely the estimate

(4.5) $$ \begin{align} \Phi(s)< 1 -\tfrac{1}{3}e^{-2s}. \end{align} $$

This estimate is valid provided $s>\tfrac 12\log {2}$ , and is easily proved with the help of Taylor’s formula.

4.3 Nonlinearity and conformal distortion

We will also need certain well-known results concerning the geometric distortion of holomorphic univalent maps. For details and some background, we recommend [15, §3.8].

Let $\varphi : V\to \mathbb {C}$ be a holomorphic univalent map defined on an open set $V\subset \mathbb {C}$ . Then we have Koebe’s pointwise estimate on the nonlinearity $\varphi "/\varphi '$ ; to wit, for every $z\in V$ , we have

(4.6) $$ \begin{align} \bigg|\frac{\varphi"(z)}{\varphi'(z)}\bigg| \leq \frac{4}{\mathrm{dist}(z,\partial V)}, \end{align} $$

where $\mathrm {dist}(\cdot ,\cdot )$ denotes Euclidean distance. This form of pointwise control of the nonlinearity of $\varphi $ has the following geometric consequence. Suppose $D\subset V$ is a compact convex subset, and write

(4.7) $$ \begin{align} N_{\varphi}(D) = \mathrm{diam}(D)\,\sup_{z\in D}{\bigg|\frac{\varphi"(z)}{\varphi'(z)}\bigg|}. \end{align} $$

Then for all $z, w\in D$ , we have

(4.8) $$ \begin{align} e^{-N_{\varphi}(D)} \leq \bigg|\frac{\varphi'(z)}{\varphi'(w)}\bigg| \leq e^{N_{\varphi}(D)}. \end{align} $$

When D is not convex, we can still get an estimate like equation (4.8) by covering D with small disks. The following result is by no means the sharpest of its kind, but it will be quite sufficient for our purposes.

Lemma 4.4. Let $\varphi :V\to \mathbb {C}$ be holomorphic univalent, and let $W\subset V$ be a non-empty compact connected set. Suppose $M>1$ is such that $1\le \mathrm {diam}(V)\leq M$ and $\mathrm {dist}(\partial V,\partial W)\geq M^{-1}$ . Also, let $z_0\in W$ be given. Then the following assertions hold.

1. (i) There exists $K_1=K_1(M)>1$ such that for all $z, w\in W$ , we have

   (4.9) $$ \begin{align} \frac{1}{K_1} \leq \bigg|\frac{\varphi'(z)}{\varphi'(w)}\bigg| \leq K_1. \end{align} $$
   In fact, we can take $K_1=e^{32\pi M^4}$ .

2. (ii) There exists $K_2=K_2(M)>0$ such that $\max \{\|\varphi '|_{W}\|_{C^0}, \|\varphi "|_{W}\|_{C^0}\}\leq K_2 |\varphi '(z_0)|$ .

Proof. Cover W with a finite number m of non-overlapping closed squares $Q_j$ , $1\leq j\leq m$ , with each $Q_j$ having the same side $\ell =(2\sqrt {2} M)^{-1}$ , and take m to be the smallest possible. Then , the diameter of $Q_j$ is $(2M)^{-1}$ and $\mathrm {dist}(Q_j, \partial V)\geq (2M)^{-1}$ for each $1\leq j\leq m$ . Since the total area of these squares cannot exceed the area of V, which is less than $\pi M^2$ , we see that $m<8\pi M^4$ . Moreover, from Koebe’s estimate in equation (4.7), we have for each j,

$$ \begin{align*} N_\varphi(Q_j)\leq (2M)^{-1}\cdot \frac{4}{(2M)^{-1}} = 4. \end{align*} $$

Now, since W is connected, given any pair of points $z,w\in W$ , we can join them by a chain of pairwise distinct squares $Q_{j_1}, Q_{j_2}, \ldots , Q_{j_n}$ such that , with $z\in Q_{j_1}$ and $w\in Q_{j_n}$ , say. Choose $z_k\in Q_{j_k}\cap Q_{j_{k+1}}$ for $k=1,2,\ldots , n-1$ , and set $z_0=z, z_n=w$ . Use equation (4.8) to get

$$ \begin{align*} \bigg|\frac{\varphi'(z)}{\varphi'(w)}\bigg| &= \prod_{k=0}^{n-1} \bigg|\frac{\varphi'(z_k)}{\varphi'(z_{k+1})}\bigg| \\ & \leq \exp{\bigg(\sum_{k=1}^{n} N_{\varphi}(Q_{j_k})\bigg)} \leq e^{4m}. \end{align*} $$

This establishes the upper bound in equation (4.9); the lower bound is obtained in the same way, or simply by interchanging z and w. Hence, assertion (i) is proved. Assertion (ii) follows from assertion (i) and the inequality in equation (4.6).

4.4 Quasiconformality and holomorphic motions

We need some non-trivial facts from the theory of quasiconformal mappings. Good references for what follows are [1, 3]. Given a quasiconformal homeomorphism $\phi $ , we write $\mu _\phi (z)$ for the Beltrami form of $\phi $ at z, and $K_\phi (z)=(1+|\mu _\phi (z)|)/(1-|\mu _\phi (z)|)$ for the dilatation of $\phi $ at z. We also denote by $K_\phi $ the maximal dilatation of $\phi $ , namely the supremum of $K_\phi (z)$ over all z in the domain of $\phi $ .

Lemma 4.5. Let $\phi : \mathbb {C}\to \mathbb {C}$ be a K-quasiconformal homeomorphism. Then for each $z\in \mathbb {C}$ and all $r>0$ and $s>0$ , we have

$$ \begin{align*} \frac{\max_{|\zeta-z|=rs}|\phi(\zeta)-\phi(z)|}{\min_{|\zeta-z|=s}|\phi(\zeta)-\phi(z)|} \leq e^{\pi K}\max\{ r^{K}, r^{1/K}\}. \end{align*} $$

For a proof of this lemma, see [3, pp. 312–313].

Lemma 4.6. Let $\phi :\mathbb {D}\to \mathbb {C}$ be a quasiconformal embedding of the disk with $\phi (0)=0$ , and let $0<r<1$ . Then the restriction $\phi |_{D(0,r)}$ admits a homeomorphic K-quasiconformal extension to the entire plane, where $K = ({1+r})/({1-r}) K_\phi $ .

This lemma and its proof can be found in [3, pp. 310]. We shall need also the following rather non-trivial result due to Slodkowski (a proof of which can be found in [3, §12.3]). Recall that a holomorphic motion of a set $E\subseteq \widehat {\mathbb {C}}$ is a map $F: \Delta \times E\to \widehat {\mathbb {C}}$ , where $\Delta \subset \mathbb {C}$ is a disk, such that: (i) for each $z\in E$ , the map $t\mapsto F(t,z)$ is holomorphic in $\Delta $ ; (ii) for each $t\in \Delta $ , the map $\varphi _t:E\to \widehat {\mathbb {C}}$ given by $\varphi _t(z)=F(t,z)$ is injective; (iii) for a certain $t_0\in \Delta $ , we have $\varphi _{t_0}(z)=z$ for all $z\in E$ . The point $t_0$ is called the base point of the motion.

Theorem 4.7. (Slodkowski’s theorem)

Let $F: \Delta \times E\to \widehat {\mathbb {C}}$ be a holomorphic motion of a set $E\subseteq \widehat {\mathbb {C}}$ with base point $t_0\in \Delta $ . Then there exists a continuous map $\widehat {F}: \Delta \times \widehat {\mathbb {C}}\to \widehat {\mathbb {C}}$ with the following properties.

1. (i) The map $\widehat {F}$ is a holomorphic motion of $\widehat {\mathbb {C}}$ which extends F (in the sense that $\widehat {F}(t,z)=F(t,z)$ for all $z\in E$ and all $t\in \Delta $ ).

2. (ii) For each $t\in \Delta $ , the map $\psi _t(z)=\widehat {F}(t,z)$ is a global $K_t$ -quasiconformal homeomorphism with $K_t\leq \exp \{d_{\Delta }(t,t_0)\}$ (where $d_{\Delta }$ denotes the hyperbolic metric of $\Delta $ ).

The following lemma contains a well-known result stating that every quasiconformal homeomorphism can be embedded in a holomorphic motion (see [3, Ch. 12]). It will be used in combination with Slodkowski’s theorem.

Lemma 4.8. Let $\psi {\kern-1pt}:{\kern-1pt} \mathbb {C}{\kern-1pt}\to{\kern-1pt} \mathbb {C}$ be a quasiconformal homeomorphism with $k{\kern-1pt}={\kern-1pt}\|\mu _\psi \|_\infty {\kern-1pt}\neq{\kern-1pt} 0$ , and let $z_0\in \mathbb {C}$ be such that $\psi (z_0)=z_0$ .

1. (i) There exists a holomorphic motion $\psi _t:\mathbb {C}\to \mathbb {C}$ , $t\in \mathbb {D}$ such that $\psi _k=\psi $ and $\psi _t(z_0)=z_0$ for all t.

2. (ii) If $0<r_0<1$ and $M>1$ are such that $\psi (D(z_0,r_0))\subseteq D(z_0,Mr_0)$ , then for all $0\leq r<1$ and all t with $|t|<\tfrac 12$ , we have $\psi _t(D(z_0,r))\subseteq D(z_0,R)$ , where

   (4.10) $$ \begin{align} R = \frac{2Me^{6\pi}r^{1/3}}{kr_0^2}. \end{align} $$

Proof. We may assume that $z_0=0$ (otherwise we simply conjugate $\psi $ by the translation $z{\kern-1pt}\mapsto{\kern-1pt} z{\kern-1pt}-{\kern-1pt}z_0$ and work with the resulting map, which fixes $0$ ). For each $t{\kern-1pt}\in{\kern-1pt} \mathbb {D}$ , let $\varphi _t{\kern-1pt}:{\kern-1pt} \mathbb {C}{\kern-1pt}\to{\kern-1pt} \mathbb {C}$ be the unique solution to the Beltrami equation

normalized so that $\varphi _t$ fixes $0,1$ and $\infty $ . Define $\psi _t: \mathbb {C}\to \mathbb {C}$ by the formula

(4.11) $$ \begin{align} \psi_t(\zeta) =\bigg[ 1+ \frac{t}{k}( \psi(1)-1 ) \bigg] \varphi_t(\zeta). \end{align} $$

Note that $\psi _t(0)=0$ for all t. Also, for $t=k$ , we have $\psi _k(\zeta )=\psi (1)\varphi _k(\zeta )$ , so $\psi _k(1)=\psi (1)$ . Since the Beltrami form of $\psi _k$ is the same as the Beltrami form of $\varphi _k$ , which is $\mu _\psi $ , it follows from uniqueness of normalized solutions to the Beltrami equation that $\psi _k=\psi $ . This proves item (i).

Applying Lemma 4.5 to $\phi =\varphi _t$ , $z=0$ and $s=1$ , we see that for all $0<r<1$ ,

$$ \begin{align*} \max_{|\zeta|=r}|\varphi_t(\zeta)| \leq e^{\pi K_t}r^{1/K_t}, \end{align*} $$

where $K_t$ is the maximal dilatation of $\varphi _t$ , which satisfies

$$ \begin{align*} K_t \leq \frac{1+|t|}{1-|t|}. \end{align*} $$

In particular, since $K_t<3$ for all t with $|t|<\tfrac 12$ , we have

(4.12) $$ \begin{align} \varphi_t(D(0,r))\subseteq D(0,e^{3\pi}r^{1/3}). \end{align} $$

Let us now estimate the scaling factor multiplying $\varphi _t(\zeta )$ on the right-hand side of equation (4.11). Applying Lemma 4.5 with $\phi =\psi $ , $z=0$ , $s=r_0$ and $r=r_0^{-1}$ , and taking into account that the maximal dilatation of $\psi $ is less than $3$ , we get

$$ \begin{align*} \max_{|\zeta|=1}{|\psi(\zeta)|}&\leq e^{3\pi}\frac{1}{r_0^3}\min_{|\zeta|=r_0}{|\psi(\zeta)|} \\ &\leq e^{3\pi}\frac{1}{r_0^3}(Mr_0) = \frac{Me^{3\pi}}{r_0^2}. \end{align*} $$

In particular, $|\psi (1)-1|\leq 2Me^{3\pi }r_0^{-2}$ , and therefore

$$ \begin{align*} \bigg|1+\frac{t}{k}(\psi(1)-1)\bigg| \leq \frac{2Me^{3\pi}}{kr_0^2} \end{align*} $$

for all t with $|t|\leq \tfrac 12$ . Combining this fact with equation (4.12), it follows that for all such t, we have

$$ \begin{align*} \max_{|\zeta|\leq r}|\psi_t(\zeta)| \leq \frac{2Me^{6\pi}r^{1/3}}{kr_0^2}. \end{align*} $$

Therefore, $\psi _t(D(0,r)) \subseteq D(0,R)$ for all t with $|t|\leq \tfrac 12$ and all $0<r<1$ , where R is given by equation (4.10). This proves item (ii).

4.5 Quasi-isometry estimates for almost conformal maps

Our goal in this subsection is to make more precise a somewhat vague but intuitive assertion, namely that if a self-map of a hyperbolic domain (or Riemann surface) is almost conformal, then it is an almost isometry of the hyperbolic metric. For the sake of the dynamical applications we have in mind, what is needed is an infinitesimal version of this statement.

The desired infinitesimal quasi-isometry property will be presented in two versions. In the first version, we deal with the case when the quasiconformal map has small dilatation everywhere, and the quasi-isometry bounds we get are in terms of this global small dilatation. In the second version, we deal with the situation when the map is K-quasiconformal (with K not necessarily small) but the quasi-isometry bounds we get are local, near any point $z\in \mathbb {D}$ where the dilatation is bounded by some fixed power of the distance between z and $\partial \mathbb {D}$ . This last version is precisely what we need when studying the metric distortion properties of maps which are asymptotically holomorphic. Both versions are first established for quasiconformal diffeomorphisms of the unit disk, but at the end of this subsection, we show how to transfer these results to the kind of simply connected regions that matter to us.

First, let us introduce some notation. We denote by $\rho _{\mathbb {D}}(z)=2(1-|z|^2)^{-1}$ the Poincaré density of the unit disk, as before. We also denote by $\Delta _z\subset \mathbb {D}$ the closed Euclidean disk $\{\zeta :|\zeta -z|\leq \tfrac 12(1-|z|)\}$ . Given a $C^2$ map $\phi :{\mathbb {D}}\to {\mathbb {D}}$ , we denote by $m_\phi (z)$ the $C^2$ norm of $\phi |_{\Delta _z}$ . We write $J_\phi (z)=\det {D\phi (z)}$ for the Euclidean Jacobian of $\phi $ at z, and

$$ \begin{align*} J_{\phi}^{h}(z) = J_{\phi}(z)\bigg(\frac{\rho_{\mathbb{D}}(\phi(z))}{\rho_{\mathbb{D}}(z)}\bigg)^2 \end{align*} $$

for the hyperbolic Jacobian of $\phi $ at z.

Proposition 4.9. For each $0<\theta < 1$ , there exists a universal continuous function $A_{\theta }:(1,\infty )\times \mathbb {R}^+\to \mathbb {R}^+$ for which the following holds. Let $0<\epsilon <1$ and $\alpha>1$ be given, and suppose $\phi : {\mathbb {D}}\to {\mathbb {D}}$ is a $C^2$ quasiconformal diffeomorphism with $K_\phi \leq 1+\epsilon $ . If $z\in \mathbb {D}$ is such that

(4.13) $$ \begin{align} \alpha^{-1} \leq \frac{\rho_{\mathbb{D}}(\phi(z))}{\rho_{\mathbb{D}}(z)} \leq \alpha, \end{align} $$

then

$$ \begin{align*} J_{\phi}^{h}(z) \leq 1+ A_{\theta}(\alpha,m_\phi(z))\epsilon^{1-\theta}. \end{align*} $$

The proof, given later in this subsection, will use the following three lemmas.

Lemma 4.10. Let $z\in \mathbb {D}$ and let $0< r<1-|z|$ . Then,

(4.14) $$ \begin{align} \mathrm{mod}(\mathbb{D}\setminus D(z,r)) \leq \log{\bigg(\frac{1-|z|^2+|z|r}{r} \bigg)}. \end{align} $$

Proof. We may assume that z is real and non-negative, say $z=x\in [0,1)$ . Let $\varphi \in \mathrm {Aut}(\mathbb {D})$ be given by

$$ \begin{align*} \varphi(\zeta) = \frac{\zeta -x}{1-x\zeta} \end{align*} $$

and define

$$ \begin{align*} \alpha = \varphi(x-r) =\frac{-r}{1-x^2+rx};\quad \beta = \varphi(x+r) =\frac{r}{1-x^2-rx}. \end{align*} $$

Then $D_r'=\varphi (D(x,r))$ is a disk with diameter $(\alpha ,\beta )\subset (-1,1)$ . Since $|\alpha |\leq \beta $ , we see that $D_r'\supseteq D(0, |\alpha |)$ . Therefore,

$$ \begin{align*} \mathrm{mod}(\mathbb{D}\setminus D(x,r))&= \mathrm{mod}(\mathbb{D}\setminus D_r') \\ &\leq \mathrm{mod}(\mathbb{D}\setminus D(0,|\alpha|)) = \log{\frac{1}{|\alpha|}}\\ &=\log{\frac{1-x^2+rx}{r}}, \end{align*} $$

and this finishes the proof.

Remark 4.11. It follows from equation (4.14) that $\mathrm {mod}(\mathbb {D}\setminus D(z,r))\leq \log {({2}/{r})}$ . This estimate will be useful when r is small compared to the distance from z to $\partial \mathbb {D}$ . If $r=\tfrac 12\delta (1-|z|)$ with $0<\delta \leq 1$ , then an easy manipulation of the right-hand side of equation (4.14) yields the estimate $\mathrm {mod}(\mathbb {D}\setminus D(z,r))\leq \log {({5}/{\delta })}$ . This remark will be used in the proof of Lemma 4.13 below.

Lemma 4.12. Let $\alpha>1$ and suppose $z,w\in \mathbb {D}$ are such that

(4.15) $$ \begin{align} \alpha^{-1} \leq \frac{\rho_{\mathbb{D}}(z)}{\rho_{\mathbb{D}}(w)} \leq \alpha. \end{align} $$

Then there exists $\psi \in \mathrm {Aut}(\mathbb {D})$ with $\psi (z)=w$ such that the following inequalities hold for all $\zeta \in \Delta _z$ :

1. (i) ${{1}/{2\alpha } \leq |\psi '(\zeta )|\leq 4\alpha ^2}$ ;

2. (ii) $\displaystyle {|\psi "(\zeta )| \leq 16\alpha ^3}$ .

Proof. Write $a=|z|$ and $b=|w|$ so that $0\leq a,b<1$ . We have $1-a^2=\rho _{\mathbb {D}}(z)^{-1}$ and $1-b^2=\rho _{\mathbb {D}}(w)^{-1}$ , so equation (4.15) tells us that

(4.16) $$ \begin{align} \alpha^{-1} \leq \frac{1-a^2}{1-b^2} \leq \alpha. \end{align} $$

Let $\varphi \in \mathrm {Aut}(\mathbb {D})$ be the hyperbolic translation with axis $(-1,1)\subset \mathbb {D}$ such that $\varphi (a)=b$ . Then,

$$ \begin{align*} \varphi(\zeta) = \frac{\zeta-c}{1-c\zeta} , \end{align*} $$

where $c=(a-b)/(1-ab)\in (-1,1)$ , as a simple calculation shows. Moreover, we have

(4.17) $$ \begin{align} \varphi'(\zeta) =\frac{1-c^2}{(1-c\zeta)^2} \end{align} $$

as well as

(4.18) $$ \begin{align} \varphi"(\zeta) =\frac{2c(1-c^2)}{(1-c\zeta)^3}. \end{align} $$

Since $1-c^2 = (1-a^2)(1-b^2)/(1-ab)^2$ and since $\min \{1-a^2,1-b^2\}\leq 1-ab\leq \max \{1-a^2,1-b^2\}$ , it follows from equation (4.16) that

(4.19) $$ \begin{align} \alpha^{-1} \leq 1-c^2 \leq 1. \end{align} $$

Now, if $\zeta \in \Delta _a$ , then $|\zeta |\leq (1+a)/2$ . Hence,

$$ \begin{align*} |1-c\zeta| \geq 1-|c|\bigg(\frac{1+a}{2}\bigg) = \frac{1-|c|}{2}+\frac{1-|c|a}{2}> \frac{1-|c|a}{2}. \end{align*} $$

Here, there are two cases to consider. If $a{\kern-1pt}\geq{\kern-1pt} b$ , then $c{\kern-1pt}\geq{\kern-1pt} 0$ and $1{\kern-1pt}-{\kern-1pt}|c|a{\kern-1pt}={\kern-1pt}1{\kern-1pt}-{\kern-1pt}ca{\kern-1pt}={\kern-1pt}(1{\kern-1pt}-{\kern-1pt}a^2)/ (1-ab)$ , so from equation (4.16), we deduce that $1-|c|a\geq {\alpha ^{-1}}$ . If however $a< b$ , then $c< 0$ , and in this case, we see that

$$ \begin{align*} 1-|c|a = \frac{1-b^2+(b-a)^2}{1-ab}> \frac{1-b^2}{1-a^2} \geq \alpha^{-1}, \end{align*} $$

where once again we have used equation (4.16). Thus, in either case, we have

(4.20) $$ \begin{align} \frac{1}{2\alpha} \leq|1-c\zeta|<2 \quad\textrm{for all } \zeta\in \Delta_a. \end{align} $$

Using both equations (4.19) and (4.20) in equations (4.17) and(4.18), we easily arrive at inequalities (i) and (ii) with $\varphi $ replacing $\psi $ (and $\Delta _a$ replacing $\Delta _z$ ). Finally, we define $\psi =R_b\circ \varphi \circ R_a$ , where $R_a$ is the rigid rotation around $0$ with $R_a(z)=a$ , and $R_b$ is the rigid rotation around $0$ with $R_b(b)=w$ . Then $\psi (z)=w$ , and since $R_a,R_b$ are Euclidean isometries and $R_a(\Delta _z)=\Delta _a$ , the inequalities (i) and (ii) for $\psi $ follow from the corresponding inequalities for $\varphi $ .

For our final lemma, we introduce further notation. Given a $C^2$ map $\phi :\mathbb {D}\to \mathbb {D}$ , a point $z\in \mathbb {D}$ and $0<\delta \leq 1$ , we denote by $m_\phi (z,\delta )$ the $C^2$ norm of the restriction of $\phi $ to the disk $\{\zeta :|\zeta -z|\leq \delta r_z\}$ , where $r_z=\tfrac 12(1-|z|)$ . In particular, $m_\phi (z,1)=m_\phi (z)$ .

Lemma 4.13. For each $0<\theta <1$ , there exists a universal, continuous monotone function $B_{\theta }:\mathbb {R}^+\to \mathbb {R}^+$ such that the following holds. Given $0<\epsilon <1$ , let $\phi : {\mathbb {D}}\to {\mathbb {D}}$ be a $C^2$ quasiconformal diffeomorphism with $K_\phi \leq 1+\epsilon $ , and suppose that $z\in \mathbb {D}$ is a fixed point of $\phi $ . Then for each $0<\delta \leq 1$ , we have

(4.21) $$ \begin{align} J_\phi^h(z) \leq 1+ B_{\theta}\bigg(\frac{m_\phi(z,\delta)}{\delta}\bigg)\epsilon^{1-\theta}. \end{align} $$

Proof. The basic geometric idea behind the proof is to use macroscopic estimates on the moduli of certain annuli to bound a microscopic quantity, namely the hyperbolic Jacobian at z. Rotating the coordinate axes if necessary, we may also assume that $D\phi (z)=S\cdot T$ , where $S=\rho I=(\begin {smallmatrix} \rho & 0\\0 & \rho \end {smallmatrix})$ , for some $\rho>0$ , and $T=(\begin {smallmatrix} \unicode{x3bb} & b\\ 0 & \unicode{x3bb} ^{-1}\end {smallmatrix})$ , where $\unicode{x3bb} \geq 1$ and $b\in \mathbb {R}$ . Here, we obviously have $\rho ^2=\det {D\phi (z)}=J_\phi (z)=J_\phi ^h(z)$ . We shall prove the lemma only in the case when $b=0$ and $\unicode{x3bb}>1$ . The cases when $b\neq 0$ and/or $\unicode{x3bb} =1$ are similarly handled. Note that the linear map $D\phi (z)$ maps the circle of radius $1$ about the origin onto an ellipse with major axis $\rho \unicode{x3bb} $ and minor axis $\rho /\unicode{x3bb} $ . Since $\phi $ is $(1+\epsilon )$ -qc, we have $\unicode{x3bb} ^2\leq 1+\epsilon $ . In what follows, we assume that $\rho>\unicode{x3bb} +\epsilon $ , as otherwise, $\rho ^2\leq (\unicode{x3bb} +\epsilon )^2\leq 1+6\epsilon $ and there is nothing to prove.

If $\zeta $ is such that $|\zeta -z|\leq \delta r_z$ , we can write, using Taylor’s formula and the fact that $\phi (z)=z$ ,

(4.22) $$ \begin{align} \phi(\zeta) =z+D\phi(z)\cdot (\zeta -z) + R_\phi(\zeta), \end{align} $$

where the remainder $R_\phi (\zeta )$ satisfies $|R_\phi (\zeta )|\leq C|\zeta -z|^2$ , with $C=C_0 m_\phi (z,\delta )>0$ (and $C_0>0$ an absolute constant). Let us choose $0< r\leq \delta r_z$ so small that

(4.23) $$ \begin{align} \frac{\rho}{\unicode{x3bb}}r -Cr^2> \frac{\rho}{\unicode{x3bb}+\epsilon}r. \end{align} $$

For definiteness, we take

(4.24) $$ \begin{align} r=\min\bigg\{\delta r_z,\frac{\rho\epsilon}{C\unicode{x3bb}^2(\unicode{x3bb}+\epsilon)}\bigg\}. \end{align} $$

Then equations (4.22) and (4.23) tell us that $\phi $ maps the disk $D(z,r)$ onto a Jordan domain $V_r$ which contains that disk and also the round annulus . Setting , we have , and so

(4.25)

Consider the images of under the forward iterates of $\phi $ , i.e. , $n\geq 0$ . The annuli are pairwise disjoint, and . By sub-additivity of the modulus, we have

(4.26)

Now, since $\phi $ is $(1+\epsilon )$ -qc, we know that $\phi ^n$ is $(1+\epsilon )^n$ -qc, and therefore,

(4.27)

Putting together equations (4.25), (4.26) and (4.27), we get

(4.28) $$ \begin{align} \log{\bigg( \frac{\rho}{\unicode{x3bb}+\epsilon} \bigg)} \sum_{n=0}^{\infty} \frac{1}{(1+\epsilon)^n} \leq\mu_r. \end{align} $$

Applying Lemma 4.10 and Remark 4.11 to our r, as defined in equation (4.24), we see that

(4.29) $$ \begin{align} \mu_r \leq\begin{cases} \log{\bigg(\dfrac{5}{\delta} \bigg)} & {\textrm{ when }} r=\delta r_z;\\[8pt] \log{\bigg(\dfrac{2C\unicode{x3bb}^2(\unicode{x3bb}+\epsilon)}{\rho\epsilon} \bigg)} & {\textrm{ when }} r=\dfrac{\rho\epsilon}{C\unicode{x3bb}^2(\unicode{x3bb}+\epsilon)}. \end{cases} \end{align} $$

Regardless of which of the two cases occur, we certainly have

(4.30) $$ \begin{align} \mu_r \leq \log{\bigg(\frac{10C\unicode{x3bb}^2(\unicode{x3bb}+\epsilon)}{\delta\rho\epsilon}\bigg)}< \log{\bigg(\frac{60C}{\delta\epsilon}\bigg)}, \end{align} $$

where in the last step, we have used that $\unicode{x3bb} ^2(\unicode{x3bb} +\epsilon )<6$ and $\rho>1$ . Combining equations (4.28) and (4.30), we deduce that

(4.31) $$ \begin{align} \log{\bigg( \frac{\rho}{\unicode{x3bb}+\epsilon} \bigg)}&\leq \frac{\epsilon}{1+\epsilon}\log{\bigg(\frac{60C}{\delta\epsilon}\bigg)} \nonumber \\ & < \epsilon \log{\bigg(\frac{60C}{\delta}\bigg)} + \epsilon \log{\frac{1}{\epsilon}}. \end{align} $$

Since $0<\epsilon <1$ , we have $\epsilon <\epsilon ^{1-\theta }$ and $\epsilon ^{\theta }\log {({1}/{\epsilon })}\leq (\theta e)^{-1}$ . Using these facts in equation (4.31), we get

(4.32) $$ \begin{align} \rho&\leq (\unicode{x3bb}+\epsilon)\exp\bigg\{\bigg(\frac{1}{\theta e}+ \log{\frac{60C}{\delta}}\bigg)\epsilon^{1-\theta}\bigg\} \end{align} $$

(4.33) $$ \begin{align} &\leq 1+ \bigg(2+180e^{{1}/{\theta e}}\frac{C}{\delta}\bigg)\epsilon^{1-\theta}, \end{align} $$

where we have used that $\unicode{x3bb} +\epsilon \leq 1+2\epsilon $ . From this, and the fact that $C=C_0 m_\phi (z,\delta )$ , it readily follows that

$$ \begin{align*} J_\phi^h(z) = \rho^2\leq 1+ 3\bigg(2+180e^{{1}/{\theta e}}C_0\frac{m_\phi(z,\delta)}{\delta}\bigg)^2 \epsilon^{1-\theta}. \end{align*} $$

This proves equation (4.21), provided we take $B_\theta (t)=3(2+ 180e^{{1}/{\theta e}}C_0t)^2$ .

We are now ready for the proof of the first main result of this subsection.

Proof of Proposition 4.9

The idea, of course, is to reduce the required estimate to the case treated in Lemma 4.13. Let $\psi \in \mathrm {Aut}(\mathbb {D})$ be the conformal automorphism given by Lemma 4.12, with $\psi (z)=w=\phi (z)$ . Then the diffeomorphism $F=\psi ^{-1}\circ \phi :\mathbb {D}\to \mathbb {D}$ has a fixed point at z. Since $\psi ^{-1}$ is an isometry of the hyperbolic metric, we certainly have $J_F^h(z)=J_\phi ^h(z)$ . We would like to estimate $J_F^h(z)$ using Lemma 4.13. For this, we need an estimate on the $C^2$ norm of the composition $\psi ^{-1}\circ \phi $ in a suitable disk around z. By Koebe’s one-quarter theorem, $\psi (\Delta _z)$ contains the disk

$$ \begin{align*} D =\big\{ \zeta: |\zeta -w|<\tfrac{1}{4}|\psi'(z)|\cdot r_z \big\}. \end{align*} $$

Since we know from Lemma 4.12(i) that $|\psi '(z)|\geq (2\alpha )^{-1}$ , it follows that $\psi (\Delta _z)\supset D(w,R)$ , where $R=r_z/8\alpha $ . Now let us define

$$ \begin{align*}\delta= \frac{1}{8\alpha m_\phi(z)}\quad \textrm{and}\quad M=\sup_{\zeta\in \Delta_z}|D\phi(\zeta)| \leq m_\phi(z).\end{align*} $$

Then we have $\phi (D(z,\delta r_z))\subset D(w,M\delta r_z)\subseteq D(w,R)\subset \psi (\Delta _z)$ . We can now estimate the $C^2$ norm of F restricted to the disk $D(z,\delta r_z)$ , i.e. we can estimate $m_F(z,\delta )$ , with the help of Lemma 4.12. We do this by means of the following two steps.

1. (i) By the chain rule for first derivatives, we have $DF=D\psi ^{-1}\circ \phi \cdot D\phi $ . Since $\psi ^{-1}$ is holomorphic, for each $\zeta \in D(z,\delta r_z)$ , we have

   (4.34) $$ \begin{align} \|D\psi^{-1}(\phi(\zeta))\|\leq |(\psi^{-1})'(\phi(\zeta))| = |\psi'(\psi^{-1}\circ\phi(\zeta))|^{-1} \leq 2\alpha. \end{align} $$
   Hence, the $C^0$ norm of $DF$ in $D(z,\delta r_z)$ is bounded by $2\alpha m_\phi (z)$ .

2. (ii) By the chain rule for second derivatives, we have

   (4.35)

   
   Again, since $\psi ^{-1}$ is holomorphic, a simple calculation shows that
   $$ \begin{align*} (\psi^{-1})" = -\frac{\psi"\circ \psi^{-1}}{(\psi'\circ \psi^{-1})^3}. \end{align*} $$
   Therefore, for each $\zeta \in D(z,\delta r_z)$ , we have, with the help of Lemma 4.12,
   (4.36) $$ \begin{align} \|D^2\psi^{-1}(\phi(\zeta))\| \leq |(\psi^{-1})"(\phi(z))| \leq 128\alpha^6. \end{align} $$
   Using equations (4.34), (4.36) and the fact that in equation (4.35), we deduce that the $C^0$ norm of $D^2F$ in the disk $D(z,\delta r_z)$ is bounded by $(128\alpha ^6+2\alpha )m_\phi (z)<130\alpha ^6m_\phi (z)$ .

From steps (i) and (ii) above, we deduce that $m_F(z,\delta )\leq 130\alpha ^6m_\phi (z)$ . Therefore, applying Lemma 4.13 for F yields

$$ \begin{align*} J_\phi^h(z) =J_F^h(z) \leq 1+ B_{\theta}\bigg(\frac{m_F(z,\delta)}{\delta}\bigg)\epsilon^{1-\theta} \leq 1+ B_{\theta}\bigg(1040\alpha^7(m_\phi(z))^2\bigg)\epsilon^{1-\theta}. \end{align*} $$

This completes the proof of our theorem, provided we take $A_{\theta }(s,t)=B_{\theta }(1040s^7t^2)$ .

Proposition 4.14. For each $0<\theta < 1$ , there exists a universal continuous function $C_{\theta }:(1,\infty )\times (1,\infty )\times \mathbb {R}^+\times \mathbb {R}^+\to \mathbb {R}^+$ for which the following holds. Let $\alpha>1$ and $\beta>1$ be given, and suppose $\phi : {\mathbb {D}}\to {\mathbb {D}}$ is a $C^2$ quasiconformal diffeomorphism. If $z\in \mathbb {D}$ is such that

(4.37) $$ \begin{align} \alpha^{-1} \leq \frac{\rho_{\mathbb{D}}(\phi(z))}{\rho_{\mathbb{D}}(z)} \leq \alpha, \end{align} $$

and

(4.38) $$ \begin{align} \sup_{\zeta\in \Delta_z} |\mu_\phi(\zeta)|\leq b_0(1-|z|)^{\,\beta}, \end{align} $$

then

(4.39) $$ \begin{align} J_{\phi}^{h}(z) \leq 1+ C_{\theta}(\alpha,\beta,b_0,m_\phi(z))(1-|z|)^{\,\beta(1-\theta)}. \end{align} $$

Proof. We present the proof of the required estimate under the additional assumption that z is a fixed-point of $\phi $ . The general case can be reduced to this one by post-composing $\phi $ with a suitable conformal automorphism of the unit disk, and proceeding just as in the proof of Proposition 4.9, mutatis mutandis. For the sake of clarity of exposition, we divide the proof into a series of steps.

1. (i) First, we introduce some notation. Throughout the proof, we denote by $c_0,c_1,\ldots $ positive constants that are either absolute or depend on the given constants $\alpha , \beta , b_0, M$ , where $M=m_\phi (z)$ . Let us write $\epsilon =b_0(1-|z|)^{\,\beta }= (b_02^{\,\beta })r_z^{\,\beta }$ . Also, let $k_0=\sup _{\zeta \in \Delta _z}|\mu _\phi (\zeta )|\leq \epsilon $ , and set $r_0=\epsilon r_z$ . We may assume without loss of generality that $\epsilon $ is small, say $\epsilon <1/{32}$ .

2. (ii) The restricted map $\phi |_{\Delta _z}: \Delta _z\to \mathbb {D}$ is a $({1+k_0})/({1-k_0})$ -quasiconformal embedding. By Lemma 4.6, the further restriction $\phi |_{D(z,r_0)}$ can be extended to a global quasiconformal homeomorphism $\psi :\mathbb {C}\to \mathbb {C}$ with $k=\|\mu _\psi \|_{\infty }$ satisfying

   $$ \begin{align*} \frac{1+k}{1-k}\leq \frac{1+\epsilon}{1-\epsilon}\cdot \frac{1+k_0}{1-k_0} \leq \bigg(\frac{1+\epsilon}{1-\epsilon}\bigg)^2. \end{align*} $$

3. (iii) In particular, $k\leq 16\epsilon <\tfrac 12$ (by our assumption on $\epsilon $ in step (i)). We may assume that $k\neq 0$ (if this is not the case, it is easy to perturb $\psi $ slightly in a neighbourhood of infinity). By Lemma 4.8(i), there exists a global holomorphic motion $\psi _t : \mathbb {C}\to \mathbb {C}$ with $\psi _k=\psi $ and $\psi _t(z)=z$ for all $t\in \mathbb {D}$ . Now choose $r_1>0$ so small that

   $$ \begin{align*} R = \frac{2Me^{6\pi}}{k_0r_0^2}\cdot r_1^{1/3}< r_z. \end{align*} $$
   For definiteness, take $r_1=c_1k^3r_z^{6\,\beta +9}$ , where $c_1=b_0^6/(M^3e^{18\pi })$ . Then, by Lemma 4.8(ii), we have $\psi _t(D(z,r_1))\subset D(z,R)$ for all t with $|t|<\tfrac 12$ (note that this includes the time $t=k$ ).

4. (iv) We may now define, for each $t{\kern-1pt}\in{\kern-1pt} D(0,\tfrac 12)$ , the map $\widetilde {\psi _t}{\kern-1pt}:{\kern-1pt} D(z,r_1)\cup (\mathbb {C}{\kern-1pt}\setminus{\kern-1pt} \mathbb {D})\to \mathbb {C}$ by

   $$ \begin{align*} \widetilde{\psi_t}(\zeta) = \begin{cases} {\psi_t(\zeta)} &\textrm{for } \zeta\in D(z,r_1), \\ {\zeta} &\textrm{for } \zeta\in\mathbb{C}\setminus \mathbb{D}. \end{cases} \end{align*} $$
   Since $D(z,R)\subset \mathbb {D}$ , we have from step (iii) that . Hence, $\widetilde {\psi _t}$ , $|t|<\tfrac 12$ , is a holomorphic family of injections, that is, a holomorphic motion of the set $D(z,r_1)\cup (\mathbb {C}\setminus \mathbb {D})$ .

5. (v) Now apply Slodkowski’s Theorem 4.7 to get a global extension $\widehat {\psi }_t: \mathbb {C}\to \mathbb {C}$ of the motion $\widetilde {\psi _t}$ , with time parameter t in $D(0,\tfrac 12)$ . In particular, the map $\widehat {\psi }=\widehat {\psi }_k$ is K-quasiconformal with $K=({1+2k})/({1-2k})$ , and it maps the unit disk onto itself. Moreover, we have

   $$ \begin{align*} \widehat{\psi}|_{D(z,r_1)}=\psi|_{D(z,r_1)}=\phi|_{D(z,r_1)}. \end{align*} $$
   Thus, $\widehat {\psi }$ is the desired modification of $\phi $ away from z.

6. (vi) We are now in a position to use the same annulus trick we employed in the proof of Lemma 4.13. Let $\rho>0$ , $\unicode{x3bb}>1$ and the absolute constant $C_0>0$ be as in the proof of that Lemma. In particular, $\rho ^2=J_\phi ^h(z)=J_{\widehat {\psi }}^h(z)$ , and thus our goal is to bound $\rho $ from above. We have $\unicode{x3bb} \leq 1+\epsilon $ , and we may assume that $\rho>\unicode{x3bb} +\epsilon $ , otherwise there is nothing to prove. Now let $r_2>0$ be given by

   $$ \begin{align*} r_2 = \frac{\epsilon}{3C_0M}< \frac{\rho\epsilon}{C_0M\unicode{x3bb}^2(\unicode{x3bb}+\epsilon)}. \end{align*} $$
   Then for all $r\leq r_2$ , the inequality in equation (4.23) holds. Let us choose $r=\min \{r_1,r_2\}$ . With this choice of r, using the Taylor expansion in equation (4.22) as in the proof of Lemma 4.13, we see that is a conformal annulus, with
   (4.40)

   

7. (vii) Now define for all $n\geq 0$ , and note that

   (4.41)

   
   Since , we deduce from equations (4.40) and (4.41) that
   (4.42) $$ \begin{align} \log{\bigg(\frac{\rho}{\unicode{x3bb}+\epsilon}\bigg)} \sum_{n=0}^{\infty} \bigg(\frac{1-2k}{1+2k}\bigg)^n \leq \log{\frac{2}{r}}, \end{align} $$
   where we have used the estimate on $\mod {(\mathbb {D}\setminus D(z,r))}$ given by Lemma 4.10 (and Remark 4.11). From equation (4.42), it follows that
   (4.43) $$ \begin{align} \log{\bigg(\frac{\rho}{\unicode{x3bb}+\epsilon}\bigg)} \leq \frac{4k}{1+2k}\log{\frac{2}{r}}<4k\log{\frac{2}{r}}. \end{align} $$

8. (vii) However, from our choices of $r_1$ and $r_2$ , we see that $r=\min \{r_1,r_2\}=c_2k^3r_z^{6\,\beta +9}$ for some constant $c_2>0$ . Hence,

   $$ \begin{align*} \log{\frac{2}{r}} \leq \log{\frac{2}{c_2}} +3\log{\frac{1}{k}}+(6\,\beta+9)\log{\frac{1}{r_z}}. \end{align*} $$
   Putting this back into equation (4.43) and using that $k\leq \mathrm {(const.)}r_z^{\,\beta }$ , we deduce that for each $0<\theta <1$ ,
   $$ \begin{align*} \log{\bigg(\frac{\rho}{\unicode{x3bb}+\epsilon}\bigg)} &\leq c_3k+c_4k\log{\frac{1}{k}} + c_5k\log{\frac{1}{r_z}} \\ &\leq c_6r_z^{\,\beta(1-\theta)} + c_7r_z^{\,\beta}\log{\frac{1}{r_z}} \\ &\leq c_8 r_z^{\,\beta(1-\theta)}. \end{align*} $$
   Here, the constants $c_6, c_7, c_8$ depend on $M,\beta , b_0$ and also on $\theta $ . From this, it follows that
   $$ \begin{align*} \rho \leq 1+ c_9r_z^{\,\beta(1-\theta)}, \end{align*} $$
   and therefore,
   $$ \begin{align*} J_\phi^h(z) = \rho^2 \leq 1 + c_{10}r_z^{\,\beta(1-\theta)}, \end{align*} $$
   where the constant $c_{10}$ depends on $M,\beta , b_0$ and $\theta $ .

Hence, we have established equation (4.39), with $c_{10}$ playing the role of $C_\theta $ , in the case when z is a fixed-point of $\phi $ . As we already remarked, the general case follows from this one by post-composition of $\phi $ with a suitable automorphism of the disk, using the same procedure given in the proof of Proposition 4.9. It is here, and only here, that equation (4.37) is used. Hence, the final constant $C_\theta $ indeed depends on $M, \alpha , \beta , b_0$ and of course also on $\theta $ . This finishes the proof.

As we informally said in the beginning of this subsection, our goal is to develop bounds on the infinitesimal distortion, by a self-map (diffeomorphism) of a hyperbolic Riemann surface, of the underlying hyperbolic metric in terms of the local quasiconformal distortion of the map. So far, we have only shown how to bound in such terms the hyperbolic Jacobian of these maps. Can we use such estimates on the Jacobian to bound the infinitesimal distortion of the hyperbolic metric? The answer is yes, and the reason lies in the fact that there is a simple relationship between the two concepts. More precisely, let $\phi :Y\to Y$ be a quasiconformal diffeomorphism. Then for each $z\in Y$ and each non-zero tangent vector $v\in T_zY$ , we have

(4.44) $$ \begin{align} \frac{1}{K_\phi(z)}\,J_\phi^h(z)\leq \bigg( \frac{|D\phi(z)v|_Y}{|v|_Y}\bigg)^2\leq K_\phi(z)\,J_\phi^h(z). \end{align} $$

This fact is classical (see for instance [51, pp. 17]).

Theorem 4.15. Let $U,V\subset \mathbb {C}$ be Jordan domains, symmetric about the real axis, with , and let $Y=V\setminus \mathbb {R}$ . Let $\phi :V\to V$ be a $C^r$ diffeomorphism which is symmetric about the real axis, and write

$$ \begin{align*} M=\max\{ \mathrm{diam}(V),(\mathrm{dist}(\partial V,\partial U))^{-1}\,,\,\|\phi\|_{C^2}\,,\,\|\phi^{-1}\|_{C^2}\}>0. \end{align*} $$

Then the following facts hold true for each $0<\theta <1$ .

1. (i) If $\phi $ is $(1+\delta )$ -quasiconformal ( $\delta>0$ ), then for each $z\in U\cap Y$ with $\phi (z)\in U\cap Y$ and all non-zero tangent vectors $v\in T_zY$ , we have

   (4.45) $$ \begin{align} (1+C_\theta\delta^{1-\theta})^{-1} \leq \frac{|D\phi(z)v|_Y}{|v|_Y} \leq 1+C_\theta\delta^{1-\theta}, \end{align} $$
   where $C_\theta>0$ depends only on $\theta $ and M.

2. (ii) If $\phi $ is asymptotically holomorphic of order r, so that $|\mu _\phi (z)|\leq b_0|\mathrm {Im}\,z|^{r-1}$ for all $z\in Y$ , then for each $z\in U\cap Y$ with $\phi (z)\in U\cap Y$ and all non-zero tangent vectors $v\in T_zY$ , we have

   (4.46) $$ \begin{align} (1+C_\theta|\mathrm{Im}\,z|^{(r-1)(1-\theta)})^{-1} \leq \frac{|D\phi(z)v|_Y}{|v|_Y} \leq 1+C_\theta|\mathrm{Im}\,z|^{(r-1)(1-\theta)}, \end{align} $$
   where $C_\theta>0$ depends only on $\theta $ , M and $b_0$ .

Proof. The hard work has already been done in Propositions 4.9 and 4.14, and all we have to do is to show, with the help of equation (4.44), how to reduce the present theorem to the situation in those auxiliary results. There is no loss of generality in assuming that $\phi $ preserves $Y^+=Y\cap \mathbb {C}^+$ (and therefore also $Y^{-}=Y\cap \mathbb {C}^{-}$ ). Also, it suffices to establish the upper estimates in equations (4.45) and (4.46), since the lower estimates follow by replacing $\phi $ with its inverse. Moreover, by symmetry, we only need to establish these upper estimates for points $z\in U\cap Y^+$ .

Let $(a,b)=V\cap \mathbb {R}$ , and let $\varphi :V\to \widehat {\mathbb {C}}$ be a holomorphic univalent map with $\varphi (Y^+)=\mathbb {D}$ , , normalized so that $\varphi (a)=-1$ , $\varphi (b)=+1$ . Let $W^*=\bigcup _{\zeta \in \varphi (U^+)} \Delta _\zeta \subset \mathbb {D}$ , and consider $W=\varphi ^{-1}(W^*)\subset Y^+$ . Note that $W\supset U^+$ . By Lemma 4.4(ii), the $C^2$ norms of the restrictions $\varphi |_{W}$ and $\varphi ^{-1}|_{\varphi (W^*)}$ are both bounded by a constant that depends only on $\mathrm {dist}(\partial V, \partial W)$ , and it is not difficult (albeit a bit laborious) to see that this last distance is bounded by a constant that depends only on M. These bounds also imply that there exists a constant $K_1>1$ depending only on M such that

(4.47) $$ \begin{align} \frac{1}{K_1}(1-|\varphi(z)|) \leq |\mathrm{Im}\,z| \leq K_1(1-|\varphi(z)|) \end{align} $$

for all $z\in W$ .

Now consider the $C^2$ diffeomorphism $\psi :\mathbb {D}\to \mathbb {D}$ given by $\psi =\varphi \circ \phi \circ \varphi ^{-1}$ . Note that, by the chain rule and the bounds on $\varphi $ , $\varphi ^{-1}$ stated above, the $C^2$ norm of $\psi |_{W^*}$ is also bounded by a constant that depends only on M.

Given a point $z\in Y^+$ and a vector $v\in T_zY^+\equiv T_zY$ , let $\zeta =\varphi (z)\in \mathbb {D}$ and $w=D\varphi (z)v\in T_\zeta \mathbb {D}$ . Since $\varphi $ yields an isometry between the hyperbolic metric of $Y^+$ (that is, of Y) and the hyperbolic metric of $\mathbb {D}$ , we have $|v|_Y=|w|_{\mathbb {D}}$ . Moreover, by the chain rule, we have

$$ \begin{align*} |D\phi(z)v|_Y = |D\varphi^{-1}(\psi(\zeta))\,D\psi(\zeta)w|_Y = |D\psi(\zeta)w|_{\mathbb{D}}, \end{align*} $$

where, in the last step, we have used that $\varphi ^{-1}$ yields an isometry between the hyperbolic metric of $\mathbb {D}$ and the hyperbolic metric of $Y^+$ (and therefore the derivative $D\varphi ^{-1}(\psi (\zeta ))$ is an infinitesimal isometry between corresponding tangent spaces). This shows that for each $z\in Y^+$ and each non-zero tangent vector $v\in T_zY$ , we have

(4.48) $$ \begin{align} \frac{|D\phi(z)v|_Y}{|v|_Y} = \frac{|D\psi(\zeta)w|_{\mathbb{D}}}{|w|_{\mathbb{D}}}. \end{align} $$

In addition, since $\varphi $ and $\varphi ^{-1}$ are conformal, we have that $\psi $ and $\phi $ have the same dilatation at corresponding points, i.e. $K_\psi (\zeta )=K_\phi (z)$ for all $z\in Y^+$ . Also, since $\varphi $ and $\varphi ^{-1}$ are hyperbolic isometries, the hyperbolic Jacobians of $\psi $ and $\phi $ agree on corresponding points, that is, $J_\psi ^h(\zeta )=J_\phi ^h(z)$ .

Putting these facts together, we see that the assertions (i) and (ii) in the statement (that is, the estimates in equations (4.45) and (4.46)) will be proved for $\phi $ as soon as the corresponding assertions for $\psi $ are proved. However, assertion (i) for $\psi $ follows by putting together Proposition 4.9 and equation (4.44), whereas assertion (ii) for $\psi $ follows by putting together Proposition 4.14 and equation (4.44). To see why this is so, we need to check that, in each case, the hypotheses of the corresponding propositions are satisfied by $\psi $ .

Case (i). If $\phi $ is $(1+\delta )$ -quasiconformal, as in assertion (i), then $\psi $ is $(1+\delta )$ -quasiconformal as well. The hypotheses on $\phi $ imply that there exists a constant $K_2>1$ depending only on M such that

(4.49) $$ \begin{align} \frac{1}{K_2} \leq \frac{|\mathrm{Im}\,z|}{|\mathrm{Im}\,\phi(z)|} \leq K_2 \end{align} $$

for all $z\in W$ . Applying this with $z=\varphi ^{-1}(\zeta )$ for $\zeta \in W^*$ and using equation (4.47), we deduce that there exists $K_3>1$ depending only on M such that

$$ \begin{align*} \frac{1}{K_3} \leq \frac{\rho_{\mathbb{D}}(\zeta)}{\rho_{\mathbb{D}}(\psi(\zeta))} \leq K_2 \end{align*} $$

for all $\zeta \in W^*$ . This shows that the inequality of equation (4.13) in the hypothesis of Proposition 4.9 is satisfied for $\psi $ . Moreover, for each $\zeta \in \varphi (U^+)$ , we have $\Delta _\zeta \subset W^*$ , and so, in the notation introduced before, $m_\psi (\zeta )\leq \|\psi |_{W^*}\|_{C^2}\leq K_4$ , where $K_4>0$ is a constant that depends only on M. Hence, all the hypotheses of Proposition 4.9 are satisfied by $\psi $ . It follows that for each $0<\theta <1$ , there exists a constant $K_\theta $ depending only on $\theta $ and M such that

(4.50) $$ \begin{align} J_{\psi}^h(\zeta) \leq 1+ K_\theta \delta^{1-\theta} \end{align} $$

for all $\zeta \in \varphi (U^+)$ . Combining equation (4.50) with the general upper estimate in equation (4.44) (for $\psi $ ), we see that for each $0<\theta <1$ , there exists a constant $C_\theta>0$ depending only on $\theta $ and M such that

(4.51) $$ \begin{align} \frac{|D\psi(\zeta)w|_{\mathbb{D}}}{|w|_{\mathbb{D}}}\leq 1+ C_\theta \delta^{1-\theta} \end{align} $$

for all $\zeta \in \varphi (U^+)$ and each non-zero tangent vector $w\in T_\zeta \mathbb {D}$ . Putting equation (4.51) together with equation (4.48) for $z=\varphi ^{-1}(\zeta )\in U^+$ and $v=D\varphi ^{-1}(\zeta )w\in T_zY^+$ , we deduce the upper estimate in equation (4.45), as desired.

Case (ii). If $\phi $ is asymptotically holomorphic (near the real axis), then so is $\psi $ (near the boundary of the unit disk). Verifying the hypotheses of Proposition 4.14 for $\psi $ in this case is similar to what was done in case (i), and hence we omit the details.

Remark 4.16. In the application we have in mind, namely Theorem 5.4 below, the diffeomorphism $\phi $ will be the asymptotically holomorphic diffeomorphism appearing in the Stoilow decomposition of a high renormalization of an (infinitely renormalizable) AHPL-map. For such maps, we can always assume that the constant $b_0$ appearing in assertion (ii) is equal to one. The reason for this is embedded in the proof of a slightly improved version of the complex bounds (see Theorem 3.2(iv)).

5 Recurrence and expansion

This section contains a crucial step towards the proof of our main theorem (as stated in the introduction), namely Theorem 5.4 below. We show that every AHPL-map arising as a deep renormalization of an infinitely renormalizable $C^r$ unimodal map with bounded combinatorics expands the hyperbolic metric of its co-domain minus the real axis. From this, we deduce a few basic properties concerning the global dynamics of these AHPL-maps—such as the fact that all of their periodic points are expanding. The expansion property proved here will lead to much stronger results in §6, including, of course, the proof of the main theorem.

5.1 Controlled AHPL-maps

To establish the desired expansion property, we need to assume that our AHPL-maps satisfy certain geometric constraints. We call such maps controlled AHPL-maps. These geometric constraints may seem artificial, but the point is that they are always verified once we renormalize a given AHPL-map a sufficient number of times.

Let us proceed with the formal definition. First, we need some notation. Given $z=x+iy\in \mathbb {C}\setminus \mathbb {R}$ and $\alpha>1$ , let $z_\alpha =x+i\alpha y$ .

Definition 5.1. Let $\alpha , M>1$ and $0<\delta ,\theta <1$ be real constants, and let $n_0\in \mathbb {N}$ . An AHPL-map $f:U\to V$ of class $C^r$ , $r\geq 3$ , is said to be $(\alpha , \delta , \theta ,M, n_0)$ -controlled if the following conditions are satisfied:

1. (i) we have $\mathrm {diam}(V)\leq M$ and $\mathrm {mod}(V\setminus U)\geq M^{-1}$ ;

2. (ii) if $f=\phi \circ g$ is the Stoilow decomposition of f, with $\phi :V\to V$ a $C^r$ -diffeomorphism and $g:U\to V$ holomorphic, then $\|\phi \|_{C^2}, \|\phi ^{-1}\|_{C^2}\leq M$ ;

3. (iii) $\phi $ is $(1+\delta )$ -quasiconformal on V;

4. (iv) the dilatation $\mu _\phi $ satisfies $|\mu _\phi (z)|\leq M|\mathrm {Im}\,z|^{r-1}$ ;

5. (v) for all $z\in U_\alpha =U\cap \{w:\,|\mathrm {Im}\,w|\leq (\alpha M)^{-1}\}$ , we have $D(z_\alpha ,|\mathrm {Im}\,z_\alpha |)\subset Y= V\setminus \mathbb {R}$ ;

6. (vi) for all $z\in U\setminus \mathbb {R}$ , we have $M^{-1}\leq |\mathrm {Im}\,z|/|\mathrm {Im}\,\phi (z)|\leq M$ , as well as $M^{-1}\leq \rho _Y(z)/\rho _Y(\phi (z))\leq M$ ;

7. (vii) we have

   $$ \begin{align*} \Phi(\mathrm{diam}_Y(U\setminus U_\alpha)+2n_0\log{M})< 1-C_\theta\delta^{1-\theta}, \end{align*} $$
   where $\Phi $ is McMullen’s universal function in equation (4.4) and $C_\theta =C_\theta (M)$ is the constant appearing in Theorem 4.15(i).

Remark 5.2. It is possible to prove, with the help of Lemma 4.1 and the Riemann mapping theorem, that $\mathrm {diam}_Y(U\setminus U_\alpha )\leq C+\log {\alpha }$ for some positive constant $C=C(M)$ .

The following result is a straightforward consequence of the complex bounds, as given by Theorem 3.2, together with the $C^2$ bounds, as given by Theorem 3.1 and Remark 3.5.

Theorem 5.3. For each positive integer N, there exists $M=M(N)>1$ such that the following holds. Let $f: U\to V$ be an AHPL-map of class $C^r$ , $r\geq 3$ , whose restriction to the real line is an infinitely renormalizable unimodal map with combinatorics bounded by N. Then for each $\alpha>1$ and $0<\theta <1$ and each $n_0\in \mathbb {N}$ , there exist $0<\delta <1$ and $n_1=n_1(f,\alpha , \theta , n_0)\in \mathbb {N}$ such that for all $n\geq n_1$ , the nth renormalization $R^nf:U_n\to V_n$ is an $(\alpha , \delta , \theta ,M,n_0)$ -controlled AHPL map.

Now, we have the following main theorem.

Theorem 5.4. Given $M>1$ , $r>3$ and $0<\theta <1$ so small that $(r-1)(1-\theta )>2$ , there exists $\alpha _0>1$ such that the following holds for all $\alpha>\alpha _0$ . Let $f: U\to V$ be an AHPL-map of class $C^r$ and assume that f is $(\alpha , \delta , \theta ,M, n_0)$ -controlled for some $0<\delta <1$ and some $n_0\in \mathbb {N}$ . Suppose also that r, $\alpha $ , $\theta $ and $n_0$ are such that

(5.1) $$ \begin{align} r> 1 + \frac{4n_0\alpha}{(n_0-1)(1-\theta)(2\alpha-1)}. \end{align} $$

Then the following assertions hold true.

1. (a) There exists a constant $0<\eta <1$ such that $|Df^n(z)v|_Y\geq \eta |v|_Y$ for all $z\in Y\cap U$ , such that $f^i(z)\in Y$ for $0\leq i\leq n$ and all $v\in T_zY$ .

2. (b) If z is a point in the filled-in Julia set of f and its -limit set is not contained in the real axis, we have $|Df^n(z)v|_Y/|v|_Y\to \infty $ as $n\to \infty $ for each non-zero tangent vector $v\in T_zY$ .

3. (c) Every periodic orbit of f is expanding.

4. (d) The expanding periodic points are dense in the set of all recurrent points.

Proof. First, we give an informal description of the argument. For a suitable constant $0<\unicode{x3bb} <1$ , we partition the domain of $f=\phi \circ g$ into a sequence of scales, the nth scale being the set of points in the domain (off the real axis) whose distance to the real axis is of the order $\unicode{x3bb} ^n$ . The rough idea then is that at each level, the worst expansion of the hyperbolic metric of Y by g beats the best contraction of that metric by $\phi $ . In this, we are aided by Theorem 4.15 and Lemma 4.3. We warn the reader that in what follows, whenever invoking Theorem 4.15, we denote by $C_\theta $ the largest of the two constants with that name appearing in assertions (i) and (ii) of said theorem.

Let us now present the formal proof. Let us assume we are given a large number $\alpha>1$ . How large $\alpha $ must be will be determined in the course of the argument.

To start with, note that by equation (4.2) in Lemma 4.2, we have for all $z\in U_\alpha $ ,

(5.2) $$ \begin{align} \frac{1}{|\mathrm{Im}\,z|} \leq \rho_Y(z) \leq \frac{1}{|\mathrm{Im}\,z|}\bigg(1-\frac{1}{2\alpha}\bigg)^{-1}. \end{align} $$

Let us fix for the time being a real number $0<\unicode{x3bb} <1$ , which we will use to define the scales we mentioned above. For definiteness, we take $\unicode{x3bb} =M^{-1}$ . For each $n\geq 1$ , we define

$$ \begin{align*} W_n = \bigg\{z\in U_\alpha : \frac{\unicode{x3bb}^n}{\alpha M}\leq |\mathrm{Im}\,z| < \frac{\unicode{x3bb}^{n-1}}{\alpha M}\bigg\}. \end{align*} $$

Also, we set $W_0=U\setminus U_\alpha \subset Y$ . Then we have, of course, $U\setminus \mathbb {R}= \bigcup _{n=0}^\infty W_n$ .

Claim. There exists a sequence of numbers $\xi _n>1$ , $n\geq 0$ , with $\xi _n\to 1$ as $n\to \infty $ , having the following property: for each $z\in W_n$ and each tangent vector $v\in T_{z}Y$ , we have

(5.3) $$ \begin{align} |D(g\circ \phi)(z)v|_Y \geq \xi_n|v|_Y. \end{align} $$

Proof of Claim

To prove this claim, we analyse separately the expansion of the conformal map g and the (possible) contraction of the quasiconformal diffeomorphism $\phi $ . We proceed through the following steps.

1. (i) Let $X\subset Y$ be the open set containing $\phi (z)$ such that g maps X univalently onto Y. Writing $w=D\phi (z)v\in T_{\phi (z)}Y$ and applying Lemma 4.3 together with the estimatein equation (4.5), we deduce that

   (5.4) $$ \begin{align} |Dg(\phi(z))\,w|_Y \geq \big(1 +\tfrac{1}{3}e^{-2s_{X,Y}(\phi(z))}\big)|w|_Y. \end{align} $$
   Now we need to estimate $s_{X,Y}(\phi (z))$ .

2. (ii) Let us write $p=\phi (z)=x+iy$ and let $q=x+i(\alpha M)^{-1}{y}/{|y|}\in U\setminus U_\alpha $ , which lies in the same vertical as p. There are two cases to consider.

   1. (1) We have $p\in X$ but $q\notin X$ . In this case, we have $d_Y(p,Y\setminus X)\leq d_Y(p,q)$ . Using equation (5.2), we get

      $$ \begin{align*} s_{X,Y}(\phi(z))\leq d_Y(p,q)\leq \bigg(1-\frac{1}{2\alpha}\bigg)^{-1}\log{\frac{(\alpha M)^{-1}}{|\mathrm{Im}\,\phi(z)|}}.\\[-12pt] \end{align*} $$
      However, by property (vi) of Definition 5.1, we have $|\mathrm {Im}\,\phi (z)|\geq M^{-1}\unicode{x3bb} ^{n}(\alpha M)^{-1}$ . Hence,
      (5.5) $$ \begin{align} s_{X,Y}(\phi(z))\leq \bigg(1-\frac{1}{2\alpha}\bigg)^{-1}\bigg[ n\log{\frac{1}{\unicode{x3bb}}} +\log{M}\bigg]. \end{align} $$

   2. (2) We have $p\in X$ and $q\in X$ . In this case, we have

      $$ \begin{align*} d_Y(p,Y\setminus X)& \leq d_Y(p,q) + d_Y(q,Y\setminus X) \\ & \leq d_Y(p,q) + \mathrm{diam}_Y(U\setminus U_\alpha).\\[-14pt] \end{align*} $$
      Therefore,
      (5.6) $$ \begin{align} s_{X,Y}(\phi(z))\leq C_\alpha + \bigg(1-\frac{1}{2\alpha}\bigg)^{-1}\bigg[ n\log{\frac{1}{\unicode{x3bb}}} +\log{M}\bigg],\\[-15pt] \nonumber \end{align} $$
      where $C_\alpha =\mathrm {diam}_Y(U\setminus U_\alpha )$ .

   Whichever case occurs, we see that equation (5.6) always holds. Combining these facts with equation (5.4), we deduce that

   (5.7) $$ \begin{align} |Dg(\phi(z))w|_Y \geq (1 + K_1\unicode{x3bb}^{2n(1-{1}/{2\alpha})^{-1}})|w|_Y,\\[-15pt] \nonumber \end{align} $$
   where $K_1=K_1(\alpha , M)$ is the constant given by
   (5.8) $$ \begin{align} K_1 = \frac{1}{3}e^{-2C_\alpha}\,\exp\bigg\{-2\bigg(1-\frac{1}{2\alpha}\bigg)^{-1}\log{M}\bigg\} < 1.\\[-15pt] \nonumber \end{align} $$
   This gives us a lower bound on the amount of expansion of the hyperbolic metric of Y by the conformal map g for points at level n.

3. (iii) Let us now bound the amount of contraction of the hyperbolic metric by the quasiconformal diffeomorphism $\phi $ at $z\in W_n$ . First, we assume that $n\geq n_0$ . Applying Theorem 4.15(ii), we have for all $v\in T_zY$ the estimate

   (5.9) $$ \begin{align} |D\phi(z)v|_Y \geq (1-C_\theta |\mathrm{Im}\, z|^{(r-1)(1-\theta)})|v|_Y. \end{align} $$
   However, since $z\in W_n$ , we know that $|\mathrm {Im}\, z|\leq (\alpha M)^{-1}\unicode{x3bb} ^{n-1}$ . Carrying this information back into equation (5.9), we deduce that
   (5.10) $$ \begin{align} |D\phi(z)v|_Y \geq (1-K_2\unicode{x3bb}^{(n-1)(r-1)(1-\theta)}) |v|_Y, \end{align} $$
   where $K_2=K_2(\alpha , \theta , r, M)$ is the constant given by
   (5.11) $$ \begin{align} K_2= C_\theta (\alpha M)^{(1-r)(1-\theta)}. \end{align} $$

4. (iv) Note that both constants $K_1$ and $K_2$ depend on $\alpha $ . We claim that the ratio $K_2/K_1$ goes to zero as $\alpha \to \infty $ . From equations (5.8) and (5.11), we see that

   $$ \begin{align*} \frac{K_2}{K_1}< C_1 e^{2C_\alpha} \alpha^{(1-r)(1-\theta)} ,\\[-12pt] \end{align*} $$
   where $C_1 = 3C_\theta M^{(1-r)(1-\theta )}M^4$ is independent of $\alpha $ . By Remark 5.2, we have $C_\alpha < C_2 + \log {\alpha }$ for some constant $C_2$ depending only on M. Hence,
   (5.12) $$ \begin{align} \frac{K_2}{K_1}< C_3 \alpha^{2-(r-1)(1-\theta)}, \end{align} $$
   where $C_3=C_1e^{2C_2}$ . Since by hypothesis $(r-1)(1-\theta )>2$ , it follows that the right-hand side of equation (5.12) indeed goes to zero as $\alpha \to \infty $ . Hence, we assume from now on that $\alpha $ is so large that $2K_2<K_1$ .

5. (v) Thus, if for each $n\geq n_0$ we let $\xi _n$ be given by

   (5.13) $$ \begin{align} \xi_n = (1 + K_1\unicode{x3bb}^{2n(1-{1}/{2\alpha})^{-1}}) (1-K_2\unicode{x3bb}^{(n-1)(r-1)(1-\theta)}), \end{align} $$
   then we have $|D(g\circ \phi )(z)v|_Y\geq \xi _n |v|_Y$ for all $z\in W_n$ and each $v\in T_zY$ . Note that $\xi _n\to 1$ as $n\to \infty $ , because $\unicode{x3bb} <1$ . We still need to check that $\xi _n>1$ for all $n\geq n_0$ . This will be true provided
   (5.14) $$ \begin{align} K_1\unicode{x3bb}^{2n(1-{1}/{2\alpha})^{-1}}> 2K_2\unicode{x3bb}^{(n-1)(r-1)(1-\theta)} \end{align} $$
   for all $n\geq n_0$ . Note that both sides of equation (5.14) are indeed smaller than $1$ , because from equation (5.8) and step (iv), we have $2K_2<K_1<1$ and $\unicode{x3bb} <1$ . Extracting logarithms from both sides of equation (5.14), we get
   $$ \begin{align*} 2n\bigg(1-\frac{1}{2\alpha}\bigg)^{-1}\log{\unicode{x3bb}}> (n-1)(r-1)(1-\theta)\log{\unicode{x3bb}} + \log{(2K_1^{-1}K_2)}.\\[-12pt] \end{align*} $$
   Dividing both sides of the above inequality by $(n-1)(1-\theta )\log {\unicode{x3bb} } <0$ , we arrive at
   (5.15) $$ \begin{align} r> 1 + \frac{2n}{(n-1)(1-\theta)(1-{1}/{2\alpha})} + \frac{\log{(2K_1^{-1}K_2)}}{(n-1)(1-\theta)\log{({1}/{\unicode{x3bb}})}}. \end{align} $$
   However, since $2K_1^{-1}K_2<1$ (by our choice of $\alpha $ at the end of step (iv)), the third term on the right-hand side of equation (5.15) is negative and therefore can be safely ignored. Moreover, since $n\geq n_0$ , we have $2n/(n-1)\leq 2n_0/(n_0-1)$ . Therefore, the inequality in equation (5.14) will hold for all $n\geq n_0$ provided
   $$ \begin{align*} r> 1+ \frac{2n_0}{(n_0-1)(1-\theta)(1-{1}/{2\alpha})}. \end{align*} $$
   However, this is nothing but equation (5.1) in disguise! Hence, we have established that the $\xi _n$ terms given by equation (5.13) satisfy $\xi _n>1$ for all $n\geq n_0$ .

6. (vi) To establish the claim, it remains to analyse what happens when $z\in W_0\cup W_1\cup \cdots \cup W_{n_0-1}$ . On the one hand, since $\phi $ is $(1+\delta )$ -quasiconformal throughout, applying Theorem 4.15 for such z and any $v\in T_zY$ yields the lower bound

   (5.16) $$ \begin{align} |D\phi(z)v|_Y \geq (1-C_\theta \delta^{1-\theta})|v|_Y. \end{align} $$
   On the other hand, using the estimate in equation (5.6) above with $n=n_0$ , we deduce that
   $$ \begin{align*} s_{X,Y}(\phi(z))\leq C_\alpha + 2(n_0-1)\log{\frac{1}{\unicode{x3bb}}} +2\log{M} = C_\alpha + 2n_0\log{M}. \end{align*} $$
   Therefore, by McMullen’s Lemma 4.3, we have for all $w\in T_{\phi (z)}Y$ ,
   (5.17) $$ \begin{align} |Dg(\phi(z))w|_Y &\geq \Phi(s_{X,Y}(\phi(z))^{-1}|w|_Y \end{align} $$
   (5.18) $$ \begin{align} \kern92pt\geq \Phi(C_\alpha + 2n_0\log{M})^{-1}|w|_Y. \end{align} $$
   Combining equations (5.16) and (5.17) (with $w=D\phi (z)v$ ), we deduce that
   $$ \begin{align*} |D(g\circ \phi)(z)v|_Y \geq \Phi(C_\alpha + 2n_0\log{M})^{-1}(1-C_\theta \delta^{1-\theta})\,|v|_Y. \end{align*} $$
   Hence, we can take
   $$ \begin{align*} \xi_0=\xi_1=\cdots =\xi_{n_0-1}=\Phi(C_\alpha + 2n_0\log{M})^{-1}(1-C_\theta \delta^{1-\theta})>1. \end{align*} $$
   This establishes equation (5.3) for all $z\in W_n$ , for all $n\geq 0$ , and completes the proof of our claim.

With the claim at hand, we proceed to the proof of the assertions in the statement of our theorem. Let $z\in \mathcal {K}_f$ be a point whose iterates up to time $n>1$ stay off the real axis—in other words, $f^i(z)\in Y$ for all $0\leq i\leq n$ . Note that, since $f=\phi \circ g$ , we have $f^n=\phi \circ (g\circ \phi )^{n-1}\circ g$ . Write $z_1=g(z)$ and define inductively $z_{j+1}=g\circ \phi (z_j)$ for $j=1,\ldots , n-1$ . Then for each non-zero tangent vector $v\in T_zY$ , we have, by the chain rule,

(5.19) $$ \begin{align} Df^n(z)v = D\phi(z_n)\bigg[\prod_{j=1}^{n-1} Dg(\phi(z_j))D\phi(z_j)\bigg] Dg(z)v. \end{align} $$

Now, since the holomorphic map g expands the hyperbolic metric of Y, we have that $|Dg(z)v|_Y>|v|_Y$ . Moreover, the amount of possible contraction of the hyperbolic metric by the $(1+\delta )$ -quasiconformal diffeomorphism $\phi $ is bounded from below. Indeed, we have $|D\phi (\zeta )w|_Y\geq (1-C_\theta \delta ^{1-\theta })|w|_Y$ for all $\zeta \in Y$ and all $w\in T_{\zeta }Y$ . Moreover, writing $v_1=Dg(z)v\in T_{z_1}Y$ and $v_{j+1}=D(g\circ \phi )(z_j)v_j\in T_{z_{j+1}}Y$ for $j=1,\ldots , n-1$ , and applying the above claim, we get

$$ \begin{align*} |v_{j+1}|_Y = |D(g\circ \phi)(z_j)v_j|_Y \geq \xi_{k_j}|v_j|_Y, \end{align*} $$

where $k_j\geq 0$ is the unique integer such that $z_j\in W_{k_j}$ . Setting $\eta =1-C_\theta \delta ^{1-\theta }<1$ and carrying these facts back into equation (5.19), we deduce that

(5.20) $$ \begin{align} |Df^n(z)v|_Y> \eta\,\bigg[\prod_{k=1}^{\infty} \xi_k^{N_{k,n}(z)}\bigg] |v|_Y, \end{align} $$

where $N_{k,n}(z)$ is the total number of j terms in the range $1\leq j\leq n-1$ such that $z_j\in W_{k}$ (in particular, the product appearing in the right-hand side is actually finite). This proves assertion (a). Now suppose that z is such that its -limit set accumulates at a point off the real axis, say $p\in Y$ . This is the case, for instance, if z is a recurrent or periodic point for f. Then there exist $k\geq 0$ and a sequence $j_\nu \to \infty $ such that $z_{j_\nu }\to p$ as $\nu \to \infty $ and $z_{j_\nu }\in W_k$ for all $\nu $ . However, this tells us that $N_{k,n}(z)\to \infty $ as $n\to \infty $ , and therefore, from equation (5.20), we deduce at last that $|Df^n(z)v|_Y/|v|_Y\to \infty $ as $n\to \infty $ . This proves the desired expansion property stated in assertion (b), and it also proves assertion (c). Hence, it remains to prove assertion (d).

Let $z\in Y\cap \mathcal {K}_f$ be a recurrent point. Let $N\geq 1$ be such that $|Df^N(z)v|_Y\geq 3\eta ^{-1}|v|_Y$ for all $v\in T_zY$ , where $\eta $ is the constant of assertion (a). Such an N exists because of assertion (b). By continuity of $\zeta \mapsto Df^N(\zeta )$ , we can find $\epsilon _0>0$ such that $|Df^N(\zeta )v|_Y\geq 2\eta ^{-1}|v|_Y$ for all $\zeta \in B_Y(z,\epsilon _0)$ and each $v\in T_{\zeta }Y$ . Now, given $0<\epsilon <\tfrac 14\eta \epsilon _0$ , choose $m>N$ such that $f^m(z)\in B_Y(z,\epsilon )$ ; this is possible because z is recurrent. Write $\mathcal {O}=B_Y(f^m(z), 2\epsilon )\subset B_y(z,\epsilon _0)$ , and let $\mathcal {O}'\subset Y$ be the component of $f^{-m}(\mathcal {O})$ that contains z. Then, $f^m|_{\mathcal {O}'}: \mathcal {O}'\to \mathcal {O}$ is a diffeomorphism. By assertion (a), the inverse diffeomorphism $f^{-m}|_{\mathcal {O}}: \mathcal {O}\to \mathcal {O}'$ is Lipschitz with constant $\eta ^{-1}$ in the hyperbolic metric of Y. Therefore,

$$ \begin{align*} \mathcal{O}'\subset B_Y(z, \eta^{-1}\cdot(2\epsilon)) \subset B_Y(z, \epsilon_0). \end{align*} $$

Now that we know this fact, writing $f^m=f^{m-N}\circ f^N$ , we see that for all $\zeta \in \mathcal {O}'$ and each non-zero $v\in T_{\zeta } Y$ ,

$$ \begin{align*} \frac{|Df^m(\zeta)v|_Y}{|v|_Y}&= \frac{|Df^{m-N}(f^N(\zeta))Df^N(\zeta)v|_Y}{|Df^N(\zeta)v|_Y}\cdot \frac{|Df^N(\zeta)v|_Y}{|v|_Y} \\ & \geq \eta \cdot (2\eta^{-1}) = 2. \end{align*} $$

Equivalently, we have shown that $|Df^{-m}(\zeta )v|_Y\leq \tfrac 12|v|_Y$ for all $\zeta \in \mathcal {O}$ and each $v\in T_\zeta Y$ . In other words, $f^{-m}|_{\mathcal {O}}: \mathcal {O}\to \mathcal {O}'$ is, in fact, a contraction of the hyperbolic metric of Y, with contraction constant $\tfrac 12$ . In particular,

$$ \begin{align*} \mathcal{O}'= f^{-m}|_{\mathcal{O}}(\mathcal{O}) \subset B_Y(z,\epsilon) \Subset B_Y(f^m(z), 2\epsilon) =\mathcal{O}. \end{align*} $$

This means that $f^{-m}|_{\mathcal {O}}$ maps the hyperbolic ball $\mathcal {O}$ strictly inside itself (and it is a contraction of the hyperbolic metric). Hence, there exists $z_*\in \mathcal {O}'$ such that $f^m(z_*)=z_*$ , and this periodic point is necessarily expanding, by assertion (c). Thus, we have proved that for each $\epsilon>0$ , there exists an expanding periodic point $\epsilon $ -close to z. This establishes assertion (d) and completes the proof of our theorem.

It is worth pointing out that, combining Theorem 5.4 with Theorem 5.3, we already deduce the following simple properties of the dynamics of all sufficiently deep renormalizations of a given AHPL-map. Considerably stronger results will be proved in §6 below.

Corollary 5.5. Let $f: U\to V$ be an AHPL-map of class $C^r$ , with $r> 3$ , whose restriction to the real line is an infinitely renormalizable unimodal map with bounded combinatorics. There exists $n_1=n_1(f)\in \mathbb {N}$ such that for all $n\geq n_0$ , the nth renormalization $f_n=R^nf: U_n\to V_n$ is an AHPL-map with the following properties.

1. (a) Every periodic orbit of $f_n$ is expanding.

2. (b) The expanding periodic points are dense in the set of all recurrent points.

3. (c) There are no stable components of $\mathrm {int}(\mathcal {K}_{f_n})$ whose closures do not intersect the real axis.

Proof. Choose $0<\theta <1$ , as well as $n_0\in \mathbb {N}$ and $\alpha>1$ large enough so that equation (5.1) holds true. This is possible because $r>3$ . Then, by Theorem 5.3, there exists $n_1\in \mathbb {N}$ such that for all $n\geq n_1$ , the nth renormalization $f_n$ of f is an $(\alpha , \delta , \theta ,M,n_0)$ -controlled AHPL map, for some $0<\delta <1$ . Hence, assertions (a) and (b) follow from the corresponding assertions in Theorem 5.4. To prove assertion (c), suppose is a stable component of $\mathrm {int}(\mathcal {K}_{f_n})$ such that . Let $p\geq 1$ be such that . Also, consider the decomposition of the domain of $f_n$ into scales, as in Theorem 5.4. Since is compact, it is contained in the union of finitely many scales. In each scale, $f_n$ expands the hyperbolic metric of $Y_n$ by a definite amount. Hence, so does $f_n^p$ on . However, this is impossible, because has finite hyperbolic area.

6 Topological conjugacy to polynomials and local connectivity of Julia sets

In this section, we will prove that a $(\alpha , \delta ,\theta , M,n_0)$ -controlled AHPL-mapping $f:U\rightarrow V,$ which is infinitely renormalizable of bounded type, is topologically conjugate to a real polynomial in a neighbourhood of its filled Julia set, so that from the topological point of view, the dynamics of these mappings are the same as those of polynomials; in particular, such mappings do not have wandering domains. We will also prove that the Julia set of such an AHPL-mapping is locally connected. Specifically, we will assume that f satisfies the conditions of Theorem 5.4. In particular, we assume that $f:U\rightarrow V$ is a $C^r$ asymptotically holomorphic polynomial-like mapping that is $(\alpha , \delta ,\theta , M,n_0)$ -controlled,

$$ \begin{align*}r>1+\frac{4n_0\alpha}{(n_0-1)(1-\theta)(2\alpha-1)},\end{align*} $$

and that the conclusions of Theorem 5.4 all hold. By Theorems 3.2 and 5.3, for any $r>3,$ if g is a $C^r$ mapping of the interval, which is infinitely renormalizable of bounded type, then for any n sufficiently large, there is a renormalization, $R^ng:U_n\rightarrow V_n$ of $g,$ which is an AHPL-mapping that satisfies these assumptions.

6.1 Dilatation and expansion

The proof of the following lemma is implicit in the proof of Theorem 5.4; it makes the lower bound in equation (5.3) explicit.

Lemma 6.1. Let $\xi _n$ be the constant defined in equation (5.13). There exists $N\geq n_0$ such that if $n\geq N,$ then

(6.1) $$ \begin{align} 1+M\bigg(\frac{\unicode{x3bb}^{n-1}}{\alpha M}\bigg)^{r-1}\leq \xi_n. \end{align} $$

Proof. It is sufficient to show that

$$ \begin{align*} M\bigg(\frac{\unicode{x3bb}^{n-1}}{\alpha M}\bigg)^{r-1}\leq K_1\unicode{x3bb}^{2n(1-{1}/{2\alpha})^{-1}}-2K_2\unicode{x3bb}^{(n-1)(r-1)(1-\theta)}, \end{align*} $$

see equation (5.14). Factoring out $\unicode{x3bb} ^{(n-1)(r-1)}$ on the right and cancelling it with the same term on the left, this is equivalent to

(6.2) $$ \begin{align} \frac{M}{(\alpha M)^{r-1}}\leq K_1\unicode{x3bb}^{2n(1-{1}/{2\alpha})^{-1}-(n-1)(r-1)} -2K_2\unicode{x3bb}^{-\theta(n-1)(r-1)}. \end{align} $$

Since $n>n_0$ , we have that

(6.3) $$ \begin{align} 4n_0\alpha (n-1)-4n\alpha(n_0-1) = 4\alpha(n_0(n-1)-n(n_0-1))>0. \end{align} $$

Now, since

$$ \begin{align*} r\geq 1+\frac{4 n_0 \alpha}{(n_0-1)(1-\theta)(2\alpha-1)}, \end{align*} $$

we have that

(6.4) $$ \begin{align} (r-1)(1-\theta)(n-1)\geq \frac{4n_0\alpha(n-1)}{(n_0-1)(2\alpha-1)}. \end{align} $$

So,

$$ \begin{align*} (r-1)(1-\theta)(n-1)-2n\bigg(1-\frac{1}{2\alpha}\bigg)^{-1} &\geq \frac{4n_0\alpha(n-1)}{(n_0-1)(2\alpha-1)}- 2n\frac{2\alpha}{2\alpha-1}\\ &=\frac{4n_0\alpha(n-1)-4n\alpha(n_0-1)}{(n_0-1)(2\alpha-1)}\\ &>0, \end{align*} $$

where the first inequality follows from equation (6.4) and the last inequality follows from equation (6.3). Thus, we have

$$ \begin{align*}2n\bigg(1-\frac{1}{2\alpha}\bigg)^{-1}-(n-1)(r-1)\leq-\theta(n-1)(r-1).\end{align*} $$

Since both exponents on the right-hand side of equation (6.2):

$$ \begin{align*}2n\bigg(1-\frac{1}{2\alpha}\bigg)^{-1}-\theta(n-1)(r-1) \quad\mbox{and}\quad -\theta(n-1)(r-1)\end{align*} $$

are negative, equation (6.1) holds for n sufficiently large.

Let

$$ \begin{align*}K_{f^n}(z)=\frac{1+|\mu_{f^n}(z)|}{1-|\mu_{f^n}(z)|},\end{align*} $$

be the quasiconformal distortion of $f^n$ at z. A chain of domains is a sequence of domains $\{B_j\}_{j=0}^n$ where $B_j$ is a component of $f^{-1}(B_{j+1})$ for all $j=0,1,2,\ldots ,n-1$ and $B_n$ is a domain in $\mathbb C$ . To a mapping $f^n:A\rightarrow B,$ we associate the chain of domains $\{B_j\}_{j=0}^n$ , where $B_n=B$ and $B_j=\mathrm {Comp}_{f^{j}(B)}f^{-(n-j)}(B)$ for $j=0,\ldots ,n-1$ .

Recall that $W_k$ is the strip

$$ \begin{align*} W_k = \bigg\{z\in U_\alpha : \frac{\unicode{x3bb}^k}{\alpha M}\leq |\mathrm{Im}\,z| < \frac{\unicode{x3bb}^{k-1}}{\alpha M}\bigg\}. \end{align*} $$

Corollary 6.2. For each $N\in \mathbb N$ , there exists $c>0$ such that the following holds. Let A be an open domain in $\mathbb C$ . Suppose that $f^n:A\rightarrow B$ is onto and let $\{B_j\}_{j=0}^n$ be the chain with $B_0=A$ and $B_n=B$ . Assume that for each $0\leq j\leq n$ ,

Then,

$$ \begin{align*}c\cdot \sup_{z\in A} \log K_{f^n}(z)\leq \inf_{z\in A}\log|Df^n(z)v|_Y\end{align*} $$

for each unit tangent vector $v\in T_zY$ .

Proof. Let us express $f^n:B_0\rightarrow B_n$ as $\phi \circ (g\circ \cdots \circ \ g \circ \phi )\circ g$ . For each $0\leq j < n$ , $g:B_j\rightarrow \phi ^{-1}(B_{j+1})$ . Since $\phi $ is a $(1+\varepsilon (\delta ))$ -quasi-isometry in the hyperbolic metric on Y where $\varepsilon (\delta )\rightarrow 0$ as $\delta \rightarrow 0,$ we have that there exists $N_1$ , depending only on N, so that $\phi ^{-1}(B_{j})$ intersects at most $N_1$ strips $W_k$ .

For each $B_j,$ let $n_j$ be minimal so that . Then for any $g(z)\in \phi ^{-1}(B_j)$ , $1\leq j<n$ , we have that

$$ \begin{align*}\bigg|\frac{\bar\partial(g\circ\phi)}{\partial(g\circ\phi)}(g(z))\bigg| =\bigg|\frac{\bar\partial\phi}{\partial \phi}(g(z))]\bigg| \leq M\bigg(\frac{\unicode{x3bb}^{n_{j}-1}}{\alpha M}\bigg)^{r-1}.\end{align*} $$

By equation (5.3) and Lemma 6.1, we have that for all $v\in T_zY$ with $|v|_Y=1$ ,

$$ \begin{align*}|D(g\circ \phi)(z)|_Y\geq 1+M\bigg(\frac{\unicode{x3bb}^{n_{j}-1+N_1}}{\alpha M}\bigg)^{(r-1)}=1+M\unicode{x3bb}^{N_1(r-1)}\bigg(\frac{\unicode{x3bb}^{n_{j}-1}}{\alpha M}\bigg)^{r-1},\end{align*} $$

so that

$$ \begin{align*}|D(g\circ\phi)(z)v|_Y\geq \bigg(1+\unicode{x3bb}^{N_1(r-1)}\sup_{z\in B_j}\bigg|\frac{\bar\partial(g\circ\phi)}{\partial(g\circ \phi)}(g(z)) \bigg|\bigg)|v|_Y.\end{align*} $$

Thus, we have that

$$ \begin{align*}\inf_{z\in B_j}|D(g\circ\phi)(z)v|_Y\geq \bigg(1+\unicode{x3bb}^{N_1(r-1)}\sup_{z\in B_j}\bigg|\frac{\bar\partial(g\circ\phi)}{\partial(g\circ \phi)}(g(z)) \bigg|\bigg)|v|_Y.\end{align*} $$

For each i, let

and let us reindex the $B_j$ as follows. For each $i\in \mathbb N\cup \{0\}$ , let $B_{i_{0}},\ldots , B_{i_{k_i}}$ be an enumeration of all $B_j$ so that

and for all

. Notice that $n=\sum _{i=0}^\infty k_i$ .

By the chain rule and Theorem 3.1, we have that there exists a constant $c_1>0$ so that

$$ \begin{align*} \inf_{z\in B_0} |Df^n(z)v|_Y \geq c_1 \prod_{i=0}^\infty \prod_{j=0}^{k_i}(1+ \unicode{x3bb}^{N_1(r-1)}\sup_{z\in B_{i_j}} |\mu_f(z))|).\end{align*} $$

Now, there exists a constant $c_2>0$ such that

$$ \begin{align*}\log \prod_{i=0}^\infty \prod_{j=0}^{k_i}(1+ \unicode{x3bb}^{N_1(r-1)} \sup_{z\in B_{i_j}}|\mu_f(z)|) = \sum_{i=0}^\infty\sum_{j=0}^{k_i}\log (1+ \unicode{x3bb}^{N_1(r-1)}\sup_{z\in B_{i_j}}|\mu_f(z)|)\end{align*} $$

$$ \begin{align*} & \geq c_2 \sum_{i=0}^\infty\sum_{j=0}^{k_i} \unicode{x3bb}^{N_1(r-1)}\sup_{z\in B_{i_j}}|\mu_f(z)|\\ & = c_2\frac{\unicode{x3bb}^{N_1(r-1)}}{2} \sum_{i=0}^\infty\sum_{j=0}^{k_i} (\sup_{z\in B_{i_j}}(|\mu_f(z)|-(-|\mu_f(z)|)))\\ &\geq c_2\frac{\unicode{x3bb}^{N_1(r-1)}}{2} \sum_{i=0}^\infty\sum_{j=0}^{k_i} \sup_{z\in B_{i_j}}\log\bigg(\frac{1+|\mu_f(z)|}{1-|\mu_f(z)|}\bigg)\\ &=c_2\frac{\unicode{x3bb}^{N_1(r-1)}}{2}\log\prod_{i=0}^\infty \prod_{j=0}^{k_i}\sup_{z\in B_{i_j}} \bigg(\frac{1+|\mu_f(z)|}{1-|\mu_f(z)|}\bigg). \end{align*} $$

Hence, there exists a constant c so that

$$ \begin{align*}\inf_{z\in B_0} \log |Df^n(z)v|_Y\geq c\cdot \log \sup_{z\in B_0}K_{f^n}(z).\\[-40pt]\end{align*} $$

6.2 Puzzle pieces

Let us construct external rays for f. These will allow us to construct Yoccoz puzzle pieces for f where the role of equipotentials is played by the curves $f^{-i}\partial V$ . To construct these rays, we use a method analogous to the one used by Levin–Przytycki in [41] to construct external rays for holomorphic polynomial-like maps.

First, we associate to f an external map, $h_f$ , as follows. Let $X_0=V$ and for $i\in \mathbb N$ , set $X_{i+1}=f^{-1}(X_i)$ . Notice that since $U\Subset V$ , $f:U\rightarrow V$ is a branched covering of V, ramified at a single point, 0, and $f^i(0)\in U$ for all i, we have that $X_i=f^{-i}(V)$ is a connected and simply connected topological disk for all $i\in \mathbb N\cup \{0\}$ , and $X_{i+1}\Subset X_i.$ Let $M=\mod (V\setminus \mathcal K_f),$ and let

be the uniformization of $V\setminus \mathcal K_f$ by a round annulus. Let $D_i=\phi ^{-1}(X_{i}),$ we have that each annulus

is mapped as a d-to-1 covering map onto $D_{i-1}\setminus D_i$ by $h_f=\phi ^{-1}\circ f\circ \phi $ . The mapping $h_f$ extends continuously to $\partial \mathbb D$ , and by Schwarz reflection, $h_f$ can be defined as a mapping between annuli $W'\subset W$ , each with the same core curve, $\partial \mathbb D$ . We have that $h_f$ is a $C^3$ expanding mapping of $S^1$ (see the proof of [12, Lemma 10.17]) and that the dilatation of $h_f$ on $W'$ is the same as the dilatation of f. Foliate $W\setminus W'$ by $C^{r}, h_f$ invariant rays, connecting $\partial W'$ and $\partial W$ , and pull them back by $h_f$ . We obtain a foliation by $C^r$ rays of $W'\setminus \partial \mathbb D$ that is continuous on $W'$ . Pulling back this foliation of $W'$ by $\phi $ , we obtain a foliation of $V\setminus \mathcal K_f.$ The leaves of this foliation are the external rays of f.

Remark 6.3. Observe that since $h_f|S_1$ is a degree d expanding mapping of the circle, it is topologically conjugate to $z\mapsto z^d$ on a neighbourhood of $S^1$ . Consequently, one can carry out this construction simultaneously for two mappings $f:U\rightarrow V$ and $\tilde f:\tilde U\rightarrow \tilde V$ to obtain a mapping $H:V\rightarrow \tilde V$ such that $H\circ F(z)=\tilde F\circ H(z)$ for any $z\in U$ contained in an equipotential or ray.

For each $z\in V\setminus \mathcal K_f,$ we let $R_z$ denote the ray through z. Let us parametrize $R_z$ by $R_z(t), t\geq 0,$ such that for each $n\in \mathbb N$ , we have that $R_z(n)$ is the unique point on $R_z$ that passes through $\partial X_n$ . We say that a ray $R_z$ lands at a point p if $\lim _{t\rightarrow \infty } R_z(t)=p$ .

To prove that certain rays land, we will need the following lemma.

Lemma 6.4. [6, Lemma 2.3]

Let be a hyperbolic region. Let be a family of curves with uniformly bounded hyperbolic length and such that Then $\mathrm {diam}(\gamma _n)\rightarrow 0.$

Lemma 6.5. If $R_z$ accumulates on a real repelling periodic point p, then $R_z$ lands at p.

Proof. Compare [41, Lemma 2.1] and [6]. Suppose that p is a real repelling periodic point of period s. Then one can repeat the proof of linearization near repelling periodic points of holomorphic maps to prove that there exists a neighbourhood B of p such that $f^s$ is conjugate to $z\mapsto \unicode{x3bb} z$ near p, where $\unicode{x3bb} =Df^s(p)$ , see [53].

Let $R_z([n-1,n])$ be the segment of the ray connecting $\partial X_{n-1}$ and $\partial X_n.$ Let us show that $\mathrm {diam}(R_z([n-1,n]))\rightarrow 0$ as $n\rightarrow \infty $ . By Lemma 6.4, and since $\phi $ is an isometry in the hyperbolic metric, it is sufficient to show that the curves $\phi ^{-1}(R_z([n-1,n]))$ have uniformly bounded hyperbolic lengths. This follows from the fact that $\|Dh_f(z)\|>1$ in the hyperbolic metric for z sufficiently close to $\partial \mathbb D$ , which was proved in the proof of [12, Lemma 10.17]. Thus, we have that $\mathrm {diam}(R_z([n-1,n]))\rightarrow 0$ as $n\rightarrow \infty .$ So there exists $n_0\in \mathbb N$ such that for all $n\geq n_0$ , we have that $R_{z}([n,n+1])\subset (f|_B)^{-s(n-n_0)}(B).$ Since $f^s|_B$ is qc-conjugate to $z\mapsto \unicode{x3bb} z$ with $\unicode{x3bb}>1$ in a neighbourhood of 0, we have that $\bigcap _{n=n_0}^{\infty }(f|_B)^{-s(n-n_0)}(B)=\{p\}.$ So the only accumulation point of the ray is p.

We define puzzle pieces for f as follows. Let us index the renormalizations $R^nf:U_n\rightarrow V_n$ of f by $f_n:U_n\rightarrow V_n$ , so that $f_n=f^{q_n}|_{U_n}.$ Let $I_n=\mathcal K_{f_n}\cap \mathbb R$ denote the invariant interval for $f_n$ . Let $\tau :I_0\rightarrow I_0$ be the even, dynamical, symmetry about the even critical point at 0. Let $\beta _n\in \partial I_n$ be the orientation preserving fixed point of $f_n$ in $\partial I_n.$ By real-symmetry, there exist two rays, labelled $R_{\beta _n}$ and $R_{\beta _n}'$ that land at $\beta _n$ . Let $R_{\tau (\beta _n)}$ and $R_{\tau (\beta _n)}'$ denote the preimages under $f^{q_n}$ of $R_{\beta _n}$ and $R_{\beta _n}'$ , respectively, which land at $\tau (\beta _n)$ . For each $n\in \mathbb N$ , the initial configuration of puzzle pieces at level n are the components of $V\setminus (R_{\beta _n}\cup R_{\beta _n}'\cup R_{\tau (\beta _n)} \cup R_{\tau (\beta _n)}'\cup \{\beta _n,\tau (\beta _n)\})$ . We denote this union of puzzle pieces by $\mathcal {Y}^{(n)}_0$ . Given an initial configuration, $\mathcal {Y}^{(n)}_0$ , for $j\in \mathbb N\cup \{0\}$ , we define $\mathcal Y_{j}^{(n)}$ to be the union of the connected components of $f^{-j}(\mathcal {Y}^{(n)}_0)$ . Given any $z\in \mathcal K_f$ , we let $Y^{(n)}_j(z)$ denote the component of $\mathcal Y_{j}^{(n)}$ that contains z, and we let $Y^{(n)}_j=Y^{(n)}_j(0)$ be the component that contains the critical point.

Lemma 6.6. For each $n\in \mathbb N,$ there exists j, so that $\mathcal K_{f_n}\subset Y^{(n)}_j \subset U_n$ .

Proof. For all $j\in \mathbb N$ , . Let $q_n$ be the period of the renormalization $f_n$ of f. Let $K_j=\mathrm {comp}_0 f^{-q_n j}(Y_0^{(n)})$ . Since $K_j\subset K_{j-1}$ and $f^{s_n}:K_{j}\rightarrow K_{j-1}$ , and is a compact connected set, we have that .

Proposition 6.7. Suppose that $z\in \mathcal K_f$ . Then there exist arbitrarily small neighbourhoods P of z such that P is a union of puzzle pieces.

Proof. Observe that Lemma 6.6 implies that there are arbitrarily small puzzle pieces containing the critical point of f. Let us start by spreading this information throughout the filled Julia set of f. Let $z\in \mathcal K_f$ .

Case 1: Assume that . For each n, let $C_n\subset U_n$ be the puzzle piece given by Lemma 6.6. Let $r_n$ be minimal so that $f^{r_n}(x)\in C_n$ and let $C^0_n=\mathrm {comp}_xf^{-r_n}(C_n)$ . Each $C_n$ is contained in the topological disk, $\Gamma _n$ , bounded by the core curve $\gamma _n$ of the annulus . By Theorem 3.2, there exists $C>0$ such that for all $n\in \mathbb N,$ we have that Thus, the domain $\Gamma _n$ is a $K=K(C)$ -quasidisk. Let $V_n^0=\mathrm {Comp}_x f^{-r_n}(V_n)$ and $\Gamma _n^0=\mathrm {Comp}_x f^{-r_n}(\Gamma _n).$ It is not hard to see that $f^{r_n}:V_n^0\rightarrow V_n$ is a diffeomorphism: suppose that there exists $0<j<r_n$ so that but $f^j(V_n^0)$ is not contained in $U_n$ so that Since $f^{s_n}:U_n\rightarrow V_n$ is a first return mapping to $V_n$ , for all $k\in \mathbb N$ , $f^{j+ks_n}(V_n^0)$ intersects both $\mathcal K_{f_n}$ and $\partial V_n,$ and we have that there exists no $j_1{\kern-1.5pt}\in{\kern-1.5pt} \mathbb N$ such that $f^{j_1}(V_n^0){\kern-1.5pt}={\kern-1.5pt}V_n.$ Thus, we have that if for some $j,$ then $f^j(V_n^0)\subset U_n$ , but then since for all $k\in \mathbb N$ , $j=r_n$ , and so $f^{r_n}:V_n^0\rightarrow V_n$ is a diffeomorphism.

Case 1a: Suppose that and $z\in \mathbb R\cap \mathcal K_f$ . Then, by the complex bounds, we have that there exists $K>1$ for each n, the mapping $f^{r_n}:V_n^0\rightarrow V_n$ is a diffeomorphism with quasiconformal distortion bounded by K. Hence, there exists $m>0$ depending only on K and M such that for all n, . Thus, the puzzle pieces $C_n^0$ have diameters converging to $0$ .

Case 1b: Suppose that , , and for all j, $f^j(z)\notin \mathbb R$ . We consider the case when the mappings $f^j$ have uniformly bounded quasiconformal distortion near $z,$ and the case when they have unbounded quasiconformal distortion near $z,$ separately. First, suppose that there exists $K_x\geq 1$ such that for each n, the mapping $f^{r_n}:C^0_n\rightarrow C_n$ extends to a mapping from $V_n^0$ onto $V_n$ with quasiconformal distortion bounded by $K_x$ . We have that each $\Gamma _n^0$ is a $K_1$ -quasidisk for some $K_1>1$ depending on x, and there exists a constant $m>0$ such that for all n, $\mod (\Gamma _n^0\setminus \Gamma _{n+1}^0)\geq m$ , and so the puzzle pieces $C_n^0$ shrink to z.

Suppose now that the quasiconformal distortion of $f^{r_n}:V_n^0\rightarrow V_n$ tends to infinity as n tends to infinity. For each n, let $\{V_n^j\}_{j=0}^{r_n}$ be the chain with $V_n^{r_n}=V_n$ and $V_n^0=\mathrm {Comp}_z f^{-r_n}(V_n),$ and let $\{\Gamma _n^j\}_{j=0}^{r_n}$ be the chain with $\Gamma _n^{r_n}=\Gamma _n$ and $\Gamma _n^0=\mathrm {Comp}_z f^{-r_n}(\Gamma _n).$ For all n sufficiently large, there exists $0\leq j_n<r_n$ maximal so that the set $V_n^{j_n}=\mathrm {Comp}_{f^{j_{n}}(z)}f^{-(r_n-j_n)}(V_n)$ does not intersect the real line (see case 2a below). Let $\Gamma _n^{j_n}=\mathrm {Comp}_{f^{j_{n}}(z)}f^{-(r_n-j_n)}(\Gamma _n).$ Since $\partial \Gamma _n^{j_n}$ is the core curve $V_n\setminus U_n$ , and $f^{(r_n-j_n)}$ has bounded quasiconformal distortion, we have that there exists $m_1>0$ such that $\mathrm {mod}(V_n^{j_n}\setminus \Gamma _n^{j_n})>m_1.$ However, this implies that there exists $m_2>0$ such that $\mathrm {dist}(\partial V_n^{j_n},\Gamma _n^{j_n})>m_2\mathrm {diam}(\Gamma _n^{j_n})$ , which immediately gives us that there exists $m_3>0$ so that $\mathrm {dist}(\Gamma _n^{j_n},\mathbb R)> m_3\mathrm {diam}(\Gamma _n^{j_n}).$ It follows that there exists $\xi>0$ such that for all n, $\mathrm {diam}_Y(\Gamma _n^{j_n})<\xi .$

Let us inductively choose a subsequence $V_{n_i}$ of the levels $V_n$ so that the landing maps from $V_{n_i}^{j_{n_i}}$ to $V_{n_{i+1}}^{j_{n_{i+1}}}$ all have definite expansion. Let $\eta \in (0,1)$ be the constant from Theorem 5.4, so that $|Df^i(z)v|_Y\geq \eta |v|_Y$ . Then we have that if $X,$ a component of $f^{-i_0}(V_{n}^{j_n}),$ is a pullback of $V_{n}^{j_n}$ such that the quasiconformal distortion of $f^{i_0}|_X$ is bounded by $2(1+\delta )/\eta $ , then there exists $N\in \mathbb N$ such that for each $i\leq i_0$ , for each element $X_i=f^i(X)$ in the chain associated to the pullback, $X_i$ intersects at most N of the strips $W_k$ . Let $c>0$ be the constant associated to N from Corollary 6.2. Let $k_0>0$ be minimal so that

$$ \begin{align*} \sup_{z\in \Gamma^0_{k_0}} K_{f^{j_{k_0}}}(z)^c\geq \frac{2}{\eta}. \end{align*} $$

Let $0\leq j_{k_0}'<j_{k_0}$ be maximal so that

$$ \begin{align*} \sup_{z\in \Gamma^{j_{k_0}'}_{k_0}} K_{f^{j_{k_0}-j_{k_0}'}}(z)^c\geq \frac{2}{\eta}. \end{align*} $$

Then, since f is $(1+\delta )$ -qc, we have that

$$ \begin{align*} \sup_{z\in \Gamma_{k_0}^{j_{k_0}'}} K_{f^{j_{k_0}-j_{k_0}'}}(z)^c\leq (1+\delta)\frac{2}{\eta}. \end{align*} $$

Thus by Corollary 6.2, we have that

$$ \begin{align*}\mathrm{diam}_Y(\Gamma_{k_0}^{j_{k_0}'})\leq \frac{\eta}{2}\mathrm{diam}_Y (\Gamma^{j_{k_0}}_{k_0})\leq \frac{\eta\xi}{2},\end{align*} $$

and by Theorem 5.4, we have that

$$ \begin{align*}\mathrm{diam}_Y (\Gamma^0_{k_0})<\frac{\xi}{2}.\end{align*} $$

We now repeat the argument: let $k_1>k_0$ be minimal so that

$$ \begin{align*} \sup_{z\in \Gamma^{j_{k_0}}_{k_0}} K_{f^{-(j_{k_1}-j_{k_0})}}(z)^c\geq \frac{2}{\eta}, \end{align*} $$

and let $0\leq j_{k_1}'<j_{k_1}$ be maximal so that

$$ \begin{align*} \sup_{z\in \Gamma^{j_{k_1'}}_{k_1}} K_{f^{j_{k_1}-j_{k_1}'}}(z)^c\geq \frac{2}{\eta}. \end{align*} $$

Then, since f is $(1+\delta )$ -qc, we have that

$$ \begin{align*} \sup_{z\in \Gamma_{j_{k_1}}^{k_1'}} K_{f^{j_{k_1}-j_{k_1}'}}(z)^c\leq (1+\delta)\frac{2}{\eta}. \end{align*} $$

Again by Corollary 6.2, we have that

$$ \begin{align*}\mathrm{diam}_Y(\Gamma_n^{j_{k_1}'})\leq \frac{\eta}{2}\mathrm{diam}_Y(\Gamma^{j_{k_1}}_{k_1})\leq \frac{\eta\xi}{2},\end{align*} $$

and by Theorem 5.4, we have that

$$ \begin{align*}\mathrm{diam}_Y (\Gamma^{j_{k_0}}_{k_1})<\frac{\xi}{2}.\end{align*} $$

Combining this with the first step, we have that

$$ \begin{align*}\mathrm{diam}_Y(\Gamma_{k_1}^0)<\frac{\xi}{4}.\end{align*} $$

If the quasiconformal distortion of $f^n$ diverges at x, we see that we can repeat this argument infinitely many times to obtain a nest of puzzle pieces $\{C^0_{k_i}\}$ about z such that $\mathrm {diam}_Y(C^0_{k_i})\rightarrow 0$ .

Combining cases (1a) and (1b), we have that for all z such that , there are arbitrarily small puzzle pieces Now we treat the cases when .

Case 2a: Suppose that there exists $n\in \mathbb N$ such that Let $\mathcal {Y}^{(n)}_0$ be the initial configuration of puzzle pieces at level n. Let then, since the real traces of puzzle pieces shrink to points, there exist $m_0>0$ and a union of (closed) puzzle pieces of $\mathcal {Y}^{(n)}_{m_0}$ , denoted by $Q_0$ , such that and $x_0\in \mathrm {int}(Q_0)$ . Let $\mathcal {Y}^{(n)}_j(x_0)$ denote the closure of the set of puzzle pieces P in $\mathcal {Y}^{(n)}_j$ with Let $Q=\bigcap _{j=0}^\infty \mathcal {Y}^{(n)}_j(x_0).$

Let us show that $Q=\{x_0\}$ . If $\mathrm {diam}(Q)>0,$ then, since $\bigcup _n f^n(Q)$ is a bounded set, there exist $C>0$ , $x\in Q$ and a vector $v\in T_x\mathbb C$ such that $|Df^{k_i}(x)v|<C.$ If is not contained in the real-line, then in a small neighbourhood of x, the hyperbolic metric on Y is comparable to the Euclidean metric, but now $|Df^{k_i}(x)v|<C$ contradicts Theorem 5.4(b). So we can assume that but then is contained in the hyperbolic set of points that avoid $V_n$ , and we have that $|Df^{k_i}(x)v|\rightarrow \infty $ for any $v\in T_x\mathbb C,$ and so $\mathrm {diam}(Q)=0$ . Let us point out that this argument shows that if $z\in \mathbb R$ is contained in a hyperbolic set, then for any n sufficiently big, $\mathrm {diam}(\mathcal {Y}^{(n)}_{j}(z))\rightarrow 0$ as $j\rightarrow \infty $ , and indeed that $J_f$ is locally connected at any point in $J_f\cap \mathbb R$ that is contained in a hyperbolic set.

Suppose that for all $j\in \mathbb N\cup \{0\},$ $f^j(z)\notin \mathbb R.$ Let $r_0$ be the first return time of $x_0$ to $Q_0$ , and let $Q_1=\mathrm {Comp}_{x_0}f^{-r_0}(Q_0)$ . Inductively define $Q_{i+1}$ by taking $r_i$ to be the first return time of $x_0$ to $Q_i$ and setting $Q_{i+1}=\mathrm {Comp}_{x_0}f^{-r_i}(Q_i).$ Let $\varepsilon>0$ be so small that if $z=x+iy$ satisfies $\mathrm {dist}(z,\mathbb R)<\varepsilon $ and $z\notin V_n$ , then $\mathrm {dist}(x,0)>\mathrm {diam}(V_n)/2$ . Since , there exist $n_i\rightarrow \infty $ with the property that $n_i$ is minimal with $f^{n_i}(z)\in Q_i$ . It is sufficient to show that there exists a constant $c>0$ so that for all i, $\|Df^{n_i}(z)\|\geq c.$ Fix some $i\in \mathbb N$ . Let $j_0\geq n_0$ be minimal so that $\mathrm {dist}(f^{j_0}(z),\mathbb R)>\varepsilon ,$ and let $j_1\leq n_i$ be maximal so that $\mathrm {dist}(f^{j_1}(z),\mathbb R)>\varepsilon ,$ then there exists a constant $c_1>0$ so that

$$ \begin{align*}\|Df^{n_i}(z)\|\geq c_1\eta\|Df^{n_i-j_1}(f^{j_1}(z))\|\|Df^{j_0-n_0}(f^{n_0}(z))\|\|Df^{n_0}(z)\|.\end{align*} $$

Thus, it suffices to bound $\|Df^{n_i-j_1}(f^{j_1}(z))\|$ and $\|Df^{j_0-n_0}(f^{n_0}(z))\|$ from below. Let $z_0=f^{n_0}(z)$ and define $z_i=f^i(z_0)$ , $x_i=f^i(x_0)$ . Then there exist constants $c_2,c_3$ so that

$$ \begin{align*}\|Df^{j_0-n_0}(z_0)\| \geq c_2\prod_{i=0}^{j_0-n_0}\|Df(x_i)\| \prod_{i=0}^{j_0-n_0}\bigg(1-c_3\frac{|z_i-x_i|}{\|Df(x_i)\|}\bigg).\end{align*} $$

By our choice of $\varepsilon $ , and since $x_0$ is contained in a hyperbolic Cantor set, we have that there exists a constant $c_4>0$ and $\Lambda>1$ so that

$$ \begin{align*}\sum_{i=0}^{j_0-n_0}\frac{|z_i-x_i|}{\|Df(x_i)\|}\leq \frac{1}{2\mathrm{diam}(V_n)}\sum_{i=0}^{j_0-n_0}|z_i-x_i| \leq \frac{1}{2\mathrm{diam}(V_n)}\frac{c_4\varepsilon}{1-\Lambda^{-1}}.\end{align*} $$

Thus, we have that $\|Df^{j_0-n_0}(z_0)\|$ is bounded from below. The proof that $\|Df^{n_i-j_1} (f^{j_1}(z))\|$ is bounded from below is similar.

Case 2b: Suppose that . Let $z_0$ be an accumulation point of that is not contained in $\mathbb R$ . Since the real puzzle pieces shrink to points, there exist n and m and a union Q of puzzle pieces in $\mathcal {Y}^{(n)}_m$ and a sequence $k_i\rightarrow \infty $ such that and $f^{k_i}(z)\in Q$ for all i. By Theorem 5.4(b), we have that

$$ \begin{align*}\mathrm{diam}(\mathrm{Comp}_{f^{k_0}(z)}(f^{-(k_i-k_0)}(Q)))\rightarrow 0\mbox{ as }i\rightarrow \infty.\end{align*} $$

Thus by Theorem 5.4(a),

$$ \begin{align*}\mathrm{diam}(\mathrm{Comp}_{z}(f^{-k_i}(Q)))\rightarrow 0\mbox{ as }i\rightarrow\infty.\\[-35pt]\end{align*} $$

Proposition 6.7 has several important consequences.

Corollary 6.8. Suppose that $f\in C^r$ is an asymptotically holomorphic polynomial-like mapping, which is $(\alpha , \delta ,\theta , M,n_0)$ -controlled, and that

$$ \begin{align*}r>1+\frac{4n_0\alpha}{(n_0-1)(1-\theta)(2\alpha-1)}.\end{align*} $$

Then the following hold.

1. (1) $\mathcal J_f=\mathcal K_f$ .

2. (2) $f:U\rightarrow V$ is topologically conjugate to a polynomial mapping in a neighbourhood of its Julia set. In particular, $f:U\rightarrow V$ has no wandering domains.

3. (3) $\mathcal J_f$ is locally connected.

Proof. (1). To see that $\mathcal J_f=\mathcal K_f$ , observe that for each $z\in \mathcal K_f$ , there are arbitrarily small puzzle pieces containing z, so z is a limit of points whose orbits eventually land in $V\setminus U$ . Thus, $z\in \mathcal J_f$ . In particular, $\mathcal K_f$ has empty interior.

(2). Let us now show that $f:U\rightarrow V$ is topologically conjugate to a polynomial mapping in a neighbourhood of its Julia set. Let $I\subset U\cap \mathbb R$ denote the invariant interval for f. Since $f|_{I}$ has negative Schwarzian derivative, there exists a real polynomial p with a critical point of the same degree as the critical point of f such that f is topologically conjugate to p on $I.$ Let $h:I\rightarrow I$ be the continuous mapping such that $h\circ f|_I=p\circ h.$ Let $\tilde V$ be a domain containing $\mathcal J_p$ that is bounded by some level set of the Green’s function for p. Let $\tilde U=p^{-1}(\tilde V)$ .

Let $H_0:V\rightarrow \tilde V$ be a homeomorphism such that:

• • for each $z\in \partial U, H_0\circ F(z) =p\circ H_{0}(z)$ ;

• • for each $z\in \bigcup _{n}(R_{\beta _n}\cup R_{\tau (\beta _n)}),$ we have that $H_0(z)\circ f=p\circ H_0(z)$ ; and

• • $H_0|_I=h$ .

See Remark 6.3 for a description of how to construct such an $H_0$ .

Given that $H_i$ is defined, define $H_{i+1}$ by $H_{i}\circ f=p\circ H_{i+1}$ . Since each $H_i$ is conjugacy on J between f and p that maps that critical value of f to the critical value of p, this pullback is always well defined and continuous. Observe that for each $z\in U\setminus \mathcal K_f$ , $H_i$ eventually stabilizes. Let $H:V\rightarrow \tilde V$ be a limit of the $H_i$ . To see that H is continuous, take any $z\in U$ and let $\{z_n\}$ be a sequence of points such that $z_n\rightarrow z$ . If $z\notin \mathcal K_f$ , then there exists a neighbourhood W of z and $i_0\in \mathbb N,$ large, such that for all $i\geq i_0$ and $w\in W$ , $H_i(w)=H_{i_0}(w).$ Hence, $H(z_n)\rightarrow H(z)$ . So suppose that $z\in \mathcal K_f$ , then since the nests of puzzle pieces about z and $H(z)$ both shrink to points, and H maps puzzle pieces for f to corresponding puzzle pieces for p, $H(z_n)\rightarrow H(z)$ . Also, since for each $z\in U\setminus \mathcal K_f$ , $H_i$ eventually stabilizes, $H:U\rightarrow \tilde U$ satisfies $H\circ F(z)=p\circ H(z)$ for all $z\in U\setminus \mathcal K_f$ and since $\mathcal K_f$ has empty interior, we have that H is a conjugacy between f and p on U.

(3). Finally, let us show that $\mathcal J_f$ is locally connected. Let $z\in \mathcal J_f$ , and let B be any open set that contains z, by Proposition 6.7, there exists a neighbourhood $Q\subset B$ of z, such that Q is a union of puzzle pieces. Since $\mathcal J_f\cap P$ is connected for any puzzle piece P, we have that $\mathcal J_f\cap Q$ is connected too.

Let us remark that since f is topologically conjugate to a polynomial, we obtain that the repelling periodic points of f are dense in $\mathcal J_f$ . We also point out that this implies that f has no wandering domains, but that this fact can be deduced immediately from the fact that the puzzle pieces shrink to points.