QUASISYMMETRIC ROBUSTNESS OF THE COLLET-ECKMANN CONDITION IN THE QUADRATIC FAMILY

ARTUR AVILA AND CARLOS GUSTAVO MOREIRA

Abstract. We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set of quadratic maps which are either regular or Collet-Eckmann with polynomial recurrence of the critical orbit has full Lebesgue measure.

Contents

1. Introduction 1 2. General definitions 3 3. Real quadratic maps 4 4. Measure and capacities 9 5. Statistics of the principal nest 11 6. Sequence of quasisymmetric constants and trees 16 7. Estimates on time 18 8. Dealing with hyperbolicity 22 9. Main Theorems 30 References 33

1. Introduction

The intent of this work is just to be a rigorous reference for [AM2]. It is based on the LaTeX file of [AM1]: the proofs are very similar and in many places differ just by change of constants. In fact the proofs in [AM1] give easily robustness under quasisymmetric reparametrizations with small qs-constant, there are some minor differences to remove this assumption (like introduction of fast landings and bad returns).

Date: May 20, 2001.

Partially supported by Faperj and CNPq, Brazil.

Here we consider the quadratic family, f_{a} = a − x^{2}, where −1/4 ≤ a ≤ 2 is the parameter, and we consider its dynamics in the invariant interval.

In [AM1], a thorough understanding of the dynamics of typical quadratic maps was obtained. More specifically, it was shown that a typical quadratic map is either regular (with a periodic attractor) or Collet-Eckmann (positive Lyapunov exponent of the critical value) with polynomial recurrence of the critical orbit. The first possibility corresponds to a hyperbolic deterministic setting, with the well known good properties of hyperbolic systems. The second is a particularly well studied case of non-uniformly hyperbolic chaotic dynamics: in the 90’s such maps were shown to possess many hyperbolic-like properties like stochastic stability, exponential decay of correlations and others ([KN], [Y], [BV] and [BBM]). In particular it was possible to answer affirmatively Palis Conjecture [Pa] for the quadratic family.

It was shown in [ALM] that the parameter space of general analytic families of unimodal maps (with negative Schwarzian derivative) can be related to the parameter space of quadratic maps through a quasisymmetric ‘holonomy map’. It becomes then feasible to transfer results from the quadratic family to other families, but there is one obstruction: quasisymmetric maps are not absolutely continuous.

Here we show that the set of ‘good’ parameters has not only full Lebesgue measure, but is resistent to a quasisymmetric reparametrization:

Theorem A. Consider a quasisymmetric reparametrization of the parameter space of the quadratic family. The set of parameters which are either regular or Collet-Eckmann:

lim inf_{n→∞} \frac{ln(|Df^{n}(f(0))|)}{n} > 0.

has full Lebesgue measure.

Theorem B. Consider a quasisymmetric reparametrization of the parameter space of the quadratic family. The set of parameters which are either regular or have polynomial recurrence of the critical orbit

0 < lim sup_{n→∞} \frac{− ln(f^{n}(0))}{ln(n)} < ∞.

has full Lebesgue measure.

In [AM2] this result is used to obtain a proof of Palis Conjecture for unimodal maps with negative Schwarzian derivative. Another approach to this result (with stronger estimates) allows to obtain more general results, the proof is however more elaborate [A].

2. General definitions

2.1. Maps of the interval. Let f : I → I be a C^{1} map defined on some interval I ⊂ ℝ. The orbit of a point p ∈ I is the sequence {f^{k}(p)}_{k=0}^{∞}. We say that p is recurrent if there exists a subsequence n_{k} → ∞ such that lim f^{n_{k}}(p) = p.

We say that p is a periodic point of period n of f if f^{n}(p) = p, and n ≥ 1 is minimal with this property. In this case we say that p is hyperbolic if |Df^{n}(p)| is not 0 or 1. Hyperbolic periodic orbits are attracting or repelling according to |Df^{n}(p)| < 1 or |Df^{n}(p)| > 1.

We will often consider the distortion of an iterate f^{n} restricted to some interval J ⊂ I, such that f^{n}|_{J} is a diffeomorphism. In this case we will be intrested on the distortion of f^{n}|_{J},

dist(f^{n}|_{J}) = \frac{sup_{J} Df^{n}}{inf_{J} Df^{n}} .

This is always a number bigger than or equal to 1, we will say that it is small if it is close to 1.

2.2. Trees. We let Ω denote the set of finite sequences of non zero integers (including the empty sequence). Let Ω_{0} denote Ω without the empty sequence.

We denote σ^{+} : Ω_{0} → Ω by σ^{+}(j_{1}, ..., j_{m}) = (j_{1}, ..., j_{m−1}) and σ^{−} : Ω_{0} → Ω by σ^{−}(j_{1}, ..., j_{m}) = (j_{2}, ..., j_{m}).

2.3. Growth of functions. Let X be a class of functions g : ℕ → ℝ such that lim_{n→∞} g(n) = ∞. We say that a function f : ℕ → ℝ grows at least rate X if there exists a function g ∈ X such that f(n) ≥ g(n) for n sufficiently big. We say that it grows at rate X if there are g_{1}, g_{2} ∈ X such that g_{1}(n) ≤ f(n) ≤ g_{2}(n) for n sufficiently big. We say that f decreases with rate (at least) X if 1/f grows at rate (at least) X.

Standard classes are the following. Linear for linear functions with positive slope. Polynomial for functions g(n) = n^{k}, k > 0. Exponential for functions g(n) = e^{kn}, k > 0.

The standard torrential function T is defined recursively by T (1) = 1, T (n + 1) = 2^{T (n)}. The torrential class is the set of functions g(n) = T (max{n + k, 1}), k ∈ ℤ.

Torrential growth can be detected from recurrent estimates easily. A sufficient condition for a function which is unbounded from above to grow at least torrentially is an estimate as

f(n + 1) > e^{f(n)^{a}}

for some a > 0. Torrential growth is implied by an estimate as

e^{f(n)^{a}} < f(n + 1) < e^{f(n)^{b}}

with 0 < a < b.

2.4. Quasisymmetric maps. Let γ ≥ 1 be given. We say that a homeomorphism f : ℝ → ℝ is γ-quasisymmetric (γ-qs) if it has a quasiconformal symmetric extension to ℂ with dilatation bounded by γ. Notice that quasisymmetric maps form a group under composition and if h_{1} is γ_{1}-qs and h_{2} is γ_{2}-qs then h_{2} ∘ h_{1} is γ_{2}γ_{1}-qs.

If X ⊂ ℝ and h : X → ℝ has a γ-quasisymmetric extension to ℝ we will also say that h is γ-qs.

Let QS(γ) be the set of γ-qs maps of ℝ.

3. Real quadratic maps

If λ ∈ ℂ we let f_{λ} : ℂ → ℂ denote the (complex) quadratic map λ − z^{2}. If λ ∈ ℝ is such that −1/4 ≤ λ ≤ 2 there exists an interval I_{λ} = [−β, β] with

β = \frac{−1 − \sqrt{1 + 4λ}}{2}

such that f_{λ}(I_{λ}) ⊂ I_{λ} and f_{λ}(∂I_{λ}) ⊂ ∂λ. For such a λ, the map f = f_{λ}|_{I_{λ}} is unimodal, that is, is a self map of I_{λ} with a unique turning point.

We will keep the notation f_{λ} to refer to a quadratic map when we discuss its complex extension and f to denote a fixed quadratic map when we discuss its unimodal restriction.

3.1. The combinatorics of unimodal maps. In this subsection we fix a real quadratic map f and define some objects related to it.

3.1.1. Return maps. Given an interval I we define the first return map R_{I} : X → I where X ⊂ I is the set of points x such that there exists n > 0 with f^{n}(x) ∈ I, and R_{I}(x) = f^{n}(x) for the minimal n with this property.

3.1.2. Nice intervals. An interval I is nice if it is symmetric and the iterates of ∂I never intersect int I. Given a nice interval I we notice that the domain of the first return map R_{I} decomposes in a union of intervals I^{j}, indexed by integer numbers. We reserve the index 0 to denote the component of 0 if it exists: 0 ∈ I^{0}, in this case we say I is proper.

If I is nice, it follows that for all j ∈ ℤ, R_{I}(∂I^{j}) ⊂ ∂I. It follows that if j ≠ 0 then R_{I}|_{I^{j}} is a diffeomorphism onto I and if I is proper, R_{I}|_{I^{0}} is symmetric with a unique critical point 0. As a consequence, I^{0} is also a nice interval.

If R_{I}(0) ∈ I^{0}, we say that R_{I} is central.

3.1.3. Landing maps. Given a proper interval I we define the landing map L_{I} : X → I^{0} where X ⊂ I is the set of points x such that there exists n ≥ 0 with f^{n}(x) ∈ I^{0}, and L_{I}(x) = f^{n}(x) for the minimal n with this property. We notice that L_{I}|_{I^{0}} = id.

3.1.4. Trees. If I is a proper interval, the first return map to I naturally relates to the first landing to I^{0} in the following way.

If \underline{d} ∈ Ω, we define I\underline{d} inductively in the following way. I\underline{d} = I if \underline{d} is empty and if \underline{d} = (j_{1}, ..., j_{m}) we let I\underline{d} = (R_{I}|_{I_{j_{1}}})^{−1}(I^{σ^{−}(\underline{d})}).

We denote R\underline{d} ; I = R_{I}^{|\underline{d}|}|_{I\underline{d}} which is always a diffeomorphism over I. We denote by C\underline{d} = (R\underline{d} ; I)^{−1}(I^{0}). We notice that the domain of the

first landing map L_{I} coincides with the union of the C\underline{d}, and further-

more L_{I}|_{C\underline{d}} = R\underline{d} ; I.

Notice that this allows us to relate R_{I} to R_{I^{0}}, since R_{I^{0}} = L_{I} ∘ R_{I}.

3.1.5. Renormalization. We say that f is renormalizable if there is an interval I with and m > 1 such that f^{m}(T) ⊂ T and f^{j}(T) ∩ I = ∅ for 1 ≤ j < n. The maximal such interval is called the renormalization interval of period m, it has the property that f^{m}(∂T) ⊂ ∂T.

The set of renormalization periods of f gives an increasing (possibly empty) sequence of numbers m_{i}, each related to a unique renormalization interval T^{(i)} which form a nested sequence of intervals.

We say that f is finitely renormalizable if there is a smallest renormalization interval T^{(k)}. We say that f ∈ F if f is finitely renormalizable and 0 is recurrent but not periodic.

3.1.6. Principal nest. Let Δ_{k} denote the set of all maps f which have (at least) k renormalizations and which have an orientation reversing non-attracting periodic point of period m_{k} which we denote p_{k} (that is, f^{m_{k}}|_{T^{(k)}} has a fixed point p_{k} and Df^{n}(p_{k}) ≤ −1). In this case we denote by T_{1}^{(k)} = [−p_{k}, p_{k}]. We define by induction a (possibly finite) sequence T_{i}^{(k)}, such that T_{i+1}^{(k)} is the component of the domain of R_{T_{i}^{(k)}} containing 0. If this sequence is infinite, then either it converges to a point or to an interval.

In the former case, f has a recurrent critical point which is not periodic, and it is possible to show that f is not k +1 times renormalizable, obviously in this case we have f ∈ F, and we write F_{k} for the set of such maps. If the limit set is an interval, it is possible to show that f is k + 1 times renormalizable.

We can of course write F as a disjoint union ∪_{i=0}^{∞}F_{i}. For a map f ∈ F_{k} we refer to the sequence {T_{i}^{(k)}}_{i=1}^{∞} as the principal nest.

It is important to notice that the domain of the first return map to T_{i}^{(k)} is always dense in T_{i}^{(k)}. Moreover, outside a very special case, the return map has a hyperbolic structure.

Lemma 3.1. Assume T_{i}^{(k)} does not have a non-hyperbolic periodic orbit in its boundary. For all T_{i}^{(k)} there exists C > 0, λ > 1 such that if x, f(x), ..., f^{n−1}(x) do not belong to T_{i}^{(k)} then |Df^{n}(x)| > Cλ^{n}.

This theorem is a simple consequence of a general theorem of Guckenheimer on hyperbolicity of maps of the interval without critical points and non-hyperbolic orbits (Guckenheimer assumes negative Schwarzian derivative, so this applies directly to our case, the general case is also true by Mañé’s Theorem, see [MvS]). Notice that the existence or a non-hyperbolic periodic orbit in the boundary of T_{i}^{(k)} depends on a very special combinatorial setting, in particular, all T_{j}^{(k)} must coincide (with [−p_{k}, p_{k}]), and the k-th renormalization of f is in fact renormalizable of period 2.

This Lemma shows that the complement of the domains of the return map to T_{i}^{(k)} (outside the special case) form a regular Cantor set. This has many useful consequences (for instance, the image of a regular Cantor set by a quasisymmetric map has always 0 Lebesgue measure.

3.1.7. Lyubich’s Regular or Stochastic dichotomy. A map f ∈ F_{k} is called simple if the principal nest has only finitely many central returns, that is, there are only finitely many j such that R|_{T_{j}^{(k)}} is central.

In [L2], it was proved that almost every quadratic map is either regular or simple or infinitely renormalizable. It was then shown in [L3] that infinitely renormalizable maps have 0 Lebesgue measure.

In [ALM] it is remarked that Lyubich’s estimates actually prove quasisymmetric robustness of the set of regular or simple maps, that is, they are still typical after reparametrization.

3.1.8. Strategy. Both Theorems A and B will be proved using the same strategy.

Our strategy is to describe the dynamics of the principal nest, this is our phase analysis. From time to time, we transfer the information from the phase space to the parameter, following the description of the parapuzzle nest which we will make in the next subsection.

Due to Lyubich’s results, we can completely forget about infinitely renormalizable maps, we just have to prove the claimed estimates for almost every (after reparametrization) simple map. During our discussion, for notational reasons, we will actually consider a fixed renormalization level k, that is, we will analyse maps in Δ_{k}.

This allow us to fix some convenient notation: given g ∈ Δ_{k} we define I_{i}[g] = T_{i}^{(k)}[g], so that {I_{i}[g]} is a sequence of intervals (possibly finite). We use the notation R_{i}[g] = R_{I_{i}[g]}, L_{i}[g] = L_{I_{i}[g]} and so on. When doing phase analysis (working with fixed f) we usually drop the dependence on the map and write R_{i} for R_{i}[f].

3.2. Parameter partition. Part of our work is to transfer information from the phase space of some map f ∈ F to a neighborhood of its parameter space. This is done in the following way. We consider the first landing map to I_{i}. The domains of this map partition the interval I_{i}, the complement of this set is a hyperbolic Cantor set K_{i} = I_{i} \ ∪C\underline{d} ; i. This Cantor set persists in a small parameter neighborhood J_{i} of f, changing in a continuous way.

Along J_{i}, the first landing map is topologically the same (in a way will be clear soon). However the critical value R_{i}[g](0) moves relative to the partition (when g moves in J_{i}). This allows us to partition the parameter piece J_{i} in smaller pieces, each corresponding to a region where R_{i}(0) belongs to some fixed domain of the first landing map.

For a discussion of the next Theorem, see [AM1].

Topological Phase-Parameter relation. Let f ∈ F be exactly k times renormalizable. Then there is a sequence {J_{i}}_{i∈ℕ} of nested parameter intervals (the principal parapuzzle nest of f) with the following properties.

(1) J_{i} is the maximal interval containing f such that for all g ∈ J_{i} the interval T_{i+1}^{(k)}[g] is defined and changes in a continuous way. Since the first return map to R_{i}[g] has a central domain, so that the landing map L_{i}[g] : ∪C\underline{d} ; i[g] → I_{i}[g] is defined.

(2) L_{i}[g] is topologically the same along J_{i}: there exists homeomorphisms H_{i}[g] : I_{i} → I_{i}[g], such that H_{i}[g](C\underline{d} ; i) = C\underline{d} ; i[g]. The maps H_{i}[g] change continuously.

(3) There exists a homeomorphism Ξ_{i} : I_{i} → J_{i} such that Ξ_{i}(C\underline{d} ; i) is the set of g such that R_{i}[g](0) belongs to C\underline{d} ; i[g].

The homeomorphisms H_{i} and Ξ_{i} are not uniquely defined, it is easy to see that we can modify then inside each C\underline{d} ; i window. H_{i} and Ξ_{i} are well defined maps if restricted to K_{i}.

With this result we can define for any f ∈ F intervals J_{i}^{j} = Ξ_{i}(I_{i}^{j}) and J\underline{d} ; i = Ξ_{i}(I\underline{d} ; i). From the description we gave it immediately follows that two intervals J_{i}[f] and J_{i}[g] associated to different maps f and g are either disjoint or nested, and the same happens for intervals J_{i}^{j} or

J\underline{d} ; i.

We will concentrate on the analysis of the regularity of Ξ_{i} for the special class of simple maps f: one of the good properties of the class of simple maps is better control of the phase-parameter relation. Even for simple maps, however, the regularity of Ξ_{i} is not great: there is too much dynamical information contained in it. A solution to this problem is to forget some dynamical information. With this intent we introduce an interval which will be used to erase information.

3.2.1. Gape interval. If i > 1, we define the gape interval Ĩ_{i+1} as follows.

We have that R_{i}|_{I_{i+1}} = L_{i−1} ∘ R_{i−1} = R\underline{d} ; i−1 ∘ R_{i−1} for some \underline{d}, so

that I_{i+1} = (R_{i−1}|_{I_{i}})^{−1}(C\underline{d} ; i−1). We define the gape interval Ĩ_{i+1} =

(R_{i−1}|_{I_{i}})^{−1}(I\underline{d} ; i−1).

We notice that for each I_{i}^{j}, the gape interval Ĩ_{i+1} either contains or is disjoint from I_{i}^{j}.

3.2.2. The phase-parameter relation. As we discussed before, the dynamical information contained in Ξ_{i} is entirely given by Ξ_{i}|_{K_{i}}: a map obtained by Ξ_{i} by modification inside a C\underline{d} ; i window has still the same properties. Therefore it makes sense to ask about the regularity of Ξ_{i}|_{K_{i}}. As we anticipated before we must erase some information to obtain good results.

Let f be a simple map and τ_{i} be such that R_{i}(0) ∈ I_{i}^{τ_{i}}. We define two Cantor sets, K_{i}^{τ} = K_{i} ∩ I_{i}^{τ_{i}} which contains refined information restricted to the I_{i}^{τ_{i}} window and K̃_{i} = I_{i} \ ∪I_{i}^{j} \ Ĩ_{i+1}, which contains global information, at the cost of erasing information inside each I_{i}^{j} window and in Ĩ_{i+1}.

The following Theorem is one of the main steps in [AM1].

Phase-Parameter relation. Let f be a simple map. For all γ > 1 there exists i_{0} such that for all i > i_{0} we have Quadratic PhPa1: Ξ_{i}|_{K_{i}^{τ}} is γ-qs, Quadratic PhPa2: Ξ_{i}|K̃_{i} is γ-qs, PaPa1: H_{i}[g]|_{K_{i}} is γ-qs if g ∈ J_{i}^{τ_{i}}, PaPa2: the map H_{i}[g]|K̃_{i} is γ-qs if g ∈ J_{i}.

3.3. Reparametrization. To prove Theorems A and B, we have to show that for any γ_{0}, if h is γ_{0}-qs then almost every quadratic map after reparametrization by h have some good properties. Therefore, we fix now such γ_{0} and such a reparametrization. We keep otherwise the same notation:

Reparametrized Phase-Parameter relation. Let f be a simple map. For all λ > 1 there exists i_{0} such that for all i > i_{0} we have

PhPa1: h ∘ Ξ_{i}|_{K_{i}^{τ}} is λγ_{0}-qs, PhPa2: h ∘ Ξ_{i}|K̃_{i} is λγ_{0}-qs, PaPa1: H_{i}[g]|_{K_{i}} is λ-qs if g ∈ J_{i}^{τ_{i}}, PaPa2: the map H_{i}[g]|K̃_{i} is λ-qs if g ∈ J_{i}.

Notice that PaPa1 and PaPa2 did not change after reparametrization, since they are just Phase-Phase estimates.

All references to the parameter space from now on will assume this reparametrization.

4. Measure and capacities

4.1. Quasisymmetric maps. If X ⊂ ℝ is measurable, let’s denote |X| its Lebesgue measure. Let’s describe more metric properties of γ-qs maps.

To each γ, there exists a constant k ≥ 1 such that for all f ∈ QS(γ), for all J ⊂ I intervals,

\frac{1}{k} (\frac{|J|}{|I|})^{k} ≤ \frac{|f(J)|}{|f(I)|} ≤ (\frac{k|J|}{|I|})^{1/k} .

Furthermore lim_{γ→1} k(γ) = 1. So for each ϵ > 0 there exists γ > 1 such that k(2γ − 1) < ϵ/3.

4.2. Capacities and trees. The γ-capacity of a subset X of an interval I is defined as follows.

p_{γ}(X|I) = sup_{h∈QS(γ)} \frac{|h(X)|}{|h(I)|}.

This geometric quantity is well adapted to our context, since it is well behaved under tree decompositions of sets. In other words, if I^{j} are disjoint subintervals of I and X ⊂ ∪I_{j} then

p_{γ}(X|I) ≤ p_{γ}(∪I_{j}|I) sup p_{γ}(X ∩ I^{j}|I^{j}).

4.3. A measure theoretical lemma. Our usual procedure consists in picking a class X of maps which we show is full measure among non-regular maps and then for each map in this class we describe what happens for the principal nest, showing finally that a subset Y of X is still full measure. The first such step was to consider the class of simple maps.

We describe here our usual argument (a variation of Borel-Cantelli lemma). Assume at some point we know how to prove that almost every map belong to some set X. Let Q_{n} be a (bad) property that a map may have (usually related to the n-th stage of the principle nest). Suppose we prove that if f ∈ X then the probability that a map in J_{n}(f) has the property Q_{n} is bounded by q_{n}(f) which is shown to be summable for all f ∈ X. We then conclude that almost every map does not have property Q_{n} for n big enough.

Sometimes we also apply the same argument, proving instead that q_{n}(f) is summable where q_{n}(f) is the probability that a map in J_{n}^{τ_{n}}(f) has property Q_{n}, where τ_{n} is such that f ∈ J_{n}^{τ_{n}}(f).

In other words, we use the following lemma.

Lemma 4.1. Let X ⊂ ℝ be a measurable set such that for each x ∈ X is defined a sequence J_{i}(x) of nested intervals converging to x such that for all x_{1}, x_{2} ∈ X and any i, J_{i}(x_{1}) is either equal or disjoint to J_{i}(x_{2}). Let Q_{n} be measurable subsets of ℝ and q_{n}(x) = |Q_{n} ∩ J_{n}(x)|/|J_{n}(x)|. Let Y be the set of x in X which belong to finitely many Q_{n}. If ∑ q_{n}(x) is finite for almost any x ∈ X then |Y | = |X|.

Proof. Let Y_{n} = {x ∈ X| ∑_{k=n}^{∞} q_{k}(x) < 1/2}. It is clear that Y_{n} ⊂ Y_{n+1} and | ∪ Y_{n}| = |X|.

Let Z_{n} = {x ∈ Y_{n}||Y_{n} ∩ J_{m}(x)|/|J_{m}(x)| > 1/2, m ≥ n}. It is clear that Z_{n} ⊂ Z_{n+1} and | ∪ Z_{n}| = |X|.

Let T_{n}^{m} = ∪_{x∈Z_{n}}J_{m}(x). Let K_{n}^{m} = T_{n}^{m} ∩ Q_{m}. Of course

|K_{n}^{m}| ≤ ∫_{T_{n}^{m}} q_{m} ≤ 2 ∫_{Y_{n}} q_{m}.

And of course

∑_{m≥n} ∫_{Y_{n}} q_{m} ≤ \frac{1}{2}|Y_{n}|.

This shows that ∑_{m≥n} |K_{n}^{m}| ≤ |Y_{n}|, so almost every point in Z_{n} belong to finitely many K_{n}^{m}. We conclude then that almost every point in X belong to finitely many Q_{m}.

□

In practice, we will estimate the capacity of sets in the phase space: that is, given a map f we will obtain subsets Q̃_{n}(f) in the phase space, corresponding to bad branches of return or landing maps. We will the show that for some γ > γ_{0} we have ∑ p_{γ}(Q̃_{n}(f)|J_{n}(f)) < ∞ or ∑ p_{γ}(Q̃_{n}(f)|J_{n}^{τ_{n}}(f)) < ∞. We will then use PhPa1, PhPa2 and the measure-theoretical lemma to conclude that with total probability among non-regular maps, for all n sufficiently big, R_{n}(0) does not belong to a bad set.

From now on when we prove that almost every non-regular parameter has some property, we will just say that with total probability (without specifying) such property holds.

5. Statistics of the principal nest

Through the end of this paper we fix some constant γ > γ_{0}. We also fix b̃ ≫ 1000k(2γ − 1)^{1}000 and ã = b^{−1}, and set b = b̃^{1000b̃}, a = b^{−1}.

5.1. Decay of geometry. Let as before τ_{n} ∈ ℤ such that R_{n}(0) ∈ I_{n}^{τ_{n}}. An important parameter in our construction will be the scaling factor

c_{n} = \frac{|I_{n+1}|}{|I_{n}|}.

This variable of course changes inside each J_{n}^{τ_{n}} window, however, not by much. From PaPa1, for instance, we get that with total probability

lim_{n→∞} sup_{g_{1},g_{2}∈J_{n}^{τn}} \frac{ln(c_{n}[g_{1}])}{ln(c_{n}[g_{2}])} = 1.

This variable is by far the most important on our analysis of the statistics of return maps. We will often consider other variables (say, return times): we will show that the distribution of those variables is concentrated near some average value. Our estimates will usually give a range of values near the average, and c_{n} will play an important role. Due (among other issues) to the variability of c_{n} inside the parameter windows, the ranges we select will depend on c_{n} up to an exponent (say, in a range from a to b). From the estimate we just obtained, for big n the variability (margin of error) of c_{n} will fall confortably in such range, and we won’t elaborate more.

If x ∈ I_{n}^{j} we let j^{(n)}(x) = j and if x ∈ C\underline{d} ; n we let \underline{d}^{(n)}(x) = \underline{d}.

Lemma 5.1. With total probability, for all n sufficiently big we have

p_{2γ−1}(|\underline{d}^{(n)}(x)| ≤ k|x ∈ I_{n}) < kcã ; n,

p_{2γ−1}(|\underline{d}^{(n)}(x)| ≥ k|x ∈ I_{n}) < e^{−kcb̃ ; n}.

We also have

p_{2γ−1}(|\underline{d}^{(n)}(x)| ≤ k|x ∈ I_{n}^{τ_{n}}) < kcã ; n,

p_{2γ−1}(|\underline{d}^{(n)}(x)| ≥ k|x ∈ I_{n}^{τ_{n}}) < e^{−kcb̃ ; n}.

Proof. Let’s compute the first two estimates.

Since I_{n}^{0} is in the middle of I_{n}, we have as a simple consequence of the real Schwarz Lemma (see [L1]) that

\frac{c_{n}}{4} < \frac{|C\underline{d} ; n|}{|I\underline{d} ; n|} < 4c_{n}.

As a consequence

p_{2γ−1}(|\underline{d}^{(n)}(x)| = m|x ∈ I_{n}) < (4c_{n})^{2ã}

and we get the estimate summing up on 0 ≤ m ≤ k.

For the same reason, we get that

p_{2γ−1}(|\underline{d}^{(n)}(x) > m|x ∈ I_{n}) < 1 − (c_{n}/4)^{b̃/2}p_{2γ−1}(|\underline{d}^{(n)}(x)| ≥ m|x ∈ I_{n}).

This implies

p_{2γ−1}(|\underline{d}(x)| ≥ m|x ∈ I_{n}) ≤ (1 − (c_{n}/4)^{b̃/2})^{m}.

The second estimate follows from

(1 − (\frac{c_{n}}{4} )^{b̃/2})^{k} < (1 − cb̃ ; n)^{k} < ((1 − c_{n}^{b})^{c_{n}^{−b̃}})^{kcb̃ ; n} < e^{−kcb̃ ; n}.

The two remaining estimates are analogous. □

Transferring the result (more precisely the second pair of estimates) to the parameter in each J_{n}^{τ_{n}} window using PhPa1 we get (noticing that the measure of the complement of the set of parameters in J_{n}^{τ_{n}} such that c_{n}^{−ã/2} < s_{n} < c_{n}^{−2b̃} can be estimated by 2c_{n}^{ã/2} for n big, which is summable.

Lemma 5.2. With total probability,

c_{n}^{−ã/2} < \frac{ln(s_{n})}{ln(c_{n}^{−1})} < c_{n}^{−2b̃}.

Remark 5.1. The parameter s_{n} influences the size of c_{n+1} in a determinant way. It is easy to see (using for instance the real Schwarz Lemma, see [L1]) for instance that ln(c_{n+1}^{−1}) > Ks_{n} for some universal constant K, which in general is bounded from below (real a priori bounds), but since we have decay of geometry, actually goes to infinity.

As an easy consequence we get Corollary 5.3. With total probability,

\frac{ln(ln(c_{n+1}^{−1}))}{ln(c_{n}^{−1})} > ã/3.

In particular, c_{n} decreases at least torrentially fast.

5.2. Fine partitions. We use Cantor set K_{n} and K̃_{n} to partition the phase space. In many circumstances we are directly concerned with intervals of this partition. However, sometimes we just want to exclude an interval of given size (usually around 0). This size does not usually correspond to a union of gaps, so we instead should substitute in applications an interval which is union of gaps, with approximately the given size. The degree of relative approximation will always be torrentially good (in n), so we usually won’t elaborate on this. In this section we just give some results which will imply that the partition induced by the Cantor sets are fine enough to allow torrentially good approximations.

The following Lemma summarizes the situation. The proof is based on estimates of distortion using the real Schwarz Lemma and the Koebe Principle (see [L1]) and is very simple, so we skip it.

Lemma 5.4. The following estimates hold.

• |I_{n}^{j}|/|I_{n}| = O(\sqrt{c_{n−2}}),

• |I\underline{d} ; n|/|I_{n}^{σ^{+}(\underline{d})}| = O(\sqrt{c_{n−2}}).

• |C\underline{d} ; n|/|I\underline{d} ; n| ≈ c_{n},

• |Ĩ_{n+1}|/|I_{n}| = O(e^{1−s_{n−1}}).

In other words, distances in I_{n} can be measured with precision \sqrt{c_{n−2}}|I_{n}| in the partition induced by K̃_{n}, due to first and last items (since e^{1−s_{n−1}} = O(c_{n−1})).

Distances can be measured much more precisely with respect to the partition induced by K_{n}, in fact we have good precision in each I\underline{d} ; n scale. In other words, inside I\underline{d} ; n, the central gap C\underline{d} ; n is of size O(c_{n}|I\underline{d} ; n|) (third item) and the other gaps have size O(\sqrt{c_{n−2}}|C\underline{d} ; n|) (second and third item).

Remark 5.2. We need to consider intervals which are union of gaps due to our phrasing of the phase-parameter relation, which only gives information about such gaps. However, this is not absolutely necessary, and we could have proceeded in a different way: our proof of the phaseparameter relation actually shows that there is a holonomy map with good qs estimates between phase and parameter intervals (and not only Cantor sets). While this map is not canonical, the fact that it is a holonomy map for a motion with good phase-phase estimates would allow our proofs to work.

5.3. Initial estimates on distortion. To deal with the distortion control we need some preliminary known results. We won’t get too much in details here, those estimates are related to the estimates on gaps of Cantor sets and the Koebe Principle, and can be concluded easily.

Proposition 5.5. For any j, if R_{n}|_{I_{n}^{j}} = f^{k} then dist(f^{k−1}|_{f(I_{n}^{j})}) = 1 + O(c_{n−1}).

For any \underline{d}, dist(R_{n}^{σ^{+}(\underline{d})}|_{I\underline{d} ; n}) = 1 + O(\sqrt{c_{n−2}}).

We will use the following immediate consequence for the decomposition of certain branches.

Lemma 5.6. With total probability, • R_{n}|_{I_{n}^{0}} = ϕ ∘ f where ϕ has torrentially small distortion,

• R\underline{d} ; n = ϕ_{2} ∘ f ∘ ϕ_{1} where ϕ_{2} and ϕ_{1} have torrentially small distortion and ϕ_{1} = R_{n}^{σ^{+}(\underline{d})}.

5.4. Estimating derivatives.

Lemma 5.7. With total probability,

d(R_{n}(0), ∂I_{n} ∪ {0}) < |I_{n}|n^{−b̃}.

In particular R_{n}(0) ∉ Ĩ_{n+1} for all n large enough.

Proof. This is a simple consequence of PhPa2, using that n^{−\sqrt{b̃}} is summable. □

From now on we suppose that f satisfies the conditions of the above lemma.

Lemma 5.8. With total probability, for n big enough and j ≠ 0

dist(f|_{I_{n}^{j}}) ≤ nb̃.

Proof. Notice that the gaps of the Cantor sets K_{n} inside I\underline{d} ; n which are different from C\underline{d} ; n are torrentially (in n) smaller then C\underline{d} ; n.

Denote by P \underline{d} ; n a |C\underline{d} ; n|/nb̃ neighborhood of C\underline{d} ; n.

It is clear that the image by a γ-qs homeomorphism of P \underline{d} ; n \ C\underline{d} ; n is of

order n^{−\sqrt{b̃}}|C\underline{d} ; n| (this is true both on I_{n} or restricting to I_{n}^{τ_{n}}). Since C\underline{d} ; n are disjoint,

p_{γ}(I_{n}^{τ_{n}} ∩ ∪(P \underline{d} ; n \ C\underline{d} ; n)|I_{n}^{τ_{n}})

is summable.

Transferring the result to the parameter using PhPa1 we see that the critical point will never be in a n^{−b̃} neighborhood of any I_{n+1}^{j} with j ≠ 0. □

Applying Lemma 5.6 we get

Lemma 5.9. With total probability, for n big enough and for all \underline{d}

dist(R\underline{d} ; n) ≤ nb̃.

In particular, for n big enough, sup_{\underline{d}≠∅} dist(R\underline{d} ; n) is bounded by 2^{n} and inf |R′_{n}|_{∪_{j≠0}I_{n}^{j}}| > 2.

Remark 5.3. Lemma 5.8 has also an application for approximation of intervals. The result implies that if I_{n}^{j} = (c, d) and j ≠ 0, we have 1/2^{n} < d/c < 2^{n}. As a consequence, for all symmetric interval I_{n+1} ⊂ X ⊂ I_{n}, there exists a symmetric interval X ⊂ X̃, which is union of I_{n}^{j} and such that |X̃|/|X| < 2^{n} (approximation by union of C\underline{d} ; n, with |tildeX|/|X| torrentially close to 1, follows more easily from the discussion on fine partitions).

We will also need to estimate derivatives of iterates of f, and not only of return branches.

Lemma 5.10. With total probability, if n is sufficiently big and if x ∈ ∪_{j≠0}I_{n}^{j} and R_{n}|_{I_{n}^{j}} = f^{r}, then for 1 ≤ k ≤ r, |(Df^{k}(x))| > |x|c_{n−1}^{3}.

Proof. Let n_{0} be minimum such that if n ≥ n_{0} then

inf |R′_{n}|_{∪_{j≠0}I_{n}^{j}}| > 1.

From hyperbolicity of f, from Lemma 3.1, restricted to the complement of I_{n_{0}}, there exists a constant C > 0 such that if f^{s}(x) ∉ I_{n_{0}}^{0}, r ≤ s ≤ k then |Df^{k−r}(f^{r}(x))| > C.

If k = r, the result follows from Lemma 5.9. We assume k < r.

Let’s define d(s), 1 ≤ s ≤ k such that f^{s}(x) ∈ I_{d(s)} \ I_{d(s)}^{0}. Let m(s) = max_{s≤t≤k} d(s). Let k_{0} = 0. Supposing k_{j} < k define k_{j+1} as k_{j+1} = max{k_{j} < s ≤ k|d(s) = m_{s}}.

If d(k_{1}) = n, k = r and from Lemma 5.9 |Df^{k}(x)| > 1.

We have then a sequence 0 = k_{0} < k_{1} < ... < k_{l} = k. Let k_{i} be maximal with d(k_{i}) ≥ n_{0}. We have

|(Df^{k−k_{i}}(f^{k_{i}}(x))| > C|f^{k_{i}}(x)|

and for 1 < j ≤ i,

|Df^{k_{j}−k_{j−1}}(f^{k_{j−1}}(x))| > |Df(f^{k_{j−1}})(x)| > c_{d(k_{j−1})}.

We also have |Df^{k_{1}}(x)| ≥ |x| if i ≥ 1. If i = 0, |Df^{k}(x)| ≤ C|x|. Combining it all we get

|Df^{k}(x)| > C|x| ∏_{1≤j≤i} c_{d(k_{j})} > |x|c_{n−1}^{3}.

□

6. Sequence of quasisymmetric constants and trees

6.1. Preliminary estimates. We will need from now on to consider not only γ-capacities with γ fixed, but a sequence γ_{n} converging to γ.

We define the sequences ρ_{n} = (n+1)/n and ρ̃_{n} = (2n+3)/(2n+1), so that ρ_{n} > ρ̃_{n} > ρ_{n+1} and lim ρ_{n} = 1. We define the sequence γ_{n} = γρ_{n} and an intermediate sequence γ̃_{n} = γρ̃_{n}.

As we know, renormalization proccess has two phases, first R_{n} to L_{n} and then L_{n} to R_{n+1}. The following remarks shows why it is useful to consider the sequence of quasisymmetric constants due to losses related to distortion.

Remark 6.1. Let S be an interval contained in I\underline{d} ; n. Using Lemma 5.6 we have R\underline{d} ; n|_{S} = ψ_{2}∘f∘ψ_{1}, where the distortion of ψ_{2} and ψ_{1} are torrentially small and ψ_{1}(S) is contained in some I_{n}^{j}, j ≠ 0. If S is contained in I_{n}^{0} we may as well write R_{n}|_{S} = ϕ ∘ f, with dist(ϕ) torrentially small.

In either case, if we decompose S in 2km intervals S_{i} of equal length, where k is the distortion of either R\underline{d}|_{S} or R|_{S} and m is subtorrentially big (say m < 2^{n}), the distortion obtained restricting to any interval S_{i} will be bounded by 1 + 1/m.

Remark 6.2. We have the following estimate for the effect of the pullback of a subset of I_{n} by the central branch R_{n}|_{I_{n}^{0}}. With total probability, for all n sufficiently big, if X ⊂ I_{n} satisfies

pγ̃_{n}(X|I_{n}) < δ < n^{−b} ≤ n^{−b̃^{10b̃}}

then

p_{γ_{n+1}}((R_{n}|_{I_{n+1}})^{−1}(X)|I_{n}) < δ^{ã^{3}} < δ^{−a}.

Indeed, let V be a δã|I_{n+1}| neighborhood of 0. Then R_{n}|_{I_{n+1}\V} has distortion bounded by 2δã.

Let W ⊂ I_{n} an interval of size λ|I_{n}|. Of course

pγ̃_{n}(X ∩ W |W) < δλ^{−b̃}.

Let’s decompose each side of I_{n+1} \ V as a union of nb̃δ^{−ã} intervals. Let W be such an interval. From Lemma 5.7, it is clear that the image of W covers at least δ^{2ã}n^{−3b̃}|I_{n}|. It is clear then that

pγ̃_{n}(X ∩ R_{n}(W)|R_{n}(W)) < δ(δ^{2ã}n^{−3b̃})^{−b̃} = δ^{3}n^{−3b̃^{2}}.

So we conclude (since the distortion of R_{n}|_{W} is of order 1 + n^{−3}) that

p_{γ_{n+1}}((R_{n}|_{I_{n+1}})^{−1}(X) ∩ W |W) < δ^{3}n^{−3b̃^{2}}

(we use the fact that the composition of a γ_{n+1}-qs map with a map with small distortion in γ̃_{n}-qs). Since

p_{γ_{n+1}}(V |I_{n+1}) < 2δ^{ã^{2}},

we get the required estimate.

6.2. More on trees. Let’s see an application of the above remarks.

Lemma 6.1. With total probability, for all n sufficiently big

pγ̃_{n}((R\underline{d} ; n)^{−1}(X)|I\underline{d} ; n) < 2^{n}p_{γ_{n}}(X|I_{n}).

Proof. Decompose I\underline{d} ; n in n^{ln(n)} intervals of equal length and apply Remark 6.1. □

By induction we get

Lemma 6.2. With total probability, for n is big enough, if X_{1}, ..., X_{m} ⊂ ℤ \ {0}

pγ̃_{n}(\underline{d}^{(n)}(x) = (j_{1}, ..., j_{m}), j_{i} ∈ X_{i}|x ∈ I_{n})

≤ 2^{mn} ∏_{i=1}^{m} p_{γ_{n}}(j^{(n)}(x) ∈ X_{i}|x ∈ I_{n}).

There is also a variation fixing the start of the sequence.

Lemma 6.3. With total probability, for n is big enough, if X_{1}, ..., X_{m} ⊂ ℤ \ \{0\}, and if |\underline{d}| = (j_{1}, ..., j_{k}) we have

pγ̃_{n}(\underline{d}^{(n)}(x) = (j_{1}, ..., j_{m+k}), j_{i+k} ∈ X_{i}|x ∈ I\underline{d} ; n)

≤ 2^{mn} ∏_{i=1}^{m} p_{γ_{n}}(j^{(n)}(x) ∈ X_{i}|x ∈ I_{n}).

In particular, with \underline{d} = (τ_{n}),

pγ̃_{n}(\underline{d}^{(n)}(x) = (τ_{n}, j_{1}, ..., j_{m}), j_{i} ∈ X_{i}|x ∈ I_{n}^{τ_{n}})

≤ 2^{mn} ∏_{i=1}^{m} p_{γ_{n}}(j^{(n)}(x) ∈ X_{i}|x ∈ I_{n}).

The last part of the above Lemma will be often necessary in order to apply PhPa1.

Sometimes we are more interested in the case where the X_{i} are all equal.

Let Q ⊂ ℤ \ \{0\}. Let Q(m, k) denote the set of \underline{d} = (j_{1}, ..., j_{m}) such that #{i|j_{i} ∈ Q} ∩ {1, ..., m} ≥ k.

Define q_{n}(m, k) = pγ̃_{n}(∪_{\underline{d}∈Q(m,k)}I\underline{d} ; n|I_{n}).

Let q_{n} = p_{γ_{n}}(∪_{j∈Q}I_{n}^{j}|I_{n}).

Lemma 6.4. With total probability, for n large enough,

q_{n}(m, k) ≤ \binom{m}{k}(2^{n}q_{n})^{k}.

Proof. We have the following recursive estimates for q_{n}(m, k): q_{n}(1, 0) = 1, q_{n}(1, 1) ≤ q_{n} ≤ 2^{n}q_{n}. q_{n}(m + 1, 0) = 1, q_{n}(m + 1, k + 1) ≤ q_{n}(m, k + 1) + 2^{n}q_{n}q_{n}(m, k). Indeed, if (j_{1}, ..., j_{m+1}) ∈ Q(m + 1, k + 1) then either (j_{1}, ..., j_{m}) ∈

Q(m, k + 1) or (j_{1}, ..., j_{m}) ∈ Q(m, k) and j_{m+1} ∈ Q. By the estimate 2^{n} on the distortion of all non central branches, we get our result. □

We recall that by Stirling Formula,

\binom{m}{qm} < \frac{m^{qm}}{(qm)!} < (\frac{3}{q})^{qm} .

So we can get the following estimate. For q ≥ q_{n},

q_{n}(m, (6 · 2^{n})qm) < (\frac{1}{2})^{(6·2^{n}qm)} .

It is also used in the following form. If q^{−1} > 6 · 2^{n} (it is usually the case, since q_{n} will be torrentially small)

∑_{k>q^{−2}} q_{n}(k, (6 · 2^{n})qk) < 2^{−n}q^{−1} (\frac{1}{2})^{(6·2^{n})q^{−1}} .

7. Estimates on time

Our aim in this section is to estimate the distribution of return times to I_{n}: they are concentrated around c_{n−1}^{−1} up to an exponent in some range given by a and b.

The basic estimate is a large deviation estimate and is proven in the next subsection (Corollary 7.4) and states that for k ≥ 1 the set of branches with time larger then kc_{n}^{−4b̃} has capacity less then e^{−k}.

7.1. A Large Deviation lemma for times. Let r_{n}(j) be such that R_{n}|_{I_{n}^{j}} = f^{r_{n}(j)}. We will also use the notation r_{n}(x) = r_{n}(j^{(n)}(x)), the n-th return time of x (there should be no confusion for the reader, since we consistently use j for an integer index and x for a point in the phase space.

Let

A_{n}(k) = p_{γ_{n}}(r_{n}(x) ≥ k|x ∈ I_{n})

Since f restricted to the complement of I_{n+1} is hyperbolic, from Lemma 3.1, it is clear that A_{n}(k) decays exponentially with k.

Let ζ_{n} be the maximum ζ < c_{n−1} such that for all k > ζ^{−1} we have

A_{n}(k) ≤ e^{−ζk}

and finally let α_{n} = min_{1≤m≤n} ζ_{m}.

Our main result in this section is to estimate α_{n}. We will show that with total probability, for n big we have α_{n+1} ≥ c_{n}^{4b̃}. For this we will have to do a simultaneous estimate for landing times, which we define now.

Let l_{n}(\underline{d}) be such that L_{n}|_{I\underline{d} ; n} = f^{l_{n}(\underline{d})}. We will also use the notation l_{n}(x) = l_{n}(\underline{d}^{(n)}(x)).

Let

B_{n}(k) = pγ̃_{n}(l_{n}(x) > k|x ∈ I_{n}).

Lemma 7.1. If k > c_{n}^{−2b̃}α_{n}^{−2b̃} then

B_{n}(k) < e^{−c_{n}^{2b̃}α_{n}^{2b̃}k}.

Proof. Let k > c_{n}^{−2b̃}α_{n}^{−2b̃} be fixed. Let m_{0} = α_{n}^{2b̃}k.

Notice that by Lemma 5.1

pγ̃_{n}(|\underline{d}^{(n)}(x)| ≥ m_{0}|x ∈ I_{n}) ≤ e^{−cb̃ ; nα_{n}^{2b̃}k}.

Fix now m < m_{0}. Let’s estimate

pγ̃_{n}(|\underline{d}^{(n)}(x)| = m, l_{n}(x) > k|x ∈ I_{n}).

For each \underline{d} = (j_{1}, ..., j_{m}) we can associate a sequence of m positive integers r_{i} such that r_{i} ≤ r_{n}(j_{i}) and ∑ r_{i} = k. The average value of r_{i} is at least k/m so we conclude that

∑_{r_{i}≥k/2m} r_{i} > k/2.

Recall also that

\frac{k}{2m} > \frac{1}{(2α_{n}^{2b̃})} > α_{n}^{−1}.

20 ARTUR AVILA AND CARLOS GUSTAVO MOREIRA

Given a sequence of m positive integers r_{i} as above we can do the following estimate using Lemma 6.2

pγ̃_{n}(\underline{d}^{(n)}(x) = (j_{1}, ..., j_{m}), r_{n}(j_{i}) > r_{i}|x ∈ I_{n})

≤ 2^{mn} ∏_{j=1}^{m} p_{γ_{n}}(r_{n}(x) ≥ r_{j}|x ∈ I_{n})

≤ 2^{mn} ∏_{r_{j}≥α_{n}^{−1}} p_{γ_{n}}(r_{n}(x) ≥ r_{j}|x ∈ I_{n})

≤ 2^{mn} ∏_{r_{j}≥k/2m} e^{−α_{n}r_{j}}

≤ 2^{mn}e^{−α_{n}k/2}.

The number of sequences of m positive integers r_{i} with sum k is

\binom{k + m − 1}{m − 1} ≤ \frac{1}{(m − 1)!}(k + m − 1)^{m−1}

≤ \frac{1}{m!}(k + m)^{m} ≤ (\frac{2ek}{m} )^{m} .

Notice that

2^{mn} (\frac{2ek}{m} )^{m} ≤ (\frac{2^{n+3}k}{m} )^{\frac{m}{k2^{n+3}} k2^{n+3}}

≤ (\frac{2^{n+3}k}{m_{0}} )^{\frac{m_{0}}{k2^{n+3}} k2^{n+3}} (since x^{1/x} is decreases for x > e)

≤ (\frac{2^{n+3}}{α_{n}^{2b̃}} )^{m_{0}} ≤ e^{αb̃ ; nk}.

So we can finally estimate

pγ̃_{n}(|\underline{d}^{(n)}(x)| = m, l_{n}(x) ≥ k|x ∈ I_{n}) ≤ 2^{mn} (\frac{2ek}{m} )^{m} e^{−α_{n}k/2}
< e^{(α_{n}^{b̃−1}−1/2)α_{n}k}.

Summing up on m we get

pγ̃_{n}(|\underline{d}^{(n)}(x)| < m_{0}, l_{n}(x) ≥ k|x ∈ I_{n})

≤ m_{0}e^{(α_{n}^{b̃−1}−1/2)α_{n}k}

< e^{(2α_{n}^{b̃−1}−1/2)α_{n}k}(since\frac{ln(m_{0})}{k} ≤ \frac{ln(k)}{k} ≤ αb̃ ; n)

≤ e^{−α_{n}k/3}.

As a direct consequence we get

B_{n}(k) < e^{−α_{n}k/3} + e^{−cb̃ ; nα_{n}^{2b̃}k} < e^{−c_{n}^{2b̃}α_{n}^{2b̃}k}.

□

Lemma 7.2. With total probability, for n large enough,

v_{n+1} < c_{n}^{−3b̃}α_{n}^{−3b̃}.

Proof. Let \underline{d} such that R_{n}(0) ∈ C\underline{d} ; n.

Using Lemma 7.1, with total probability, for n large enough, l_{n}(\underline{d}) < nα_{n}^{−2b̃}c_{n}^{−2b̃}, and v_{n+1} < v_{n}+nα_{n}^{−2b̃}c_{n}^{−2b̃}. Using torrential (and monotonic) decay of α_{n}c_{n} we get for n large enough v_{n+1} < c_{n}^{−3b̃}α_{n}^{−3b̃}/2. □

Lemma 7.3. With total probability, for n large enough,

α_{n+1} ≥ min{α_{n}^{4b̃}, c_{n}^{4b̃}}.

Proof. Let k ≥ max{α_{n}^{−4b̃}, c_{n}^{−4b̃}}. From Lemma 7.2 one immediately sees that if r_{n+1}(j) ≥ k then R_{n}(I_{n+1}^{j})) is contained on some C\underline{d} ; n with l_{n}(\underline{d}) ≥ k/2 ≥ α_{n}^{−2b̃}c_{n}^{−2b̃}.

Applying Lemma 7.1 we have B_{n}(k/2) < e^{−α_{n}^{2b̃}c_{n}^{2b̃}k/2}.

Applying Remark 6.2 we get

A_{n+1}(k) < e^{−kα_{n}^{2b̃}c_{n}^{2b̃}/ã^{3}} < e^{−k min{α_{n}^{4b̃},c_{n}^{4b̃}}}.
□

Since c_{n} decreases torrentially, we get

Corollary 7.4. With total probability, for n large enough α_{n+1} ≥ c_{n}^{4b̃}.

Remark 7.1. In particular, using Lemma 7.2, for n big, v_{n} < c_{n−1}^{−6b̃}.

7.2. Consequences. The Lemma bellow is just a convenient way to summarize our results on the distribution of times. We also take the opportunity to state them in terms of constants a and b.

Lemma 7.5. With total probability, for all n sufficiently large we have (1) pγ̃_{n}(l_{n}(x) < c_{n}^{−s}|x ∈ I_{n}) < c_{n}^{\sqrt{ã}−s} < c_{n}^{a−s}, with s > 0,

(2) pγ̃_{n}(l_{n}(x) < c_{n}^{−s}|x ∈ I_{n}^{τ_{n}}) < c_{n}^{a−s}, with s > 0,

(3) pγ̃_{n}(l_{n}(x) > c_{n}^{−s}|x ∈ I_{n}) < e^{−c_{n}^{b−s}}, with s > b,

(4) pγ̃_{n}(l_{n}(x) > c_{n}^{−s}|x ∈ I_{n}^{τ_{n}}) < e^{−c_{n}^{b−s}}, with s > b,

(5) p_{γ_{n}}(r_{n}(x) < c_{n−1}^{−s}|x ∈ I_{n}) < c_{n−1}^{\sqrt{ã}−s} < c_{n−1}^{a−s}, with s > 0,

(6) p_{γ_{n}}(r_{n}(x) > c_{n−1}^{−s}|x ∈ I_{n}) < e^{−c_{n−1}^{\sqrt{b}−s}} < e^{−c_{n−1}^{b−s}} with s > b.

(7) c_{n−1}^{−a} < r_{n}(τ_{n}) < c_{n−1}^{−b}.

(8) c_{n−1}^{−a} < v_{n} < c_{n−1}^{−b}.

Proof. The first and third estimates are contained in Lemma 5.1 (after noticing l_{n}(x) ≥ |\underline{d}^{(n)}(x)|) and the second follows from Lemma 7.1. 6 is contained in Corollary 7.4, while 5 can be obtained by pulling back estimate 1 (using Remark 6.2). 5 and 6 imply 7 by PhPa2. In view of 7, 4 follows from the proof of Lemma 7.1 (7 is needed to avoid r_{n}(τ_{n}) to be too big). 8 is easily obtained from Lemma 5.2 and Remark 7.1. □

Remark 7.2. It is clear (using Lemma 5.7) that 8 in Lemma 7.5 implies that with total probability, for all n big enough, c_{n+1} ≥ 4^{−c_{n}^{−b}}. This implies together with Corollary 5.3 that

a < \frac{ln(ln(c_{n}^{−1}))}{ln(c_{n−1}^{−1})} < b,

so c_{n}^{−1} grows torrentially.

8. Dealing with hyperbolicity

In this section we show by an inductive proccess that the great majority of branches are reasonably hyperbolic. In order to do that, in the following subsection, we define some classes of branches with ‘good’ distribution of times and which are not too close to the critical point. The definition of ‘good’ distribution of times has an inductive component: they are composition of many ‘good’ branches of the previous level. The fact that most branches are good is related to the validity of some kind of Law of Large Numbers estimate.

8.1. Some kinds of branches and landings.

8.1.1. Standard landings. We define the set of standard landings at time n, LS(n) ⊂ Ω as the set of all \underline{d} = (j_{1}, ..., j_{m}) satisfying the following.

LS1: (m not too small or large) c_{n}^{−a/2} < m < c_{n}^{−2b}, LS2: (No very large times) r_{n}(j_{i}) < c_{n−1}^{−3b} for all i. LS3: (Short times are sparse in not too small initial segments)

For c_{n−1}^{−2b} ≤ k ≤ m

#{r_{n}(j_{i}) < c_{n−1}^{−a/2}} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{a/2}k.

We also define the set of fast landings at time n, LF (n) ⊂ Ω by the following conditions

LF1: (m small) m < c_{n}^{−a/2}.

LS2: (no very large times) r_{n}(j_{i}) < c_{n−1}^{−3b} for all i.

Lemma 8.1. With total probability, for all n sufficiently big, (1) pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LS(n)|x ∈ I_{n}) < c_{n}^{a/3}/2,

(2) pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LS(n) ∪ LF (n)|x ∈ I_{n}) < c_{n}^{n^{2}}/2,

(3) pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LS(n)|x ∈ I_{n}^{τ_{n}}) < c_{n}^{a/3}/2,

(4) pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LS(n) ∪ LF (n)|x ∈ I_{n}^{τ_{n}}) < c_{n}^{n^{2}}/2.

Proof. The proof is immediate from our time estimates, which can be applyed directly to estimate the losses of LS1, LS2, LF1, or under large deviation form for LS3, following §6.2. Let’s estimate the complement of the sets which do not satisfy some properties:

LS1: c_{n−1}^{2a/5},

LS1+LF1: e^{c_{n}^{b}} ≪ c_{n}^{n^{2}},
LS2: e^{c_{n−1}^{−2b}} ≪ c_{n}^{n^{2}},
LS3: let q = (6 · 2^{n})c_{n−1}^{a/2}, we get

∑_{k>c_{n−1}^{−2b}} (\frac{1}{2})^{qk} ≪ c_{n}^{n^{2}}.

This gives immediately 1 and 2. For 3 and 4 the same estimates hold, using (for LS2) the estimate on r_{n}(τ_{n}). □

8.1.2. Very good returns, bad returns and excellent landings. Define the set of very good returns, V G(n_{0}, n) ⊂ ℤ \ {0}, n_{0}, n ∈ ℕ and the set of bad returns, B(n_{0}, n) ⊂ ℤ \ {0}, n_{0}, n ∈ ℕ, n ≥ n_{0} by induction as follows. We let V G(n_{0}, n_{0}) = ℤ \ {0},B(n_{0}, n_{0}) = ∅ and supposing V G(n_{0}, n) and B(n_{0}, n) defined, define the set of excellent landings LE(n_{0}, n) ⊂ LS(n) satisfying the following extra assumptions.

LE1: (Not very good moments are sparse in not too small initial segments) For all c_{n−1}^{−2b} < k ≤ m

#{i|j_{i} ∉ V G(n_{0}, n)} ∩ {1, ..., k} < 6 · 2^{n}c_{n−1}^{a^{2}}k,

LE2: (Bad moments are sparse in not too small initial segments) For all c_{n}^{−1/n} < k ≤ m

#{i|j_{i} ∉ B(n_{0}, n)} ∩ {1, ..., k} < 6 · 2^{n}c_{n−1}^{n}k,

We define V G(n_{0}, n + 1) as the set of j such that R_{n}(I_{n+1}^{j}) = C\underline{d} ; n with \underline{d} ∈ LE(n_{0}, n) and the extra condition.

VG: (distant from 0) The distance of I_{n+1}^{j} to 0 is bigger than

c_{n}^{n^{2}}|I_{n+1}|. And we define B(n_{0}, n + 1) as the set of j ∉ V G(n_{0}, n + 1) such that

R_{n}(I_{n+1}^{j}) = C\underline{d} ; n with \underline{d} ∉ LF (n).

Lemma 8.2. With total probability, for all n_{0} sufficiently big, (1) p_{γ_{n}}(j^{(n)}(x) ∉ V G(n_{0}, n)|x ∈ I_{n}) < c_{n−1}^{a^{2}},

(2) p_{γ_{n}}(j^{(n)}(x) ∈ B(n_{0}, n)|x ∈ I_{n}) < 2c_{n−1}^{2n},

(3) pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LE(n_{0}, n)|x ∈ I_{n}) < c_{n}^{2a/5},

(4) pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LE(n_{0}, n) ∪ LF (n)|x ∈ I_{n}) < c_{n}^{bn},

(5) pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LE(n_{0}, n)|x ∈ I_{n}^{τ_{n}}) < c_{n}^{2a/5}.

Proof. First notice that the validity for a given value of n of 1 and 2 implies 3, 4 and 5, using the large deviation technique of §6.2. More precisely, we estimate the complement

LE1: let q = 6 · 2^{n}c_{n−1}^{a^{2}},

∑_{k>c_{n−1}^{−2b}} (\frac{1}{2})^{qk} ≪ c_{n}^{n^{2}},

LE2: let q = 6 · 2^{n}c_{n−1}^{n},

∑_{k>c_{n}^{−1/n}} (\frac{1}{2})^{qk} ≪ c_{n}^{n^{2}}.

The validity of 3 and 4 imply 1 and 2 for n + 1 by pulling back, using Remark 6.2. The proof then follows by induction (the first step is trivial). □

Using PhPa2 we get

Lemma 8.3. With total probability, for all n_{0} big enough, for all n big enough, τ_{n} ∈ V G(n_{0}, n).

We also have the following trivial estimate for a very good return time

Lemma 8.4. With total probability, for all n_{0} big enough and for all n ≥ n_{0}, if j ∈ V G(n_{0}, n + 1) then

m < r_{n+1}(j) < mc_{n−1}^{−4b},

where as usual, m is such that R_{n}(I_{n+1}^{j}) = C\underline{d} ; n and \underline{d} = (j_{1}, ..., j_{m}).

Proof. The estimate from below is obvious, the estimate from above follows from LS2 and the estimate on v_{n}. □

And the following estimate for returns which are not very good or bad

Lemma 8.5. With total probability for all n_{0} sufficiently big, if n > n_{0}, if j ∉ V G(n_{0}, n) ∪ B(n_{0}, n) then r_{n}(j) < c_{n−1}^{−a/2}c_{n−2}^{−4b}.

Proof. If j ∉ V G(n_{0}, n)∪B(n_{0}, n) then R_{n−1}(I_{n}^{j}) ∈ LF (n_{0}, n−1). The estimate follows since a branch in LF (n_{0}, n − 1) has time bounded by c_{n−1}^{a/2}c_{n−2}^{−3b} and v_{n−1} < c_{n−2}^{−b}. □

Let j ∈ V G(n_{0}, n + 1). We can write R_{n+1}|_{I_{n+1}^{j}} = f^{r_{n+1}(j)}, that is, a big iterate of f. One may consider which proportion of those iterates belong to very good branches of the previous level. More generally, we can truncate the return R_{n+1}, that is, we may consider k < r_{n+1}(j) and ask which proportion of iterates up to k belong to very good branches.

Lemma 8.6. With total probability, for all n_{0} big enough and for all n ≥ n_{0}, the following holds.

Let j ∈ V G(n_{0}, n+1), as usual let R_{n}(I_{n+1}^{j}) = I\underline{d} ; n and \underline{d} = (j_{1}, ..., j_{m}).

Let m_{k} be biggest possible with

v_{n} + ∑_{j=1}^{m_{k}} r_{n}(j_{i}) ≤ k

(the amount of full returns to level n before time k) and let

β_{k} = ∑_{{1≤i≤m_{k}, \atop j_{i}∈V G(n_{0},n)}} r_{n}(j_{i}).

(the total time spent in full returns to level n which are very good before time k) Then 1 − β_{k}/k < c_{n−1}^{a^{2}/3} if k > c_{n}^{−2/n}.

Proof. Let’s estimate first the time i_{k} which is not spent on full returns:

i_{k} = k − ∑_{j=1}^{m_{k}} r_{n}(j_{i}).

This corresponds exactly to v_{n} plus some incomplete part of the return j_{m_{k+1}}. This part can be bounded by c_{n−1}^{−b} + c_{n−1}^{−3b} (use the estimate of v_{n} and LS2 to estimate the incomplete part).

Using LS2 we conclude now that

m_{k} > (k − c_{n−1}^{−b})c_{n−1}^{3b} > c_{n}^{−1/n}

so m_{k} is not too small.

Let’s now estimate the contribution h_{k} from bad full returns j_{i}. The number of such returns must be less than c_{n−1}^{n/2}m_{k}, using LS2 their total time is at most c_{n−1}^{(n/2)−3b}m_{k} < m_{k}.

The non very good full returns on the other hand can be estimated by LE1 (given the estimate on m_{k}), they are at most c_{n−1}^{a^{2}}m_{k}. So we can estimate the total time l_{k} of non very good full returns with time less then c_{n−1}^{−a/2}c_{n−2}^{−4b} by

c_{n−1}^{a^{2}}c_{n−1}^{−a/2}c_{n−2}^{−4b}m_{k},

while β_{k} can be estimated from below by

(1 − c_{n−1}^{a/4})c_{n−1}^{−a/2}m_{k}.

It is easy to see then that i_{k}/β_{k} ≪ c_{n−1}^{a/5}, h_{k}/β_{k} ≪ c_{n−1}^{a/5}. We also have

l_{k}/β_{k} < 2c_{n−1}^{a^{2}/2}.

So (i_{k} + h_{k} + l_{k})/β_{k} is less then c_{n−1}^{a^{2}/3}. Since i_{k} + h_{k} + l_{k} + β_{k} = k we have 1 − β_{k}/k < (i_{k} + h_{k} + l_{k})/β_{k}. □

8.1.3. Cool landings. Let’s define the set of cool landings LC(n_{0}, n) ⊂ Ω, n_{0}, n ∈ ℕ, n ≥ n_{0} as the set of all \underline{d} = (j_{1}, ..., j_{m}) in LE(n_{0}, n) satisfying.

LC1: (Starts very good) j_{i} ∈ V G(n_{0}, n), 1 ≤ i ≤ c_{n−1}^{−a^{2}/2}.

LC2: (Short times are sparse in not too small initial segments) For c_{n−1}^{−a/2} ≤ k ≤ m

#{r_{n}(j_{i}) < c_{n−1}^{−a/2}} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{a/3}k,

LC3: (Not very good moments are sparse in not too small initial segments) For all c_{n−1}^{−a^{2}/4} < k ≤ m

#{i|j_{i} ∉ V G(n_{0}, n)} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{a^{2}}k,

LC4: (Bad times are sparse in not too small initial segments) For c_{n−1}^{−n/3} ≤ k ≤ m

#{i|j_{i} ∈ B(n_{0}, n)} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{n/6}k,

LC5: (Starts with no bad times) j_{i} ∉ B(n_{0}, n), 1 ≤ i ≤ c_{n−1}^{−n/2}. As usual we obtain

Lemma 8.7. With total probability, for all n_{0} sufficiently big and all n ≥ n_{0},

pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LC(n_{0}, n)|x ∈ I_{n}) < c_{n−1}^{a^{2}/3}

and for all n big enough

pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LC(n_{0}, n)|x ∈ I_{n}^{τ_{n}}) < c_{n−1}^{a^{2}/3}.

Proof. LC1: 2^{n}c_{n−1}^{a^{2}/2}c_{n−1}^{a^{2}} < c_{n−1}^{a^{2}/3},

LC2: let q = 6 · 2^{n}c_{n−1}^{a/3},

∑_{k>c_{n}^{−a/2}} (\frac{1}{2})^{qk} ≪ c_{n−1}^{n^{2}}.

LC3: let q = 6 · 2^{n}c_{n−1}^{a^{2}},

∑_{k>c_{n−1}^{−a^{2}/4}} (\frac{1}{2})^{qk} ≪ c_{n−1}^{n^{2}},

LC4: let q = 6 · 2^{n}c_{n−1}^{n/6},

∑_{k>c_{n}^{−n/3}} (\frac{1}{2})^{qk} ≪ c_{n}^{n^{2}}.

LC5: 2^{n}c_{n−1}^{−n/2}c_{n−1}^{2n} ≪ c_{n−1}^{n}. This gives the first estimate, for the second we must use Lemma 8.3

and the estimates on r_{n}(τ_{n}).

□

Using PhPa1 we get

Lemma 8.8. With total probability, for all n_{0} big enough, for all n big enough we have R_{n}(0) ∈ LC(n_{0}, n).

8.2. Hyperbolicity.

8.2.1. Preliminaries. For j ≠ 0, we define

λ_{n}^{(j)} = inf_{x∈I_{n}^{j}} \frac{ln(|R′_{n}(x)|)}{r_{n}(j)}.

And λ_{n} = inf_{j≠0} λ_{n}^{(j)}. As a consequence of the exponential estimate on distortion, together with hyperbolicity of f in the complement of I_{n}^{0} we immediately have the following.

Lemma 8.9. With total probability, for all n sufficiently big, λ_{n} > 0.

8.2.2. Good branches. We define the set of good returns G(n_{0}, n) ⊂ ℤ \ {0}, n_{0}, n ∈ ℕ, n ≥ n_{0} as the set of all j such that

G1: (hyperbolic return)

λ_{n}^{(j)} ≥ λ_{n_{0}} \frac{1 + 2^{n_{0}−n}}{2} ,

G2: (hyperbolicity in partial return) for c_{n−1}^{−3/(n−1)} ≤ k ≤ r_{n}(j) we have

inf_{I_{n}^{j}} \frac{ln(|Df^{k}|)}{k} ≥ λ_{n_{0}} \frac{1 + 2^{n_{0}−n+1/2}}{2} − c_{n}^{2/(n−1)}.

Notice that since c_{n} decreases torrentially, for n sufficiently big G2 implies for c_{n−1}^{−3/(n−1)} ≤ k ≤ r_{n}(j) we have

inf_{I_{n}^{j}} \frac{ln(|Df^{k}|)}{k} ≥ λ_{n_{0}} \frac{1 + 2^{n_{0}−n}}{2} .

Lemma 8.10. With total probability, for n_{0} big enough, for all n > n_{0}, V G(n_{0}, n) ⊂ G(n_{0}, n).

Proof. Let’s prove that if G1 is satisfied for all j ∈ V G(n_{0}, n), then V G(n_{0}, n + 1) ⊂ G(n_{0}, n + 1). Let’s fix such a j. Notice that by definition of λ_{n_{0}} the hypothesis is satisfied for n_{0}.

Let a_{k} = inf_{I_{n}^{j}} ln(|Df^{k}|)/k.

Recall the estimate from Lemma 8.4 for t = r_{n+1}(j), we as usually let R_{n}(I_{n+1}^{j}) = C\underline{d} ; n, \underline{d} = (j_{1}, ..., j_{m}). Let’s say that j_{i} was completed before k if v_{n} + r_{n}(j_{1}) + ... + r_{n}(j_{i}) ≤ k. We let the queue be defined as

q_{k} = inf_{C\underline{d} ; n} ln(|Df^{k−r} ∘ f^{r}|)

where r = v_{n} + r_{n}(j_{1}) + ... + r_{n}(j_{m_{k}}) with j_{m_{k}} the last complete return.

Recall that v_{n} < c_{n−1}^{−b} and from VG it is clear |R′_{n}|_{I_{n+1}^{j}}| > c_{n}^{n^{2}}. Notice also that using Lemma 5.9, for any k_{0} ≤ m, the derivative of R_{n}^{k_{0}} in C\underline{d} ; n is at least 2^{k_{0}}. So for m_{0} = c_{n−1}^{−2b} we have that the derivative of R_{n}^{k_{0}+1} in I_{n+1}^{j} is at least 1. Notice that from LS2

k_{0} = ∑_{i=1}^{m_{0}} r_{n}(j_{i}) < c_{n−1}^{−2b}c_{n−1}^{−3b} ≪ k.

It is clear that any complete return before k gives derivative at least 1 from Lemma 5.9. The queue can be bound by ln(c_{n}c_{n−1}^{3}) using Lemma 5.10. We have −q_{k}/k ≪ c_{n}^{2/n}.

Now we use Lemma 8.6 and get

a_{k} > \frac{β_{k} − k_{0}}{k} \frac{λ_{n_{0}}(1 + 2^{n_{0}−n})}{2} − \frac{−q_{k}}{k}

≥ \frac{λ_{n_{0}}(1 + 2^{n_{0}−n−1/2})}{2} − \frac{−q_{k}}{k} .

Which gives G2. If k = r_{n+1}(j), q_{k} = 0 which gives G1. □

8.2.3. Hyperbolicity in cool landings.

Lemma 8.11. With total probability, if n_{0} is sufficiently big, for all n sufficiently big, if \underline{d} ∈ LC(n_{0}, n + 1) then for all c_{n−1}^{−4/(n−1)} < k ≤ l_{n}(\underline{d}),

inf_{C\underline{d} ; n} \frac{ln(|Df^{k}|)}{k} ≥ \frac{λ_{n_{0}}}{2} .

Proof. Fix such \underline{d} ∈ LC(n_{0}, n + 1).

Let

a_{k} = inf_{C\underline{d} ; n} \frac{ln(|Df^{k}|)}{k}.

Consider the sequence r_{i} = r_{n}(j_{i}) where as usual \underline{d} = (j_{1}, ..., j_{m}). As in Lemma 8.6, we define m_{k} as the biggest such that

∑_{i=1}^{m_{k}} r_{i} ≤ k.

We define

β_{k} = ∑_{{1≤i≤m_{k}, \atop j_{i}∈V G(n_{0},n+1)}} r_{i},

(counting the time up to k spent in complete very good returns) and

i_{k} = k − ∑_{i=1}^{m_{k}} r_{i}.

(counting the time in the incomplete return at k).

Let’s then consider two cases: small m_{k} (m_{k} < c_{n−1}^{−a^{2}/2}) and otherwise. The idea of the first case is that all full returns are very good by

LC1, and the incomplete time is also part of a very good return.

Since full very good returns are very hyperbolic by G1 and very good returns are good, we just have to worry with possibly losing hyperbolicity in the incomplete time. To control this, we introduce the

queue q_{k} = inf_{C\underline{d} ; n−1} ln(|Df^{i_{k}} ∘ f^{k−i_{k}}|). We have −q_{k} < − ln(c_{n−1}^{1/3}c_{n−1}^{3})

by Lemma 5.10

If the incomplete time is big (more than c_{n−1}^{−4/(n−1)}), we can use G2 to estimate the hyperbolicity of the incomplete time (which is part of a very good return). The reader can easily check the estimate in this case.

If the incomplete time is not big, we can not use G2 to estimate q_{k}, but in this case i_{k} is much less than k: since k > c_{n−1}^{−4/(n−1)}, at least one return was completed (m_{k} ≥ 1), and since it must be very good we conclude that k > c_{n−1}^{−a/2} by LS1, so

a_{k} > λ_{n_{0}} \frac{(1 + 2^{n−n_{0}})}{2} · \frac{k − i_{k}}{k} − \frac{−q_{k}}{k} > \frac{λ_{n_{0}}}{2} .

Let’s consider now the case m_{k} > c_{n−1}^{−a^{2}/2}. For an incomplete time we still have −q_{k} < − ln(c_{n}c_{n−1}^{3}), so −q_{k}/k < c_{n−1}^{−a^{2}/3}.

Arguing as in Lemma 8.6, we split k − β_{k} − i_{k} (time of full returns which are not very good) in part relative to bad returns h_{k} and in part relative to fast returns (not very good or bad) l_{k}. Using LC4 and LC5 we get

h_{k} < c_{n−1}^{−3b}c_{n−1}^{n/7}m_{k},

and using LC1 and LC3 we have

l_{k} < c_{n−1}^{−a/2}c_{n−2}^{−4b}(6 · 2^{n})c_{n−1}^{a^{2}}m_{k},

By LC1 and LC3 we have β_{k} > c_{n−1}^{−a/2}m_{k}/2 so we have

\frac{h_{k} + l_{k}}{β_{k}} < c_{n−1}^{a^{2}/2}.

By LC1, β_{k} > c_{n−1}^{a/2}m_{k}. Now β_{k} > c_{n−1}^{−a/2}c_{n−1}^{−a^{2}}/2 by hypothesis on m_{k}. If i_{k} < c_{n−1}^{−a/2}c_{n−2}^{−4b}, i_{k}/β_{k} is small and we are done. Otherwise by Lemma 8.5, i_{k} must be either very good or bad. If i_{k} is very good we can reason as before that G2 applies for the estimate of the queue and we are done.

If i_{k} is bad, by LC5 we have that m_{k} > c_{n−1}^{n/2}, but i_{k} < c_{n−1}^{3b} by LS2, so i_{k}/β_{k} is very small again and we are done. □

9. Main Theorems

9.1. Proof of Theorem A. We must show that with total probability, f is Collet-Eckmann. We will use the estimates on hyperbolicity of cool landings to show that if the critical point always fall in a cool landing then there is uniform control of the hyperbolicity along the critical orbit.

Let

a_{k} = \frac{ln(|Df^{k}(f(0))|)}{k}

and e_{n} = a_{v_{n}−1}.

It is easy to see that if n_{0} is big enough such that both Lemmas 8.8 and 8.10 are valid and using Lemma 5.7 so that |R_{n}(0)| > |I_{n}|/2^{n} for n big enough, we obtain using Lemma 8.11 that

e_{n+1} ≥ e_{n} \frac{v_{n} − 1}{v_{n+1} − 1} + \frac{λ_{n_{0}}}{2} \frac{v_{n+1} − v_{n}}{v_{n+1} − 1}

and so

lim inf e_{n} ≥ \frac{λ_{n_{0}}}{2} .

Let now v_{n} − 1 < k < v_{n+1} − 1. Define

q_{k} = ln(|Df^{k−v_{n}}(f^{v_{n}}(0))|).

If k < v_{n} + c_{n−1}^{−4/(n+1)} from LC1 and LS1 we know that the time of the R_{n} branch of R_{n}(0) is at least c_{n−1}^{−a/2}, so k is in the middle of this branch. Using |R_{n}(0)| > |I_{n}|/2^{n}, we get that −q_{k} < − ln(2^{−n}c_{n−1}c_{n−1}^{−3}). We then get from v_{n} > c_{n−1}^{−a} that

a_{k} ≥ e_{n} \frac{v_{n} − 1}{k} − \frac{−q_{k}}{k} > (1 − 1/2^{n})e_{n} − 1/2^{n}.

If k > v_{n} + c_{n−1}^{−4/(n+1)} using Lemma 8.11 we get

a_{k} ≥ e_{n} \frac{v_{n} − 1}{k} + \frac{λ_{n_{0}}}{2} · \frac{k − v_{n} + 1}{k}.

Those two estimates imply that lim inf a_{k} ≥ λ_{n_{0}}/2 and so f is Collet-Eckmann.

9.2. Proof of Theorem B. We must obtain, with total probability, upper and lower (polynomial) bounds for the recurrence of the critical orbit. It will be easier to first study the recurrence with respect to iterates of return branches, and then estimate the total time of those iterates.

9.2.1. Recurrence in terms of return branches.

Lemma 9.1. With total probability, for n big enough and for 1 ≤ i ≤ c_{n−1}^{−2},

\frac{ln(|R_{n}^{i}(0)|)}{ln(c_{n−1})} < b^{2}.

Proof. Notice that due to torrential (and monotonic) decay of c_{n}, we can estimate |I_{n}| = c_{n−1}^{1+δ_{n}}, with δ_{n} decaying torrentially fast.

For i = 1 it follows from Lemma 5.7.

Let X ⊂ I_{n} be a c_{n−1}^{b^{2}} neighborhood of 0. For n big, we can estimate (due to the relation between |I_{n}| and c_{n−1})

\frac{|X|}{|I_{n}|} < c_{n−1}^{b^{2}−2}

(we of course consider X_{n} a union of C\underline{d} ; n, so that its size is near the required size, the precision is high enough for our purposes due to Lemma 5.3).

We have to make sure that the critical point does not land in X for 1 < i ≤ c_{n−1}^{−2}. This requirement can be translated on R_{n}(0) not belonging to a certain set Y ⊂ I_{n} such that

Y = ⋃_{1≤|\underline{d}|<c_{n−1}^{−2}}(R\underline{d} ; n)^{−1}(X).

It is clear that

p_{γ}(I_{n}^{τ_{n}} ∩ Y |I_{n}^{τ_{n}}) ≤ c_{n−1}^{−2}c_{n−1}^{(b^{2}−2)/b} < c_{n−1}^{b−3}.

Applying PhPa1, the probability that for 1 ≤ i ≤ c_{n−1}^{−2} we have |R_{n}^{i}(0)| < c_{n−1}^{b^{2}} is at most c_{n−1}^{b−3}, which is summable. □

Lemma 9.2. With total probability, for n big enough and for c_{n−1}^{−2} < i ≤ s_{n},

\frac{ln(|R_{n}^{i}(0)|}{ln(c_{n−1}^{−1})} < b^{2} (1 + \frac{ln(i)}{ln(c_{n−1}^{−1})}) .

Proof. The argument is the same as for the previous Lemma, but the decomposition has a slight different geometry.

For j ≥ 0, let X_{j} ⊂ I_{n} be a

c_{n−1}^{b^{2}(b^{j+1})}

neighborhood of 0 (approximated as union of C\underline{d} ; n, if X_{j} ⊂ I_{n+1}, it won’t be relevant for the proof, the reader can take then X_{j} = ∅). Let Y_{j} ⊂ I_{n} be such that

Y_{j} = ⋃_{c_{n−1}^{−b^{j}}≤|\underline{d}|<c_{n−1}^{−b^{j+1}}}(R\underline{d} ; n)^{−1}(X_{j}).

It is clear that

p_{γ}(I_{n}^{τ_{n}} ∩ Y_{j}|I_{n}^{τ_{n}}) ≤ c_{n−1}^{−b^{j+1}}c_{n−1}^{b^{j+2}} < c_{n−1}^{b^{j+1}}

and

p_{γ}(I_{n}^{τ_{n}} ∩ ∪Y_{j}|I_{n}^{τ_{n}}) < ∑_{j=0}^{∞} c_{n−1}^{−b^{j}} < 2c_{n−1}.

Applying PhPa1, with total probability, the critical point does not belong to any Y_{j}. Let’s see what this means. If c_{n−1}^{−b^{j}} < i ≤ c_{n−1}^{−b^{j+1}}, R_{n}^{i}(0) ∉ X_{j}, that is |R_{n}^{i}(0)| > c_{n−1}^{b^{j+3}}. This clearly implies the statement.

□ Corollary 9.3. With total probability, for n big enough and for 1 ≤ i ≤ s_{n},

\frac{ln(|R_{n}^{i}(0)|)}{ln(c_{n−1})} < b^{2} (1 + \frac{ln(i)}{ln(c_{n−1}^{−1})}) .

9.2.2. Total time of full returns. For 1 ≤ i ≤ s_{n}, let k_{i} such that R_{n}^{i}|_{I_{n+2}} = f^{k_{i}}.

Lemma 9.4. With total probability, for n big enough and for c_{n−1}^{−1} < i < s_{n}, k_{i}/i > c_{n−1}^{−a/2}/2.

Proof. This follows from condition LC2 and Lemma 8.8. □

Using that v_{n} > c_{n−1}^{−a} and k_{i} > v_{n} we get

Corollary 9.5. With total probability, for n big enough and for 1 ≤ i ≤ s_{n},

\frac{ln(k_{i})}{ln(c_{n−1}^{−1})} > a/3 (1 + \frac{ln(i)}{ln(c_{n−1}^{−1})}) .

9.2.3. Upper and lower bounds. Considering |R_{n}(0)| = |f^{v_{n}}(0)| < c_{n−1} and using v_{n} < c_{n−1}^{−b} we get

lim sup_{n→∞} \frac{− ln |f^{n}(0)|}{ln(n)} > a.

Let now v_{n} ≤ k < v_{n+1}. If |f^{k}(0)| < k^{−3b^{3}} we have f^{k}(0) ∈ I_{n} and so k = k_{i} for some i. It follows from Corollaries 9.3 and 9.5 that

|f^{k}(0)| > k^{−3b^{3}}.

References

[A] A. Avila. Bifurcations of unimodal maps: the topological and metric picture. Thesis IMPA (2001) (http://www.math.sunysb.edu/∼artur/).

[ALM] A. Avila, M. Lyubich and W. de Melo. Regular or stochastic dynamics in real analytic families of unimodal maps. In preparation.

[AM1] A. Avila and C. G. Moreira. Statistical properties of unimodal maps: the quadratic family. Preprint (http://www.arXiv.org).

[AM2] A. Avila and C. G. Moreira. Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative. Preprint (http://www.arXiv.org).

[BBM] V. Baladi, M. Benedicks and V. Maume. Almost sure rates of mixing for i.i.d. unimodal maps. Preprint (1999), to appear Ann. E.N.S.

[BV] V. Baladi and M. Viana. Strong stochastic stability and rate of mixing for unimodal maps. Ann. scient. Éc. Norm. Sup., v. 29 (1996), 483-517.

[KN] G. Keller and T. Nowicki. Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Comm. Math. Phys., 149 (1992), 31-69.

[L1] M. Lyubich. Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math, 140 (1994), 347-404.

[L2] M. Lyubich. Dynamics of quadratic polynomials, III. Parapuzzle and SBR measure. Asterisque, v. 261 (2000), 173 - 200.

[L3] M. Lyubich. Almost any real quadratic map is either regular or stochastic. Preprint IMS at Stony Brook, # 1997/8. To appear in Ann. Math.

[MvS] W. de Melo and S. van Strien. One-dimensional dynamics. Springer, 1993.

[NS] T. Nowicki and D. Sands. Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132 (1998), no. 3, 633–680.

[Pa] J. Palis. A global view of dynamics and a Conjecture of the denseness of finitude of attractors. Asterisque, v. 261 (2000), 335 - 348.

[Y] L.-S. Young. Decay of correlations for certain quadratic maps. Comm. Math. Phys., 146 (1992), 123-138.

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  E-mail address: avila@impa.br, gugu@impa.br