STATISTICAL PROPERTIES OF UNIMODAL MAPS: THE QUADRATIC FAMILY

ARTUR AVILA AND CARLOS GUSTAVO MOREIRA

Abstract. We prove that almost every real quadratic map is either regular or Collet-Eckmann with polynomial recurrence of the critical orbit. It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step to get the same results for more general families of unimodal maps.

Contents

1. Introduction 1 2. General definitions 5 3. Real quadratic maps 7 4. Complex dynamics 12 5. The phase-parameter relation 19 6. Measure and capacities 22 7. Statistics of the principal nest 24 8. Sequence of quasisymmetric constants and trees 29 9. Estimates on time 32 10. Dealing with hyperbolicity 36 11. Main Theorems 45 References 48

1. Introduction

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.

The quadratic family has been one of the most studied dynamical systems in the last decades. It is one of the most basic examples and exhibits a very rich behavior. It was also studied through many different techniques. Here we are interested in describing the dynamics of a typical quadratic map from the statistical point of view.

Date: May 20, 2001.

Partially supported by Faperj and CNPq, Brazil.

1.1. The probabilistic point of view in dynamics. In the last decade Palis [Pa] described a general program for (dissipative) dynamical systems in any dimension. In short, ‘typical’ dynamical systems can be modelled stochastically in a robust way. More precisely, one should show that such typical system can be described by finitely many attractors, each of them supporting a (ergodic) physical measure: time averages of Lebesgue almost every orbit should converge to spatial averages according to one of the physical measures. The description should be robust under (sufficiently) random perturbations of the system: one asks for stochastic stability.

Finally, a typical dynamical system was to be understood in the Kolmogorov sense: as a set of full measure in generic parametrized families.

Besides the questions posed by this conjecture, much more can be asked about the statistical description of the long term behavior of a typical system. For instance, the definition of physical measure is related to the validity of the Law of Large Numbers. Are other theorems still valid, like the Central Limit or Large Deviation theorems? Those questions are usually related to the rates of mixing of the physical measure.

1.2. The richness of the quadratic family. While we seem still very far away of any description of dynamics of typical maps, the quadratic family has been a remarkable exception. We describe briefly results which show the richness of the quadratic family from the probabilistic point of view.

The initial result in this direction was the work of Jakobson [J], where it was shown that for a positive measure set of parameters the behavior is stochastic, more precisely, there is an absolutely continuous invariant measure (the physical measure) with positive Lyapunov exponent: for Lebesgue almost every x, |Df^{n}(f(x))| grows exponentially fast. On the other hand, it was later shown by Lyubich [L2] and Graczyk-Swiatek [GS] that regular parameters (with a periodic hyperbolic attractor) are (open and) dense. While stochastic parameters are predominantly expanding (in particular have sensitive dependence to initial conditions), regular parameters are deterministic (given by the periodic attractor). So at least two kinds of very distinct observable behavior are present in the quadratic family, and they alternate in a complicate way.

It was later shown that stochastic behavior could be concluded from enough expansion along the orbit of the critical value: the Collet-Eckmann condition, exponential growth of |Df^{n}(f(0))|, was enough to conclude a positive Lyapunov exponent of the system. A different approach to Jakobson’s Theorem in [BC1] focused specifically in this property: the set of Collet-Eckmann maps has positive measure. After these initial works, many others studied such parameters (sometimes with extra assumptions), obtaining refined information of the dynamics of CE maps, particularly information about exponential decay of correlations^{1} (Keller and Nowicki in [KN] and Young in [Y]), and stochastic stability (Baladi and Viana in [BV]). The dynamical systems considered in those papers have generally been shown to have excellent statistical description^{2}.

Many of those results also generalized to more general families and sometimes to higher dimensions, as in the case of Hénon maps [BC2].

The motivation behind all the results on CE maps is certainly the fact that they were valid for a class of maps that was known to have positive measure. It was known, however that very different (sometimes wild) behavior coexisted ([Jo] and [HK]). It was shown for instance the existence of parameters without a physical measure or such that the physical measure is concentrated on a repelling hyperbolic fixed point. It remained to see if wild behavior was observable.

In a big project in the last decade, Lyubich [L3] together with Martens and Nowicki [MN] showed that almost all parameters have physical measures: more precisely, besides regular and stochastic behavior, only one more possibility could happen with posive measure, namely infinitely renormalizable maps (which always have a uniquely ergodic physical measure). Later Lyubich in [L5] showed that infinitely renormalizable parameters have measure 0, thus concluding the regular or stochastic duality, giving a further advance in the comprehension of the nature of the statistical behavior of typical quadratic maps. This celebrated result is remarkably linked to the progress obtained by Lyubich on the answer of the Feigenbaum conjectures ([L4]).

1CE quadratic maps are not always mixing and finite periodicity can appear in a robust way. This phenomena is related to the map being renormalizable, and this is the only obstruction: the system is exponentially mixing after renormalization.

2It is now known that weaker expansion than Collet-Eckmann is enough to obtain stochastic behavior for quadratic maps, on the other hand, exponential decay of correlations is actually equivalent to the CE condition [NS], and all current results on stochastic stability use Collet-Eckmann.

1.3. Statements of the results. In this work we describe the assymptotic behavior of the critical orbit. Our first result is an estimate of hyperbolicity:

Theorem A. Almost every real quadratic map is either regular or Collet-Eckmann:

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

The second is an estimate on the recurrence of the critical point. For regular maps, the critical point is non-recurrent (it actually converges to the periodic attractor). Among non-regular maps, however, the recurrence occurs at a precise rate which we estimate:

Theorem B. Almost every real quadratic map is either regular or has polynomial recurrence of the critical orbit:

lim sup_{n→∞} \frac{− ln(f^{n}(0))}{ln(n)} = 1.

Theorems A and B show that a typical non regular quadratic maps have enough good properties to conclude the results on exponential decay of correlations (which can be used to prove Central Limit and Large Deviation theorems) and stochastic stability in the sense of L^{1} convergence of the densities (of stationary measures of perturbed systems). Many other properties also follow, like existence of a spectral gap in [KN] and the recent results on almost sure (stretched exponential) rates of convergence to equilibrium in [BBM]. In particular, this answers positively Palis conjecture for the quadratic family.

1.4. Unimodal maps. Another reason to deal with the quadratic family is that it seems to open the doors to the understanding of unimodal maps. Its universal behavior was first realized in the topological sense, with Milnor-Thurston theory. The Feigenbaum-Coullet-Tresser observations indicated a geometric universality ([L4]).

A first result in the understanding of measure-theoretical universality was the work of Avila, Lyubich and de Melo [ALM], where it was shown a way to relate observable dynamics in the quadratic family to the observable dynamics of general analytic families of unimodal maps. In that work the method is used successfully to extend the regular or stochastic dichotomy to this context.

We are also able to adapt those methods to our setting. The techniques developed here and the methods of [ALM] are the main tools used in [AM]. All the results of this paper are shown to be valid for non trivial real analytic families of unimodal maps (with negative Schwarzian derivative and quadratic critical point). This is a rather general set of families, as trivial families form a set of infinite codimension (for a different approach which does not use negative Schwarzian derivative, see [A]).

In [AM] we also prove a version of Palis conjecture in the smooth setting. There is a residual set of k-parameter C^{3} (for the equivalent C^{2} result, see [A]) families of unimodal maps with negative Schwarzian derivative such that almost every parameter is either hyperbolic or Collet-Eckmann with subexponential bounds for the recurrence of the critical point. The final steps use techniques that date back to Jakobson, in a form advanced by Tsujii. Those techniques are needed to deal in the non-analytic setting.

Acknowledgements: We thank Viviane Baladi, Mikhail Lyubich and Marcelo Viana for helpful discussions about this work. We are grateful to Juan Rivera-Letelier for a valuable suggestion which led to the concept of gape renormalization and for listening to a first version.

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 k ≥ 1 be given. We say that a homeomorphism f : ℝ → ℝ is quasisymmetric with constant k if for all h > 0

\frac{1}{k} ≤ \frac{f(x + h) − f(x)}{f(x) − f(x − h)} ≤ k.

The space of quasisymmetric maps is a group under composition, and the set of quasisymmetric maps with constant k preserving a given interval is compact in the uniform topology of compact subsets of ℝ. It also follows that quasisymmetric maps are Hölder.

To describe further the properties of quasisymmetric maps, we need the concept of quasiconformal maps and dilatation (see §4.2 for definitions), so we just mention the main connection: any quasisymmetric map extends to a quasiconformal symmetric map of ℂ, and conversely, the restriction of a quasiconformal symmetric map of ℂ to ℝ is quasisymmetric. Furthermore, it is possible to work out upper bounds on the dilatation (of an optimal extension) depending only on k and conversely.

The constant k is awkward to work with: the inverse of a quasisymmetric map with constant k may have a larger constant. We will therefore work with a less standard constant: we will say that h is γ-quasisymmetric (qs) if h admits a quasiconformal symmetric extension to ℂ with dilatation bounded by γ. This definition behaves much better: 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. Such maps have many good features, in particular, they are stochastic (this is a consequence of [MN] and [L1]).

In [L3], it was proved that almost every quadratic map is either regular or simple or infinitely renormalizable. It was then shown in [L5] that infinitely renormalizable maps have 0 Lebesgue measure, which stablishes the Regular or Stochastic dichotomy.

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 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.

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}.

This result can be concluded from kneading theory using the monotonicity of the quadratic family by purely combinatorial arguments. It can be also concluded by M. Lyubich description of the combinatorics of the Mandelbrot set (restricted to the real line) in [L3], which we partially reproduce in this paper (existence part in the Geometric Lemma of §4.8).

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}.

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

PhPa1: Ξ_{i}|_{K_{i}^{τ}} is γ-qs, 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}.

The proof of this Theorem will be given in §5, but involves mainly complex dynamics, so we turn to this description.

4. Complex dynamics

4.1. Notation. A Jordan curve T is a subset of ℂ homeomorphic to a circle. A Jordan disk is a bounded open subset U of ℂ such that ∂U is a Jordan curve.

If r > 1, let A_{r} = {z ∈ ℂ|1 < |z| < r}. A ring A is a subset of ℂ such that there exists a conformal map from A to some A_{r}. In this case, r is uniquely defined and we denote the moduli of A as mod(A) = ln(r).

If U is a Jordan disk, we say that U is the filling of ∂U. A subset X ⊂ U is said to be bounded by ∂U.

A graph of a continuous map ϕ : Λ → ℂ is the set of all (z, ϕ(z)) ∈ ℂ^{2}, z ∈ Λ.

Let 0 : ℂ → ℂ^{2} be defined by 0(z) = (z, 0).

Let π_{1}, π_{2} : ℂ^{2} → ℂ be the coordinate projections. Given a set X ⊂ ℂ^{2} we denote its fibers X[z] = π_{2}(X ∩ π_{1}^{−1}(z)).

A fiberwise map F : X → ℂ^{2} is a map such that π_{1} ∘ F = π_{1}. We denote its fibers F [z] : X[z] → ℂ such that F(z, w) = (z, F [z](w)).

4.2. Quasiconformal maps. Let U ⊂ ℂ be a domain. A map h : U → ℂ is K-quasiconformal (K-qc) if it is a homeomorphism onto its image and for any ring A ⊂ U, mod(A)/K ≤ mod(h(A)) ≤ K mod(A). The minimum such K is called the dilatation of h.

Let h : X → ℂ be a homeomorphism and let C, ϵ > 0. An extension H : U → ℂ of h to a Jordan disk U is (C, ϵ)-qc if there exists a ring A ⊂ U with mod(A) > C such that X is contained in the bounded component of the complement of A and H is 1 + ϵ-qc.

Lemma 4.1. For all γ > 1 there exists C, ϵ > 0 such that if h : X → ℝ has a (C, ϵ)-qc extension H : U → ℂ which is symmetric then h is γ-qs.

4.3. Complex maps.

4.3.1. R-puzzle and R-maps. A R-puzzle P is a family of Jordan disks (U, U^{j}) such that \overline{U}^{j} are disjoint, \overline{U}^{j} ⊂ U for all j and 0 ∈ U^{0}.

If P = (U, U_{j}) is a R-puzzle, a R-map (or return type map) in P is a holomorphic map R : ∪U^{j} → U, surjective in each component, such that for j ≠ 0, R|_{U^{j}} extends to a homeomorphism onto \overline{U} and for R|_{U^{0}} extends to a double covering map onto \overline{U} ramified at 0.

4.3.2. From R-maps to L-maps. Given a R-map R we construct the following objects.

If \underline{d} ∈ Ω we define inductively U\underline{d} as follows. If \underline{d} = ∅, U\underline{d} = U, otherwise U\underline{d} = (R|_{U^{j_{1}}})^{−1}(U^{σ^{−}(\underline{d})}).

We define R\underline{d} = R^{|\underline{d}|}|_{U\underline{d}}.

We define W \underline{d} = (R\underline{d})^{−1}(U^{0}).

The L-map (landing type map) associated to R is defined as L(R) : ∪W \underline{d} → U^{0}, L(R)|_{W \underline{d}} = R\underline{d}. Notice that L(R)|_{W \underline{d}} extends to a homeomorphism onto \overline{U}.

4.3.3. Renormalization: from L-maps to R-maps. Given a R-map R such that R(0) ∈ W^{\underline{d}^{0}} we can defined the renormalization N(R) by N(R) = L(R) ∘ R where defined in U^{0}: its domain is the R-puzzle (V, V^{j}) such that V = U^{0} and V^{j} = (R|_{U^{0}})^{−1}(W^{\underline{d}^{j}}).

4.3.4. Truncation and gape renormalization. The \underline{d} truncation of L(R) is defined by L\underline{d}(R) = L(R) outside U\underline{d} and as R\underline{d} in U\underline{d}.

Given a R-map R such that R(0) ∈ U^{\underline{d}^{0}} we can defined the gape renormalization G\underline{d}(R) by G = L\underline{d}(R) ∘ R where defined in U^{0}.

4.4. Holomorphic motions.

4.4.1. Preliminaries. There are several ways to look at holomorphic motions. The way we describe them tries to make no commitment to a base point.

A holomorphic motion h over a domain Λ is a family of holomorphic maps defined on Λ whose graphs (called leaves of h) do not intersect. The support of h is the set X ⊂ ℂ^{2} which is the union of the leaves of h.

We have naturally associated maps h[z] : X → X[z], z ∈ Λ defined by h[z](x, y) = w if (z, w) and (x, y) belong to the same leaf.

The transition maps h[z, w] : X[z] → X[w], z, w ∈ Λ, are defined by h[z, w](x) = h[w](z, x).

Given a holomorphic motion h we may obtain other holomorphic motions by restriction of Λ or the set of leaves. The opposite situation is called extension.

Let K : [0, 1) → ℝ be defined by K(r) = (1 + ρ)/(1 − ρ) where 0 ≤ ρ < 1 is such that the hyperbolic distance between 0 and ρ in D is r.

λ-Lemma ([MSS], [BR]) If h is a holomorphic motion, h[z, w] is K(r)-qc, where r is the hyperbolic distance between z and w in Λ.

A completion of a holomorphic motion means an extension of h to the whole complex plane: X[z] = ℂ for all z ∈ Λ.

Extension Lemma ([Sl]) Any holomorphic motion over a simply connected domain can be completed.

From now on, we will always assume that Λ is a Jordan disk.

4.4.2. Meaning and notation warning. Holomorphic motions are a convenient way to work with the dependence of the phase dynamics on the parameter. The analogous of the intervals I_{i}[g] varying in intervals J_{i} (which we defined for real quadratic maps) will be domains U[z] varying in domains Λ.

We will use the following conventions. Instead of talking about the sets X[z], fixing some z ∈ Λ, we will say that h is the motion of X over Λ, where X is to be thought of as a set which depends on the point z ∈ Λ. In other words, we usually drop the brackets from the notation.

We will also use the following notation for restrictions of holomorphic motions: if Y ⊂ X, we denote Y ⊂ X as the union of leaves through Y.

4.4.3. Symmetry. For any n ∈ ℕ, we let conj : ℂ^{n} → ℂ^{n} denote the conjugacy conj(z_{1}, ..., z_{n}) = (\overline{z}_{1}, ..., \overline{z}_{n}).

A set X ⊂ ℂ^{n} is called symmetric if conj(X) = X. Let Λ ⊂ ℂ be a symmetric domain. A holomorphic motion h over Λ is called symmetric if the image of any leaf by conj is also a leaf.

The systems we are interested on are real, so they naturally possess symmetry. In many cases, we will consider a symmetric motion associated to the system, which will need to be completed using the Extension Lemma. The Extension Lemma adds ambiguity on the procedure, since the extension is non unique. In particular, this could lead to loss of symmetry. In order to avoid this problem, we will choose a little bit more carefully our extensions.

Symmetry assumption. Extensions of symmetric motions will always be taken symmetric.

Of course, to apply this Symmetry assumption we need the following

Real Extension Lemma. Any symmetric holomorphic motion can be completed to a symmetric motion.

This version of the Extension Lemma can be proved in the same way as the non-symmetric one. The reader can check for instance that in the proof of Douady [D] of the Extension Lemma there exists only one step which needs to be changed in order to obtain symmetry: in Proposition 1 we should ask the diffeomorphism F to be symmetric (the proof is the same).

4.4.4. Dilatation. If h = h_{U} is a holomorphic motion of an open set, we define Dil(h) as the suppremum of the dilatations of the maps h[z, w].

4.4.5. Proper motions. A proper motion of a set X over Λ is a holomorphic motion of X over Λ such that the map h[z] : Λ × X_{z} → X defined by h[z](w, x) = (w, h[z](w, x)) has an extension to \overline{Λ} × X_{z} which is a homeomorphism.

4.4.6. Tubes and tube maps. An equipped tube h_{T} is a holomorphic motion of a Jordan curve T. Its support is called a tube.

To each tube there is always an equipped tube associated (by definition). This association is unique however, since it is easy to see that if h_{1} and h_{2} are holomorphic motions supported on a set X with empty interior then h_{1} = h_{2} (consider a completion h of h_{1} and conclude that h|_{X} = h_{2}).

We say that an equipped tube is proper if it is a proper motion. Its support is called a proper tube.

The filling of a tube is the set U ⊂ Λ × ℂ such that U[z] is the filling of T [z], z ∈ Λ.

A motion of a set X is said to be bounded by the tube T if X is contained in the interior of the tube.

A motion of a set X is said to be well bounded by the proper tube T if it is bounded by T and the closure of eacha leaf of the motion does not intersect the closure of T .

If T is a tube over Λ, a tube map F is a fiberwise map defined on the filling U of T , which admits a continuous extension to \overline{U}.

Let h = h_{X} be a holomorphic motion such that h|_{T} is an equipped tube and F be a tube map on U such that F(T ) ⊂ X. We say that (h, F) is equivariant if the image of any leaf of h|_{T} is a leaf of h.

4.4.7. Tube pullback. Let F : V → ℂ^{2} be a tube map such that F(∂V) = ∂U is a tube over Λ and let h be a holomorphic motion supported on \overline{U} ∩ π_{1}^{−1}(Λ).

Let Γ be a parameter open set such that \overline{Γ} ⊂ Λ and W be a phase open set which moves holomorphically by h over Λ and such that \overline{W} ⊂ U. Assume that W contains critical values of F|_{V∩π_{1}^{−1}(Λ)}, that is, if z ∈ Γ, x ∈ V_{z} and F′_{z}(x) = 0 then F_{z}(x) ∈ W_{z}.

We define the lift of h by (F, Γ, W) as an extension to the closure of V of the holomorphic motion whose leaves are connected components of preimages of leaves of h|_{U\W}. It is clear that this holomorphic motion is a special motion of V over Γ and its dilatation over F^{−1}(U \ \overline{W}) is bounded by K(r) where r is the hyperbolic diameter of Γ in Λ.

Notice that in this case F^{−1}(U \ W) is a neighborhood of ∂V in \overline{V}. For this reason, there is equivariance: the image by F of a leaf of the lift of h contained in ∂V is a leaf of h contained in ∂U (although the map F is only continuous on ∂V).

4.4.8. Diagonal. Let T be a proper tube, and h the equipped proper tube supported on T . A diagonal of the tube is a holomorphic map Ψ : Λ → ℂ^{2} with the following properties.

• Ψ(z) = (z, ψ(z))),

• Ψ extends continuously to \overline{Λ},

• h_{z} ∘ Ψ|_{∂Λ} has degree one onto T_{z}.

4.4.9. Holonomy maps. Let h = h_{X∪T} be a holomorphic motion such that h|_{T} is an equipped proper tube and h|_{X} is well bounded by T . Such a motion will be called special. Let Φ be a diagonal of h|_{T}.

It is a consequence of the Argument Principle (see [L3]) that the leaves of h|_{X} intersect the graph of Φ in a unique point (with multiplicity one).

From this we can define a map χ[z] : X[z] → Λ such that χ[z](x) = w if (z, x) and (w, Φ(w)) belong to the same leaf.

It is clear that each χ_{z} is a homeomorphism onto its image, moreover, if U ⊂ X is open, χ[z]|_{U} is locally quasiconformal, its dilatation is bounded by Dil(h|_{U}).

We say that χ is the holonomy family associated to the pair (h, Φ). Sometimes the following obvious criteria is useful. Let h = h_{X_{1}∪X_{2}∪T}

be such that h|_{X_{1}∪T} is special. Suppose that for any x ∈ X_{2} there is a compact set K such that x ⊂ K and ∂K ⊂ X_{1} (in applications usually K is the closure of a Jordan disk). Then h is special.

4.5. Families of complex maps.

4.5.1. Families of R-maps. A R-family is a pair (R, h), where R is a holomorphic map R = R_{U} : ∪U^{j} → U and h is a holomorphic motion h = h_{U} with the following properties.

• P = (U, U^{j}) is a R-puzzle,

• h|_{∂U∪∂U_{j}} is special,

• R|_{U^{j}} is a tube map,

• for any z ∈ Λ, R_{z} is a R-map in P_{z},

• (R, h) is equivariant in ∂U^{j}. If additionally R ∘ 0 is a diagonal to h, we say that the R is full.

4.5.2. From R-families to L families. Given a R-family R with motion h = h\overline{U} we construct the following objects.

We define tubes U\underline{d}, \underline{d} ∈ Ω by taking U\underline{d} = U for empty \underline{d} and U\underline{d} = (R|_{U^{j_{1}}})^{−1}(U^{σ^{−}(\underline{d})}) otherwise.

We define R\underline{d} = R^{|\underline{d}|}|_{U\underline{d}}. We construct tubes W\underline{d} = (R\underline{d})^{−1}(U^{0}). We define L(R) : ∪W\underline{d} → U^{0} by L(R)|_{W\underline{d}} = R\underline{d}.

We define a holomorphic motion L(h) in the following way. The leaf through z ∈ ∂U is the leaf of h through z. If there is a smallest U\underline{d} such that z ∈ U\underline{d}, we let the leaf through z be the preimage by R\underline{d} of the leaf through R\underline{d}(z). We finally extend it to \overline{U} using the Extension Lemma.

The L-family associated to (R, h) is a pair (L(R), L(h)). It has some properties similar to R-maps.

• L(h)_{∂U∪∂U_{j}} = h and so L(h)|_{∂U∪\overline{U}_{j}} is special,

• L(R)|_{W\underline{d}} is a tube map,

• for any z ∈ Λ, the fibers of L(R) are the L-maps associated to the fibers of R,

• (L(R), L(h)) is equivariant in ∂W \underline{d}.

4.5.3. Parameter partition. Let (R, h) be a full R-family. Let χ be the holonomy family of the pair (L(h), R(0)).

We use χ to partition Λ, so we define sets Λ\underline{d} = χ(U\underline{d}) and Γ\underline{d} = χ(W \underline{d}).

4.5.4. Family renormalization. Let (R, h) be a full R-family. The \underline{d} renormalization of (R, h) is the R-family (N\underline{d}(R), N\underline{d}(h)) in defined as follows. We take N\underline{d}(h) as the lift of L(h) by (R|_{U_{0}}, Γ\underline{d}, W \underline{d}), and extend it to a holomorphic motion of \overline{U}_{0}. We let N\underline{d}(R) = L(R) ∘ R|_{U_{0}∩π_{1}^{−1}(Γ\underline{d})} where defined.

It is clear that (N\underline{d}(R), N\underline{d}(h)) is full, and its fibers are renormalizations of the fibers of (R, h).

4.5.5. Truncation, gape renormalization. Let \underline{d} ∈ Ω. We define the \underline{d} truncation of L(R) as L\underline{d}(R) = L(R) outside U\underline{d} and L\underline{d}(R) = R\underline{d} in U\underline{d}.

Let (R, h) be a full R-family. Fix some \underline{d} ∈ Ω. We define a the \underline{d} gape renormalization of (R, h) as a pair (G\underline{d}(R), G\underline{d}(h)) defined as follows. We take G\underline{d}(h) as the lift of L(h) by (R|_{U_{0}}, Λ\underline{d}, U\underline{d}), complete the motion using the Extension Lemma, and let G\underline{d}(R) = L\underline{d}(R) ∘ R|_{U_{0}∩π_{1}^{−1}(Γ\underline{d})} where defined.

It is clear that the fibers of (G\underline{d}(R), G\underline{d}(h)) are \underline{d} gape renormalizations of the fibers of (R, h).

Observe that G\underline{d}(h) is a special motion and that G(R)(z, 0) is a diagonal to it.

4.5.6. Geometric parameters. Given a full R-family (R, h) we define

mod_{Γ}(R, h) = inf_{\underline{d}∈Ω} mod(Λ \ Γ\underline{d})

and

mod_{U}(R, h) = inf_{z∈Λ,j∈ℤ}mod(U[z] \ U^{j}[z]).

4.6. Chains. Assume now we are given a full R-family, which we will denote R_{1} (over some domain Λ_{1}, with motion h_{1}) together with a parameter λ ∈ ℝ. If λ belongs to some renormalization domain (that is, there exists \underline{d} such that λ ∈ Λ\underline{d} ; 1), let R_{2} = N\underline{d} ; 1(R_{1}). Assume we can continue this process constructing R_{i+1} = N_{i}^{\underline{d}_{i}}, n ≥ 1. Then the sequence {R_{i}} will be called the R-chain over λ. It will be called a proper chain if \underline{d}_{i} ≠ ∅ for infinitely many i. It will be called admissible if \underline{d}_{i} ≠ ∅ for i big enough.

Notice that \underline{d}_{i} = ∅ corresponds to a central return, proper chains to non-renormalizable parameters with a recurrent critical point.

4.6.1. Notation. The objects (L, G, Λ and so on) associated to R_{i} will have the subscript i.

There is a unique sequence \underline{d}_{i} such that Λ_{i+1} = Γ_{i}^{\underline{d}_{i}}.

We denote Λ̃_{i+1} = Λ_{i}^{\underline{d}_{i}}.

We denote Ũ_{i+2} = (R_{i}|_{U_{i+1}})^{−1}(U^{\underline{d}^{i}}) over Λ̃_{i+1}. Notice that Ũ_{i+2} moves holomorphicaly by G(h_{i}) and that if g ∈ J_{i+1}, Ũ_{i+2} ∩ ℝ = J̃_{i+2}.

Notice that for any j, either \overline{U}_{i+1}^{j} ⊂ Ũ_{i+2} or \overline{U_{i+1}^{j}} ∩ \overline{Ũ_{i+2}}. The first case happens for j = 0. Notice also that h_{i+1}|_{∂U^{i+2}} agrees with G(h_{i}) over Λ_{i}.

4.6.2. Holonomy maps. Let χ_{i}^{0} : U_{i} → Λ_{i} denote the holonomy family associated to (h_{i}, R_{i} ∘ 0).

Let χ_{i} : U_{i} → Λ_{i} be the holonomy family associated to (L(h_{i}), R_{i}∘0). Let χ̃_{i} : U_{i−1} → Λ̃_{i} be the holonomy family of the pair

(G^{\underline{d}^{i−1}}(h_{i−1}), G^{\underline{d}^{i−1}} ∘ R_{i−1} ∘ 0).

4.7. Real chains. A fiberwise map F : X → ℂ^{2} is symmetric if X is symmetric and F ∘ conj = conj ∘F.

We will say that a chain {R_{i}} over a parameter λ ∈ ℝ is symmetric if each R_{i} and each underlying motions h_{i} is symmetric.

Because of the Symmetry assumption, we have that a chain {R_{i}} over a parameter λ ∈ ℝ is real provided the first step data R_{1} and h_{1} is symmetric.

In this case, all objects related to the chain are symmetric, including the holonomy families χ_{i}^{0}, χ_{i} and χ̃_{i} (where we say that a holonomy family is symmetric if χ_{λ} : U_{λ} → Λ if χ[λ](z) = \overline{χ[\overline{λ}](\overline{z})}, in particular, for real λ, χ[λ] is symmetric as a function).

Remark 4.1. If X is has empty interior and supports a holomorphic motion, then this holomorphic motion is unique as we observed before. In particular, if X is symmetric, this holomorphic motion is symmetric.

4.8. M. Lyubich’s theory revisited. The next Theorem is just a reinterpretation of results of M. Lyubich ([L2] and [L3]) in the language of chains.

Geometric Lemma Let λ_{0} ∈ ℝ be a parameter corresponding to a quadratic map f = f_{λ_{0}} ∈ F with principal nest {I_{i}}. Then there exists a real proper chain R_{i} over λ_{0} with the following property: for each λ ∈ Λ_{i} ∩ ℝ, the real trace of R_{i} is the first return map to I_{i} by f_{λ}.

If f is simple the geometric parameters mod_{U}(R_{i}, h_{i}), mod_{Γ}(R_{i}, h_{i}) grow at least linearly.

Remark 4.2. It is not needed to start with a simple parameter, however the geometric estimates become valid only for a subsequence of the chain. The full version is needed to prove that non-simple parameters have Lebesgue measure 0.

Remark 4.3. This Theorem is valid also for complex parameters, but to state it precisely it is necessary to introduce a lot of language related to the Mandelbrot set.

The chain is constructed (without this name) in §3.4, 3.5, 3.6 and 3.7 of [L3] using initially combinatorial properties of the Mandelbrot set which we did not discuss followed by a renormalization procedure which we reproduced here. Although our approach is totally based on his, the notation changes a lot. To obtain the existence of the chain the reader may just use the first step (§3.4) and then follow our description. The first geometric estimate is Theorem III in [L2]. The second geometric estimate is Theorem A in [L3].

We remark that the linear growth claimed above does not depend on the real map (a consequence of Complex Bounds). For complex parameters the constants would depend on finite combinatorial information.

5. The phase-parameter relation

Let R_{i} be the R-chain given by the Geometric Lemma.

5.1. Preliminary estimates.

5.1.1. Rings. If V_{1}, V_{2}, V_{3} are Jordan disks such that \overline{V}_{2}, \overline{V}_{3} ⊂ V_{1} then we define M(V_{1}, V_{2}, V_{3}) as the suppremum over all m such that there exists a ring A ⊂ V_{1}\(V_{2}∪V_{3}) of moduli m non-homotopic to a constant on V_{1} \ V_{2}.

The following are well known facts about rings.

Proposition 5.1. There exists C > 0 such that if V_{1}, V_{2}, V_{3} are as above, M(V_{1}, V_{2}, V_{3}) > C min{mod(V_{1} \ V_{2}), mod(V_{1} \ V_{3})}.

Proposition 5.2. Let V ⊂ U be Jordan disks and f : U → W be a double covering of U ramified at 0. Then if 0 ∈ V, mod(U \ V) = 1/2 mod(W \ f(V). If 0 ∉ V, mod(U \ V) ≥ 1/3 mod(W \ f(V)).

Proposition 5.3. There exists C > 0 such that if V ⊂ U are Jordan disks such that V ∩ℝ and U∩ℝ are intervals. Then ln(|V ∩ℝ|/|U∩ℝ|) ≤ −C mod(U \ V).

5.1.2. Finding space.

Lemma 5.4. For all ϵ > 0 there exists i_{0} such that if i > i_{0} the dilatation of χ_{i}^{0}|_{U_{i}\U_{i}^{0}} is less then 1 + ϵ.

Proof. Indeed, h_{i}|_{U_{i}\U_{i}^{0}} is the lift of L(h_{i−1}) by (R_{i−1}, Γ_{i−1}^{\underline{d}_{i−1}}), W_{i−1}^{\underline{d}_{i−1}}, The conclusion follows from the Geometric Lemma which gives growth of mod(Λ_{i−1} \ Λ_{i}). □

Lemma 5.5. The moduli of Λ_{i} \Λ_{i}^{j} is linearly big (independently of j).

Proof. Fix some Λ_{i}^{j}. By the Geometric Lemma and Proposition 5.1 there exists a ring A of linearly big moduli such that A ⊂ U_{i}\(U_{i}^{j} ∪U_{i}^{0}).

By Lemma 5.4, mod(χ_{i}^{0}(A)) is linearly big. The result follows from mod(Λ_{i} \ Λ_{i}^{j}) ≥ mod(A). □

Lemma 5.6. The moduli of Λ\underline{d} ; i \ Γ\underline{d} ; i grows linearly (independently of \underline{d}).

Proof. Fix some Λ\underline{d} ; i. For \underline{d} ∈ ∅, the result is contained in Lemma 5.5. Otherwise there exists j with U\underline{d} ; i ⊂ U_{i}^{j}. By the Geometric Lemma mod(U\underline{d} ; i \ W \underline{d} ; i) is linearly big.

By Lemma 5.5, we conclude that the dilatation of χ_{i}|_{U\underline{d} ; i} is converging

to 1. This implies that mod(Λ\underline{d} ; i \ Γ\underline{d} ; i) = mod(χ_{i}(U\underline{d} ; i \ W \underline{d} ; i)) is linearly big. □

Let c : (0, ∞) → (0, ∞) be defined as follows. Let c(m) be the suppremum of the hyperbolic diameter of compact subsets of 𝔻 such that 𝔻 \ K is a ring and mod(𝔻 \ K) = m. Notice that c(m) → 0 exponentially.

Lemma 5.7. Let h\overline{U} be a special motion, Φ be a diagonal and χ be the holonomy map. Let W ⊂ U be a Jordan disk such that mod(Λ \ χ(W)) = m. Then there exists a Jordan disk V ⊂ U such that mod(χ(V) \ χ(W)) = m/2 and such that χ|_{V} has dilatation bounded by K(c(m/2)).

Proof. Let Υ be a Jordan disk such that mod(Λ \ Υ) = mod(Υ \ χ(W)) = m/2. Consider V = χ^{−1}(Υ). It is clear that the dilatation of χ|_{V} is bounded by K(r) where r is the hyperbolic diameter of V. □

5.2. The phase-parameter estimates.

Lemma 5.8. For all ϵ > 0 there exists i_{0} such that for all i > i_{0} and for all λ ∈ ∪_{j}Λ_{i}^{j}, χ_{i}(λ)|_{U_{i}^{j}} has a (C, ϵ)-qc extension which is symmetric.

Proof. The extension claimed is just χ_{i}(λ)|_{U_{i}}, we just have to compute the parameters. By Lemma 5.5, if i is sufficiently big, the moduli of Λ_{i} \ Λ_{i}^{j} is big. By Lemma 5.7 it follows that if i is sufficiently big we can find a Jordan disk V_{i} ⊂ U_{i} such that V_{i} \ U_{i}^{j} is bigger than C and χ_{i}(g)|_{V_{i}} has dilatation less than 1 + ϵ. □

Lemma 5.9. For all C, ϵ > 0 there exists i_{0} such that for all i > i_{0} and for all λ ∈ Λ_{i} ∩ ℝ, χ̃_{i}(λ)|_{U_{i}} has a (C, ϵ)-qc extension which is symmetric.

Proof. The idea is the same as before: the extension claimed is just χ_{i}(g)|_{U_{i−1}}. The result will follow after estimating the image of U_{i−1} \U_{i} by χ̃_{i}(g). This is exactly Λ_{i−1}^{\underline{d}_{i−1}} \ Γ_{i−1}^{\underline{d}_{i−1}}, which is linearly big by Lemma 5.6. □

Remark 5.1. Estimates for the holonomy map between phase spaces also follow. From the estimates on moduli above (lemmas 5.5 and 5.6) we can easily deduce that for all ϵ > 0 there exists i_{0} such that if i > i_{0} we have:

(1) for λ_{1}, λ_{2} ∈ Λ_{i}^{j} ∩ ℝ, the phase holonomy map L(h_{i})[λ_{1}, λ_{2}] extends to a 1 + ϵ-qc map of the whole plane,

(2) for λ_{1}, λ_{2} ∈ Λ_{i} ∩ ℝ, the phase holonomy map G^{\underline{d}^{i−1}}(h_{i−1})[λ_{1}, λ_{2}] extends to a 1 + ϵ-qc map of the whole plane.

The fact that the phase holonomy maps can be defined on the whole plane follows from the fact that they can be completed.

5.2.1. Proof of the Phase-Parameter relation. We are basically done. By Lemma 4.1, it is enough to show that for i sufficiently big the required phase-parameter and phase-phase maps have big extensions with small dilatation. Let f = f_{λ_{0}} be a simple map. Notice that Ξ_{i}|_{K_{i}} = χ_{i}[λ_{0}], so application of Lemma 5.8 imply PhPa1. In the same way, Ξ_{i}|K̃_{i} = χ̃_{i}[λ_{0}], so Lemma 5.8 imply PhPa2. Furthermore, if g = f_{λ} ∈ J_{i} we have H_{g}|_{K_{i}} coincides with the holonomy map of the first item of Remark 5.1 which implies PaPa1 and H_{g}|K̃_{i} coincides with the holonomy map of the second item which implies PaPa2.

6. Measure and capacities

6.1. Quasisymmetric maps. If X ⊂ ℝ is measurable, let’s denote |X| its Lebesgue measure. Let’s explicit the metric properties of γ-qs maps we will use.

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) < 1 + ϵ/5. From now on, once a given γ close to 1 is chosen, ϵ will always denote a small number with this property.

6.2. 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}).

6.3. 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 6.1. 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 > 1 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 parameter in nonregular map has some property, we will just say that with total probability (without specifying) such property holds.

7. Statistics of the principal nest

7.1. 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 1 − ϵ to 1 + ϵ), where ϵ is a small, but fixed, number. 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 7.1. With total probability, for all n sufficiently big we have

p_{2γ−1}(|\underline{d}^{(n)}(x)| ≤ k|x ∈ I_{n}) < kc_{n}^{1−ϵ/2},
p_{2γ−1}(|\underline{d}^{(n)}(x)| ≥ k|x ∈ I_{n}) < e^{−kc_{n}^{1+ϵ/2}}.

We also have

p_{2γ−1}(|\underline{d}^{(n)}(x)| ≤ k|x ∈ I_{n}^{τ_{n}}) < kc_{n}^{1−ϵ/2},
p_{2γ−1}(|\underline{d}^{(n)}(x)| ≥ k|x ∈ I_{n}^{τ_{n}}) < e^{−kc_{n}^{1+ϵ/2}}.

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})^{1−ϵ/3}

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 − (\frac{c_{n}}{4} )^{1+ϵ/3}) p_{2γ−1}(|\underline{d}^{(n)}(x)| ≥ m|x ∈ I_{n}).

This implies

p_{2γ−1}(|\underline{d}(x)| ≥ m|x ∈ I_{n}) ≤ (1 − (\frac{c_{n}}{4} )^{1+ϵ/3})^{m} .

The second estimate follows from

(1 − (\frac{c_{n}}{4} )^{1+ϵ/3})^{k} < (1 − c_{n}^{1+ϵ/2})^{k}
< ((1 − c_{n}^{1+ϵ/2})^{c_{n}^{−1−ϵ/2}})^{kc_{n}^{1+ϵ/2}}
< e^{−kc_{n}^{1+ϵ/2}}.

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}^{−1+2ϵ} < s_{n} < c_{n}^{−1−2ϵ} can be estimated by 2c_{n}^{−ϵ} for n big, which is summable for all ϵ).

Lemma 7.2. With total probability,

lim_{n→∞} \frac{ln(s_{n})}{ln(c_{n}^{−1})} = 1.

Remark 7.1. 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 7.3. With total probability,

lim inf_{n→∞} \frac{ln(ln(c_{n+1}^{−1}))}{ln(c_{n}^{−1})} ≥ 1.

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

7.2. 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 7.4. 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 7.2. 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.

7.3. 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 7.5. 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 7.6. 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})}.

7.4. Estimating derivatives.

Lemma 7.7. Let w_{n} denote the relative distance in I_{n} of R_{n}(0) to 0 and to the boundary of I_{n}, that is,

w_{n} = \frac{d(R_{n}(0), ∂I_{n} ∪ \{0\})}{|I_{n}|}

(where d(x, X) denotes the usual euclidean distance between a point and a set). With total probability,

lim sup_{n→∞} \frac{− ln(w_{n})}{ln(n)} ≤ 1.

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

Proof. This is a simple consequence of PhPa2, using that n^{−1−δ} is summable, for all δ > 0. □

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

Lemma 7.8. With total probability,

lim sup_{n→∞} \frac{sup_{j≠0} ln(dist(f|_{I_{n}^{j}}))}{ln(n)} ≤ 1/2.

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|/n^{1+δ} neighborhood of C\underline{d} ; n.

It is clear that the image by γ-qs homeomorphism (γ close to 1) of P \underline{d} ; n \ C\underline{d} ; n is of order n^{−1−δ/2}|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^{−1/2−δ} neighborhood of any I_{n+1}^{j} with j ≠ 0 (we divide the exponent by two when applying the square root). □

Applying Lemma 7.6 we get

Lemma 7.9. With total probability,

lim sup_{n→∞} \frac{sup\underline{d} ln(dist(R\underline{d} ; n))}{ln(n)} ≤ 1/2.

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}}| > 1.

Remark 7.3. Lemma 7.8 has also an application for approximation of intervals. The result implies that if I_{n}^{j} = (a, b) and j ≠ 0, we have 1/2^{n} < b/a < 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 7.10. 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 7.9. 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 7.9 |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}.

□

8. Sequence of quasisymmetric constants and trees

8.1. Preliminary estimates. We will need from now on to consider not γ-capacities with γ fixed, but a sequence of γ converging to a given value. So from now on we fix a number a number γ such that k(2γ − 1) < ϵ/3, with ϵ very small.

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 8.1. Let S be an interval contained in I\underline{d} ; n. Using Lemma 7.6 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} ; n|_{S} or R_{n}|_{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 8.2. Let’s fix now γ such that the corresponding ϵ is small enough. 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^{−1000}

then

p_{γ_{n+1}}((R_{n}|_{I_{n+1}})^{−1}(X)|I_{n}) < δ^{1/5}.

Indeed, let V be a δ^{1/4}|I_{n+1}| neighborhood of 0. Then R_{n}|_{I_{n+1}\V} has distortion bounded by 2δ^{1/4}.

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

pγ̃_{n}(X ∩ W |W) < δλ^{−1−ϵ}.

Let’s decompose each side of I_{n+1} \ V as a union of n^{3}δ^{−1/4} intervals. Let W be such an interval. From Lemma 7.7, it is clear that the image of W covers at least δ^{1/2}n^{−4}|I_{n}|. It is clear then that

pγ̃_{n}(X ∩ R_{n}(W)|R_{n}(W)) < δ^{(1−ϵ)/2}n^{4+4ϵ}.

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^{(1−ϵ)/2}n^{5}

(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^{1/4})^{1−ϵ},

we get the required estimate.

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

Lemma 8.1. 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 8.1. □

By induction we get

Lemma 8.2. 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 8.3. 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 8.4. 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 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}} .

This can be easily interpreted as a large deviation estimate in this context.

9. 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 close to 1.

The basic estimate is a large deviation estimate and is proven in the next subsection (Corollary 9.4) and states that for k ≥ 1 the set of branches with time larger then kc_{n}^{−4} has capacity less then e^{−k}.

9.1. 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}^{4}. 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 9.1. If k > c_{n}^{−3/2}α_{n}^{−3/2} then

B_{n}(k) < e^{−c_{n}^{3/2}α_{n}^{3/2}k}.

Proof. Let k > c_{n}^{−3/2}α_{n}^{−3/2} be fixed. Let m_{0} = α_{n}^{3/2}k.

Notice that by Lemma 7.1

pγ̃_{n}(|\underline{d}^{(n)}(x)| ≥ m_{0}|x ∈ I_{n}) ≤ e^{−c_{n}^{5/4}α_{n}^{3/2}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}

STATISTICAL PROPERTIES OF THE QUADRATIC FAMILY 33

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}^{3/2})} > α_{n}^{−1}.

Given a sequence of m positive integers r_{i} as above we can do the following estimate using Lemma 8.2

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}^{3/2}} )^{m_{0}} ≤ e^{α_{n}^{5/4}k}.

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}^{1/4}−1/2)α_{n}k}.

34 ARTUR AVILA AND CARLOS GUSTAVO MOREIRA

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}^{1/4}−1/2)α_{n}k}

< e^{(2α_{n}^{1/4}−1/2)α_{n}k}(since\frac{ln(m_{0})}{k} ≤ \frac{ln(k)}{k} ≤ α_{n}^{5/4})

≤ e^{−α_{n}k/3}.

As a direct consequence we get

B_{n}(k) < e^{−α_{n}k/3} + e^{−c_{n}^{5/4}α_{n}^{3/2}k} < e^{−c_{n}^{3/2}α_{n}^{3/2}k}.

□

Lemma 9.2. With total probability, for n large enough,

v_{n+1} < c_{n}^{−2}α_{n}^{−2}/2.

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

Using Lemma 9.1, with total probability, for n large enough, l_{n}(\underline{d}) < nα_{n}^{−3/2}c_{n}^{−3/2}, and v_{n+1} < v_{n} +nα_{n}^{−3/2}c_{n}^{−3/2}. Using torrential (and monotonic) decay of α_{n}c_{n} we get for n large enough v_{n+1} < c_{n}^{−2}α_{n}^{−2}/2. □ Lemma 9.3. With total probability, for n large enough,

α_{n+1} ≥ min{α_{n}^{4}, c_{n}^{4}}.

Proof. Let k ≥ max{α_{n}^{−4}, c_{n}^{−4}}. From Lemma 9.2 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}^{−3/2}c_{n}^{−3/2}.

Applying Lemma 9.1 we have B_{n}(k/2) < e^{−α_{n}^{3/2}c_{n}^{3/2}k/2}.

Applying Remark 8.2 we get

A_{n+1}(k) < e^{−kα_{n}^{3/2}c_{n}^{3/2}/200} < e^{−k min{α_{n}^{4},c_{n}^{4}}}.

□

Since c_{n} decreases torrentially, we get Corollary 9.4. With total probability, for n large enough α_{n+1} ≥ c_{n}^{4}. Remark 9.1. In particular, using Lemma 9.2, for n big, v_{n} < c_{n−1}^{−4}. 9.2. Consequences. Lemma 9.5. With total probability, for all n sufficiently large we have

(1) pγ̃_{n}(l_{n}(x) < c_{n}^{−1+ϵ}|x ∈ I_{n}) < c_{n}^{ϵ/2}.

(2) pγ̃_{n}(l_{n}(\underline{d}) > c_{n}^{−1−5ϵ/3}|x ∈ I_{n}) ≤ e^{−c_{n}^{−ϵ/4}}.

Proof. We have l_{n}(\underline{d}) ≥ |\underline{d}|, and from Lemma 7.1

pγ̃_{n}(|\underline{d}^{(n)}(x)| ≤ c_{n}^{−1+ϵ}|x ∈ I_{n}) ≤ c_{n}^{ϵ/2}.

On the other hand, by the same lemma,

pγ̃_{n}(|\underline{d}^{(n)}(x)| ≥ c_{n}^{−1−ϵ}|x ∈ I_{n}) ≤ e^{−c_{n}^{−ϵ/2}}.

Defining

X_{m} = ⋃_{{\underline{d}=(j_{1},...,j_{m}), \atop r_{n}(j_{m})>c_{n}^{−ϵ/2}c_{n−1}^{−4}}} I\underline{d} ; n

we have

pγ̃_{n}(X_{m}|I_{n}) ≤ 2^{n}e^{−c_{n}^{−ϵ/2}} < e^{−c_{n}^{−ϵ/3}}.

Since

c_{n}^{−1−ϵ}c_{n}^{−ϵ/2}c_{n−1}^{−4} < c_{n}^{−1−5ϵ/3},

we conclude that if x satisfies l_{n}(x) > c_{n}^{−1−5ϵ/3} and |\underline{d}_{n}(x)| < c_{n}^{−1−ϵ} then x belongs to some X_{m} with 1 ≤ m ≤ c_{n}^{−1−ϵ}. So we get the second estimate bounded by

e^{−c_{n}^{−ϵ/2}} + c_{n}^{−1−ϵ}e^{−c_{n}^{−ϵ/3}} < e^{−c_{n}^{−ϵ/4}}.
□

Corollary 9.6. With total probability, for all n sufficiently large we have.

(1) p_{γ_{n+1}}(r_{n+1}(x) < c_{n}^{−1+2ϵ}|x ∈ I_{n+1}) < c_{n}^{ϵ/10}.

(2) p_{γ_{n+1}}(r_{n+1}(x) > c_{n}^{−1−2ϵ}|x ∈ I_{n+1}) ≤ e^{−c_{n}^{−ϵ/5}} ≤ c_{n}^{n}.

Proof. Notice that r_{n+1}(j) = v_{n} + l_{n}(\underline{d}), where R_{n}(I_{n+1}^{j}) ⊂ C\underline{d} ; n. We use Remark 9.1 to estimate v_{n} < c_{n−1}^{−4}, and estimate the distribution of r_{n+1}(j) by the distribution of l_{n}(\underline{d}) using Lemma 9.5, with a slight loss given by Remark 8.2. □

Using PhPa2 we get

Lemma 9.7. With total probability, for all n sufficiently big

lim_{n→∞} \frac{ln(r_{n}(τ_{n}))}{ln(c_{n−1}^{−1})} = 1.

Corollary 9.8. With total probability, for all n sufficiently large we have

(1) pγ̃_{n}(l_{n}(x) < c_{n}^{−1+ϵ}|x ∈ I_{n}^{τ_{n}}) ≤ c_{n}^{ϵ/10}.

(2) pγ̃_{n}(l_{n}(x) > c_{n}^{−1−5ϵ/3}|x ∈ I_{n}^{τ_{n}}) ≤ e^{−c_{n}^{−ϵ/5}}.

Proof. The same proof of Lemma 9.5 works, we just have to use Lemma 9.7 to avoid problems for falling immediately with bad time. □

Corollary 9.9. With total probability,

lim_{n→∞} \frac{ln(v_{n+1})}{ln(c_{n}^{−1})} = 1.

Proof. This is a simple consequence of PhPa1 applied to Lemma 9.8, noticing that v_{n+1} = v_{n} + l_{n}(\underline{d}) where R_{n}(0) ∈ C\underline{d} ; n and v_{n} is estimated by Remark 9.1. □

Remark 9.2. It is clear (using Lemma 7.7) that this implies that with total probability, for all ϵ > 0, for all n big enough, c_{n+1} ≥ 4^{−c_{n}^{−1−ϵ}}. This implies together with Corollary 7.3 that

lim_{n→∞} \frac{ln(ln(c_{n}^{−1}))}{ln(c_{n−1}^{−1})} = 1,

so c_{n}^{−1} grows torrentially.

10. 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.

10.1. Some kinds of branches and landings.

10.1.1. 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}^{−1/2} < m < c_{n}^{−1−2ϵ}, LS2: (No very large times) r_{n}(j_{i}) < c_{n−1}^{−14} for all i. LS3: (Short times are sparse in not too small initial segments)

For c_{n−1}^{−2} ≤ k ≤ m

#{r_{n}(j_{i}) < c_{n−1}^{−1+2ϵ}} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{ϵ/10}k,

LS4: (Large times are sparse in not too small initial segments) For c_{n}^{−1/n} ≤ k ≤ m

#{r_{n}(j_{i}) > c_{n−1}^{−1−2ϵ}} ∩ {1, ..., k} < (6 · 2^{n})e^{−c_{n−1}^{ϵ/5}}k,

Lemma 10.1. With total probability, for all n sufficiently big,

pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LS(n)|x ∈ I_{n}) < c_{n}^{1/3}/2,

and

pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LS(n)|x ∈ I_{n}^{τ_{n}}) < c_{n}^{1/3}/2.

Proof. Let’s first do the estimates on I_{n}. Let’s estimate the complement of the set defined by each item of the definition.

LS1: This was already estimated before (see Lemma 7.1), an up-

per bound is c_{n}^{1/3}/2 (using ϵ small). LS2: By Corollary 9.4 the γ_{n}-capacity of r_{n}(j) > c_{n−1}^{−14} is at most e^{−c_{n−1}^{−10}} ≪ c_{n}^{−3}. With m bounded by above as in the previous item and using Lemma 8.4, we get an estimate 2^{n}mc_{n}^{−3} ≪ c_{n}. LS3: This is a large deviation estimate, so we follow the ideas of §8.2. Fixing k we put q_{n} = (6·2^{n})c_{n−1}^{ϵ/10} and we have the estimate

(\frac{1}{2})^{q_{n}k} ≤ (\frac{1}{2})^{c_{n−1}^{−3/2}} ≪ c_{n}^{−3}.

Summing up over k we get the estimate c_{n}. LS4: Using the same method as in the previous item we put q_{n} =

(6 · 2^{n})e^{−c_{n−1}^{−ϵ/5}} and for each c_{n}^{−1/n} ≤ k ≤ m we get

(\frac{1}{2})^{q_{n}k} ≪ c_{n}^{−3}.

summing up in k (and using the estimate by above on m) we get the estimate c_{n}.

Summing up, we get the desired estimate.

To do the estimates on I_{n}^{τ_{n}}, the only item which changes is the second (since we have to avoid to have problems already because of the first iterate which would automatically give LS(n) not intersecting I_{n}^{τ_{n}}). This was taken care in Lemma 9.7. □

10.1.2. Very good returns and excellent landings. Define the set of very good returns, V G(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} and supposing V G(n_{0}, n) defined, define the set of excellent landings LE(n_{0}, n) ⊂ LS(n) satisfying the following extra assumption.

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

#{i|j_{i} ∉ V G(n_{0}, n)} ∩ {1, ..., k} < 6 · 2^{n}c_{n−1}^{1/9}k,

And 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}^{1/3}|I_{n+1}|.

Lemma 10.2. With total probability, for all n_{0} sufficiently big and all n ≥ n_{0}, if

p_{γ_{n}}(j^{(n)}(x) ∉ V G(n_{0}, n)|x ∈ I_{n}) < c_{n−1}^{1/9}

then

pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LE(n_{0}, n)|x ∈ I_{n}) < 2c_{n}^{1/9}/3,

and

pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LE(n_{0}, n)|x ∈ I_{n}^{τ_{n}}) < 2c_{n}^{1/9}/3.

Proof. We first use Lemma 10.1 to get an estimate of branches not in LS(n).

Using the hypothesis and Lemma 8.4 we first estimate for each k and them sum up (see also the proof of the last item in Lemma 10.1), so we get (with q_{n} = (6 · 2^{n})c_{n−1}^{1/9})

∑_{k≥c_{n−1}^{−2}} (\frac{1}{2})^{q_{n}k} ≪ c_{n}.

This works both in I_{n} and I_{n}^{τ_{n}}. □

Lemma 10.3. With total probability, for all n_{0} sufficiently big,

p_{γ_{n}}(j^{(n)}(x) ∉ V G(n_{0}, n)|x ∈ I_{n}) < c_{n−1}^{1/9}.

Proof. It is clear that with total probability, for n_{0} sufficiently big and n ≥ n_{0}, the set of branches I_{n}^{j} at distance at least c_{n}^{1/3} has γ_{n}-capacity bounded by c_{n}^{1/8}. The result follows from induction on n, since step n = n_{0} is clear from the definition and the induction step follows from Lemma 10.2. □

Using PhPa2 we get

Lemma 10.4. With total probability, for all n_{0} big enough, for all n big enough, τ_{n} ∈ V G(n_{0}, n).

Lemma 10.5. With total probability, for all n_{0} big enough and for all n ≥ n_{0}, if j ∈ V G(n_{0}, n + 1) then

\frac{1}{2}mc_{n−1}^{−1+2ϵ} < r_{n+1}(j) < 2mc_{n−1}^{−1−2ϵ},

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. Let’s estimate the time t = r_{n+1}(j). We have t = v_{n} + ∑ t_{i}, where we write t_{i} for r_{n}(j_{i}), where R_{n}(I_{n+1}^{j}) = C\underline{d} ; n with \underline{d} = (j_{1}, ..., j_{m}). To estimate from below we use LS3 and get

\frac{1}{2}mc_{n−1}^{−1+2ϵ} < (1 − 6 · 2^{n}c_{n}^{ϵ/10})mc_{n−1}^{−1+2ϵ} < t.

To estimate from above, we notice v_{n} < c_{n−1}^{−4} and

∑_{t_{i}>c_{n−1}^{−1−2ϵ}} t_{i} < 6 · 2^{n}c_{n−1}^{−14}e^{−c_{n−1}^{−ϵ/5}}m < m,

so

t < mc_{n−1}^{−1−2ϵ} + m + c_{n−1}^{−4} < 2mc_{n−1}^{−1−2ϵ}.
□

Remark 10.1. Using LS1 we get the estimate c_{n}^{−1/2} < r_{n+1}(j) < c_{n}^{−1−3ϵ}.

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 10.6. 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}^{1/100} 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}^{−4} + c_{n−1}^{−14} (use Corollary 9.9 to estimate v_{n} and LS2 to estimate the incomplete part).

Using LS2 we conclude now that

m_{k} > (k − c_{n−1}^{−4})c_{n−1}^{14} > c_{n}^{−1/n}

so m_{k} is not too small.

Let’s now estimate the contribution h_{k} from full returns j_{i} with time higher than c_{n}^{−1−2ϵ}. Since m_{k} is big, we can use LS4 to conclude that the number of such high time returns must be less than c_{n−1}^{n}m_{k}, so their total time is at most c_{n−1}^{n−14}m_{k}.

The non very good full returns on the other hand can be estimated by LE (given the estimate on m_{k}), they are at most c_{n−1}^{1/10}m_{k}. So we can estimate the total time l_{k} of non very good full returns with time less then c_{n−1}^{−1−2ϵ} by

c_{n−1}^{1/10}c_{n−1}^{−1−2ϵ}m_{k}.

Since m_{k} is big, we can use LS3 to estimate the proportion of branches with not too small time, so we conclude that at most c_{n−1}^{1/11}m_{k} branches are not very good or have time less than c_{n−1}^{−1+2ϵ}, so β_{k} can be estimated from below as

(1 − c_{n−1}^{ϵ/11})c_{n−1}^{−1+2ϵ}m_{k}.

It is easy to see then that i_{k}/β_{k} ≪ c_{n−1}^{1/100}, h_{k}/β_{k} ≪ c_{n−1}^{1/100}. If ϵ is small enough, we also have

l_{k}/β_{k} < 2c_{n−1}^{1/10−4ϵ} < c_{n−1}^{1/90}.

So (i_{k} + h_{k} + l_{k})/β_{k} is less then c_{n−1}^{1/100}. Since i_{k} + h_{k} + l_{k} + β_{k} = k we have 1 − β_{k}/k < (i_{k} + h_{k} + l_{k})/β_{k}. □

10.1.3. 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}^{−1/10}.

LC2: (Short times are sparse in not too small initial segments) For c_{n−1}^{−ϵ/5} ≤ k ≤ m

#{r_{n}(j_{i}) < c_{n−1}^{−1+2ϵ}} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{ϵ/10}k,

LC3: (Not very good moments are sparse in not too small initial segments) For all c_{n−1}^{−1/10} < k ≤ m

#{i|j_{i} ∉ V G(n_{0}, n)} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{1/20}k,

LC4: (Large times are sparse in not too small initial segments) For c_{n−1}^{−200} ≤ k ≤ m

#{r_{n}(j_{i}) > c_{n−1}^{−1−2ϵ}} ∩ {1, ..., k} < (6 · 2^{n})c_{n−1}^{100}k,

LC5: (Starts with no large times) r_{n}(j_{i}) < c_{n−1}^{−1−2ϵ}, 1 ≤ i ≤

e^{c_{n−1}^{−ϵ/5}/2}. Notice that LC4 and LC5 overlap, since c_{n−1}^{−200} < e^{c_{n−1}^{−ϵ/5}/2} as do LC1

and LC3, from this we can conclude that we can control the proportion of large times or non very good times in all moments (and not only for not too small initial segments).

Lemma 10.7. 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}^{1/100}

and for all n big enough

pγ̃_{n}(\underline{d}^{(n)}(x) ∉ LC(n_{0}, n)|x ∈ I_{n}^{τ_{n}}) < c_{n−1}^{1/100}.

Proof. The estimates follow the same ideas as LS2. For the first item, we only lose going from excellent to cool an amount of

LC1:

2^{n}c_{n−1}^{1/9}c_{n−1}^{−1/10} ≪ c_{n−1}^{1/100},

LC2:

∑_{k≥c_{n−1}^{−ϵ/5}} (\frac{1}{2})^{(6·2^{n})c_{n−1}^{ϵ/10}k} ≤ (2^{−n}c_{n−1}^{−ϵ/10}) (\frac{1}{2})^{(6·2^{n})c_{n−1}^{−ϵ/10}} ≪ c_{n−1}^{1/100},

LC3:

∑_{k≥c_{n−1}^{−1/10}} (\frac{1}{2})^{(6·2^{n})c_{n−1}^{1/20}k} ≤ (2^{−n}c_{n−1}^{−1/20}) (\frac{1}{2})^{2^{n}c_{n−1}^{−1/20}} ≪ c_{n−1}^{1/100},

LC4:

∑_{k≥c_{n−1}^{−200}} (\frac{1}{2})^{(6·2^{n})c_{n−1}^{100}k} ≤ (2^{−n}c_{n−1}^{−100}) (\frac{1}{2})^{2^{n}c_{n−1}^{−100}} ≪ c_{n−1}^{1/100},

LC5:

2^{n}e^{−c_{n−1}^{ϵ/5}}e^{c_{n−1}^{−ϵ/5}/2} ≪ c_{n−1}^{1/100}.

For the second item, we do the same and use Lemmas 10.4 and 9.7 to avoid problems for landing immediately in a not very good branch or with too small or large time. □

Transferring the result to parameter, using PhPa1 we get

Lemma 10.8. With total probability, for all n_{0} big enough, for all n big enough we have R_{n}(0) ∈ LC(n_{0}, n).

10.2. Hyperbolicity.

10.2.1. 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 for returns (which competes with torrential expansion from the decay of geometry), together with hyperbolicity of f in the complement of I_{n}^{0} we immediately have the following.

Lemma 10.9. With total probability, for all n sufficiently big, λ_{n} > 0.

10.2.2. 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−1}^{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 10.10. 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 10.5 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}^{−4}, so k > v_{n}. From VG it is clear |R′_{n}|_{I_{n+1}^{j}}| > 1. It is clear that any complete return before k gives derivative at least 1 from Lemma 7.9. −q_{k} can be bounded from above by − ln(c_{n}c_{n−1}^{5}) using Lemma 7.10. We have

−\frac{q_{k}}{k} < \frac{− ln(c_{n}c_{n−1}^{3})}{c_{n}^{−3/n}} ≪ c_{n}^{2/n}.

Now we use Lemma 10.6 and get

a_{k} > \frac{β_{k}}{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. □

10.2.3. Hyperbolicity in cool landings.

Lemma 10.11. 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 10.6, 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).

Using LC2 we get that if m_{k} > c_{n−1}^{−1/10},

k − i_{k} > c_{n−1}^{−1+2ϵ}m_{k}/2.

Therefore it is easy to see by LC1 that if k < c_{n−1}^{−15/14} we have m_{k} < c_{n−1}^{−1/10} (using small ϵ).

Let’s then consider two cases: small k (k < c_{n−1}^{−15/14}) 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}^{5})

by Lemma 7.10 and using that the incomplete time is very good.

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}^{−1/2} by Remark 10.1, 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 k > c_{n−1}^{−15/14}. For an incomplete time we still have −q_{k} < − ln(c_{n}c_{n−1}^{5}) < c_{n−1}^{−1−ϵ}, so −q_{k}/k < c_{n−1}^{1/100}.

Arguing as in Lemma 10.6, we split k − β_{k} − i_{k} (time of full returns which are not very good) in part relative to returns with high time h_{k} (more than c_{n−1}^{−1−2ϵ}) and in part relative to returns with low time l_{k}. Using LC4 and LC5 we get

h_{k} < c_{n−1}^{−14}c_{n−1}^{100}m_{k},

and using LC1 and LC3 we have

l_{k} < c_{n−1}^{−1−2ϵ}c_{n−1}^{1/20}m_{k} < c_{n−1}^{−39/40}m_{k},

provided ϵ is small enough. Since k > c_{n−1}^{−1+2ϵ}m_{k}/2 we have

\frac{h_{k} + l_{k}}{k} < 4c_{n−1}^{1/80},

provided ϵ is small enough.

From our estimates on i_{k} and on h_{k} and l_{k} we have 1−(β_{k}/k) < c_{n−1}^{1/90}. Now very good returns are very hyperbolic, and full returns (even not very good ones) always give derivative at least 1 from Lemma 7.9, so we have the estimate

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

□

11. Main Theorems

11.1. 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 10.8 and 10.10 are valid and using Lemma 7.7 so that |R_{n}(0)| > |I_{n}|/2^{n} for n big enough, we obtain using Lemma 10.11 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_{n→∞} 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 r_{n}(τ_{n}) > c_{n−1}^{−1/2}, 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}^{5}) < c_{n−2}^{−1−ϵ}. We then get from v_{n} > c_{n−1}^{1−ϵ} that

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

If k > v_{n} + c_{n−1}^{−4/(n+1)} using Lemma 10.11 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.

11.2. 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.

11.2.1. Recurrence in terms of return branches.

Lemma 11.1. 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})} < (1 + 2ϵ) (1 + \frac{ln(i)}{ln(c_{n−1}^{−1})}) .

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 7.7.

For 1 ≤ j ≤ 2ϵ^{−1}, let X_{j} ⊂ I_{n} be a c_{n−1}^{(1+2ϵ)(1+jϵ)} neighborhood of 0. For n big, we can estimate (due to the relation between |I_{n}| and c_{n−1})

\frac{|X_{j}|}{|I_{n}|} < c_{n−1}^{jϵ(1+2ϵ)}

(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 7.3).

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

Y_{j} = ⋃_{c_{n−1}^{(1−j)ϵ}≤|\underline{d}|<c_{n−1}^{−jϵ}}(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}^{−jϵ}c_{n−1}^{(1+ϵ)jϵ} < c_{n−1}^{ϵ^{2}}

and

p_{γ}(I_{n}^{τ_{n}} ∩ ⋃_{j=1}^{2ϵ^{−1}} Y_{j}|I_{n}^{τ_{n}}) < 2ϵ^{−1}c_{n−1}^{ϵ^{2}}.

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

Lemma 11.2. With total probability, for n big enough and for c_{n−1}^{−1−ϵ} < i ≤ s_{n},

\frac{ln(|R_{n}^{i}(0)|)}{ln(c_{n−1})} < (1 + 2ϵ) (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}^{(1+2ϵ)(1+(1+ϵ)^{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}^{−(1+ϵ)^{j}} ≤|\underline{d}|<c_{n−1}^{−(1+ϵ)^{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}^{−(1+ϵ)^{j+1}}c_{n−1}^{(1+ϵ)^{j+2}} < c_{n−1}^{ϵ(1+jϵ)}

and

p_{γ}(I_{n}^{τ_{n}} ∩ ⋃_{j=0}^{∞} Y_{j}|I_{n}^{τ_{n}}) < ∑_{j=0}^{∞} c_{n−1}^{ϵ(1+jϵ)} = \frac{c_{n−1}^{ϵ}}{1 − c_{n−1}^{ϵ^{2}}} < c_{n−1}^{ϵ/2}.

To get the result we just have to apply PhPa1. □

Corollary 11.3. With total probability, for n big enough and for 1 ≤ i ≤ s_{n},

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

11.2.2. 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 11.4. With total probability, for n big enough and for c_{n−1}^{−ϵ} < i < s_{n},

\frac{ln(k_{i})}{ln(c_{n−1}^{−1}} < (1 − 3ϵ) (1 + \frac{ln(i)}{ln(c_{n−1}^{−1}} )

Proof. By LC2 and Lemma 10.8 we have

\frac{k_{i}}{i} > c_{n−1}^{−1+3ϵ}.

This immediately gives

\frac{ln(k_{i})}{ln(c_{n−1}^{−1}} < (1 − 3ϵ) + \frac{ln(i)}{ln(c_{n−1}^{−1}} .

□

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

Corollary 11.5. With total probability, for n big enough and for 1 ≤ i ≤ s_{n},

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

11.2.3. Upper and lower bounds. Considering |R_{n}(0)| = |f^{v_{n}}(0)| < c_{n−1} and using Lemma 9.9 we get

lim sup_{n→∞} \frac{− ln |f^{n}(0)|}{ln(n)} ≥ 1.

Let now v_{n} ≤ k < v_{n+1}. If |f^{k}(0)| < k^{−1−10ϵ} we have f^{k}(0) ∈ I_{n} and so k = k_{i} for some i. It follows from Corollaries 11.3 and 11.5 that

|f^{k}(0)| > k^{−1−10ϵ}.

Varying ϵ we get

lim sup_{n→∞} \frac{− ln |f^{n}(0)|}{ln(n)} ≤ 1.

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.

[AM] A. Avila and C. G. Moreira. Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative. In preparation.

[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.

[BC1] M. Benedicks and L. Carleson. On iterations of 1 − ax^{2} on (-1,1). Ann. Math., v. 122 (1985), 1-25.

[BC2] M. Benedicks and L. Carleson. On dynamics of the Hénon map. Ann. Math., v. 133 (1991), 73-169.

[BR] L. Bers & H.L. Royden. Holomorphic families of injections. Acta Math., v. 157 (1986), 259-286.

[D] A. Douady. Prolongement de mouvements holomorphes (d’apres Slodkowski et autres). Asterisque, v. 227, 7-20.

[GS] J. Graczyk and G. Swiatek. Generic hyperbolicity in the logistic family. Ann. of Math., v. 146 (1997), 1-52.

[HK] F. Hofbauer and G. Keller. Quadratic maps without asymptotic measure. Comm. Math. Physics, v. 127 (1990), 319-337.

[J] M. Jacobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys., v. 81 (1981), 39-88.

[Jo] S. D. Johnson. Singular measures without restrictive intervals. Comm. Math. Phys., 110 (1987), 185-190.

[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, I-II. Acta Math., 178 (1997), 185-297.

[L3] M. Lyubich. Dynamics of quadratic polynomials, III. Parapuzzle and SBR measure. Asterisque, v. 261 (2000), 173 - 200.

[L4] M. Lyubich. Feigenbaum-Coullet-Tresser Universality and Milnor’s Hairiness Conjecture. Ann. Math. (1999).

[L5] M. Lyubich. Almost any real quadratic map is either regular or stochastic. Preprint IMS at Stony Brook, # 1997/8. To appear in Ann. Math.

[MN] M. Martens and T. Nowicki. Invariant measures for Lebesgue typical quadratic maps. Asterisque, v. 261 (2000), 239 - 252.

[MSS] R. Mañé, P. Sad & D. Sullivan. On the dynamics of rational maps, Ann. scient. Ec. Norm. Sup., 16 (1983), 193-217.

[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.

[Sl] Z. Slodkowsky. Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc., 111 (1991), 347-355.

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

IMPA – Estr. D. Castorina 110, 22460-320 Rio de Janeiro – Brazil.

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