STATISTICAL PROPERTIES OF UNIMODAL MAPS: SMOOTH FAMILIES WITH NEGATIVE SCHWARZIAN

DERIVATIVE

ARTUR AVILA AND CARLOS GUSTAVO MOREIRA

Abstract. We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every parameter is either regular or Collet-Eckmann with subexponential recurrence of the critical orbit, both cases allowing a detailed and robust statistical description of the dynamics. This proves a version of Palis conjecture in this setting.

1. Introduction

‘The main strategy of the study of all mathematical models is, according to Poincaré, the consideration of each model as a point of a space of different but similar admissible systems’(V. Arnold in [?]). One of the main concerns of dynamical systems is to stablish properties valid for typical systems. Since the space of such systems is usually infinite dimension, there are of course many concepts of ‘typical’. According to [?] again, ‘The most physical genericity notion is defined by Kolmogorov (1954), who suggested to call a property of dynamical systems exceptional, if it holds only on Lebesgue measure zero set of values of the parameters in every (topologically) generic family of systems, depending on sufficiently many parameters’.

In the last decade Palis [?] described a general program for (dissipative) dynamical systems in any dimension. He conjectured that a typical dynamical system has a finite number of atractors described by physical measures which are stochastically stable and whose basins have full Lebesgue measure (finitude and robustness of attractors). Typical was to be interpreted in the Kolmogorov sense: full measure in generic families. Our aim here is to give a proof of this conjecture for an important class of one-dimensional dynamical systems.

Here we consider unimodal maps, that is, continuous maps from an interval to itself which have a unique turning point. More specifically, we consider quasiquadratic unimodal maps, that is, we assume that the map is C^{3} with negative Schwarzian derivative and that the critical point is non degenerate.

Date: July 4, 2001.

Partially supported by Faperj and CNPq, Brazil.

1.1. The quadratic family. The basic model for unimodal maps is the quadratic family, p_{a}(x) = a − x^{2}, where −1/4 ≤ a ≤ 2 is a parameter. Despite its simple appearance, the dynamics of those maps presents many remarkable phenomena. Restricting to the probabilistic point of view, its richness first became apparent with the work of Jakobson [?], where it was shown that for a positive measure of parameters the behavior is stochastic, more precisely, there is an absolutely continuous invariant measure (the physical measure) with positive Lyapunov exponent. On the other hand, it was later shown by Lyubich [?] and Graczyk-Swiatek [?] that regular parameters (with a periodic hyperbolic attractor) are (open and) dense. so at least two kinds of very distinct observable behavior are present on the quadratic family, and they alternate in a complicate way.

Besides regular and stochastic behavior, different behavior was shown to exist, including examples with bad statistics, like absence of a physical measure or a physical measure concentrated on a hyperbolic repeller. Those pathologies were shown to be non-observable in [?] and [?]. Finally in [?] it was concluded the regular or stochastic dichotomy among quadratic maps.

Among stochastic maps, a specific class grabbed lots of attention in the 90’s: Collet-Eckmann maps. They are characterized by a positive Lyapunov exponent for the critical value, and gradually they were shown to have ‘best possible’ near hyperbolic properties: exponential decay of correlations, validity of central limit and large deviations theorems, good spectral properties and zeta functions ([?], [?]). Let’s call attention to the robustness of the statistical description, with a good understanding of stochastic perturbations: strong stochastic stability ([?]), rates of convergence to equilibrium ([?]).

In [?] the regular or stochastic dichotomy was extended to show that almost every stochastic map is actually Collet-Eckmann and satisfy many strong assymptotics, in particular implying the validity of the above mentioned results.

The position of the quadratic family in the borderline of real and complex dynamics made its study a convergence point of many different techniques: most of the deeper results depend on this interaction. It gradually became clear however that studying the quadratic family allows to obtain results on more general unimodal maps.

1.2. Universality. Starting with the works of Milnor-Thurston, and also through the discoveries of Feigenbaum and Coullet-Tresser, the quadratic family was shown to be not only a prototype of unimodal maps, but to present universal features both combinatorially and geometrically. More recently, the result of density of hyperbolicity among unimodal maps was obtained in [?] exploiting the validity of this result for quadratic maps.

In [?], a general method was devised to transfer information from the quadratic family to real analytic families of unimodal maps. It is shown that the decomposition of spaces of analytic unimodal maps according to combinatorial behavior is essentially a codimension-one lamination.

Thinking of two analytic families as transversals to this lamination, one may try to compare the parameter space of both families via the holonomy map. A straightforward application of this method allowed to conclude that the bifurcation pattern of a general analytic family is locally the same as the quadratic family from the topological point of view (outside of countably many ‘bad parameters’).

The ‘holonomy’ method was then succesfully applied to extend the regular or stochastic dichotomy from the quadratic family to a general analytic family. The probabilistic point of view presents new difficulties however. First, the statistical properties of two topologically conjugate maps need not correspond by the (generally not absolutely continuous) conjugacy. Fortunately many properties are preserved, in particular the criterias used by Lyubich in his result.

The second difficulty is that the holonomy map is usually not absolutely continuous, so typical combinatorics for the quadratic family may not be typical for other families: it is needed to show that the class of regular or stochastic maps is still typical after application of the holonomy map.

1.3. Results and plan of the proof. Let’s call a k-parameter family good if almost every parameter is either regular or Collet-Eckmann (and some additional technical conditions). Our goal will be to prove that good families are generic. This question naturally makes sense in different spaces of unimodal maps (corresponding to different smoothness).

We start by describing how the holonomy method of [?] can be applyed to generalize the results of [?] to general analytic families. As a consequence we conclude that essentially all analytic families are good.

To get to the smooth setting (at least C^{3}, since we are assuming negative Schwarzian derivative), our strategy is different: we show a certain robustness of good families, which together with their denseness (from the analytic case) will yield genericity. Our main tool is one of the nice properties of Collet-Eckmann maps: persistence of the Collet-Eckmann condition under generic unfolding (a result of [?]). Using an abstract scheme we reduce the global result to this local one.

Let’s mention that we have obtained the same results without the negative Schwarzian assumption, also allowing us to get to C^{2} smoothness in [?]. The techniques are very different however, since we do not use anymore the holonomy method we use here (of global nature), which is replaced by a local holonomy analysis and use of a finite version of the ‘infinitesimal perturbation’ method of [?]. For analytic maps this also allowed us to obtain more refined assymptotic estimates which have interesting consequences [?] as pathological measure-theoretical behavior of the lamination.

Acnowledgements: We thank Viviane Baladi, Mikhail Lyubich and Marcelo Viana for helpful discussions and suggestions.

2. General definitions

2.1. Notation. Let I = [−1, 1], B^{k} be the closed unitary ball in ℝ^{k}. Let C^{r}(I) denote the space of C^{r} maps f : I → ℝ.

By a unimodal map we will mean a smooth symmetric map f : I → I with a unique critical point at 0 such that f(−1) = −1, Df(−1) ≥ 1. If f is C^{3}, we define the Schwarzian derivative as

Sf = \frac{D^{3}f}{Df} − \frac{3}{2} (\frac{D^{2}f}{Df} )^{2}

defined on I \ {0}.

For a > 0, let Ω_{a} ⊂ ℂ denote an a neighborhood I.

Let A_{a} denote the space of holomorphic maps on Ω_{a} which have a continuous extension to ∂Ω_{a}, satisfying ϕ(z) = ϕ(−z), ϕ(−1) = ϕ(1) = −1 and ϕ′(0) = 0.

Notice that A_{a} is a closed affine subspace of the Banach space of bounded holomorphic maps of Ω_{a}. We endow it with the induced metric and affine structure.

We define A_{a}^{ℝ} ⊂ A_{a} the space of maps which are real symmetric.

2.2. More on unimodal maps. A C^{3} unimodal map such that Sf < 0 on I \ {0} and such that its critical point is non-degenerate (that is, D^{2}f ≠ 0) will be called a S-unimodal map.

A k-parameter family of unimodal maps is a map Γ : B^{k} × I → I such that for p ∈ B^{k}, γ_{p}(x) = Γ(p, x) is a unimodal map. Such a family is said to be C^{n} or analytic, according to Γ being C^{n} or analytic. We introduce the natural topology in spaces of smooth families (C^{n} with n = 2, ..., ∞), but do not introduce any topology in the space of analytic families.

A unimodal map f is called Collet-Eckmann (CE) if there exists constants C > 0, λ > 1 such that for every n > 0,

|Df^{n}(f(0))| > Cλ^{n}.

This means that the map is strongly hyperbolic along the critical orbit. It is also useful to study the hyperbolicity of backward iterates of the critical point, so we say that f is backwards Collet-Eckmann (BCE) if there exists C > 0, λ > 1 such that for any n > 0 and any x with f^{n}(x) = 0, we have

|Df^{n}(x)| > Cλ^{n}.

By a result of Nowicki (see [?]), for S-unimodal maps CE implies BCE, so we will mostly discuss the Collet-Eckmann condition (except for the last section where we consider C^{2} unimodal maps as well).

Very often it is useful to estimate how fast is the recurrence of the critical orbit. We will be mainly interested in two kinds of control: polynomial recurrence (P) if there exists α > 0 such that

|f^{n}(0)| > n^{−α}

for big enough n and subexponential recurrence (SE) if for all α > 0,

|f^{n}(0)| > e^{−αn}

for n big enough.

We will say that f is weakly regular (WR) if

lim_{δ→0} lim inf_{n→∞} \frac{1}{n} ∑_{{1≤k≤n \atop f^{k}(0)∈(−δ,δ)}} ln(|Df(f^{k}(0))|) = 0.

This condition is related to stochastic stability for C^{2} maps, see [?].

We will consider spaces of S-unimodal maps: we define U^{r} ⊂ C^{r}(I) the set of S-unimodal maps. Spaces of analytic unimodal maps are now easily defined: U_{a} = U^{3} ∩ A_{a}^{ℝ}.

2.3. The quadratic family. The quadratic family is the most studied family of unimodal maps. It is usually defined by

q_{t}(x) = t − x^{2},

so that for −1/4 ≤ t ≤ 2, there exists a unique symmetric interval I_{t} = [−β_{t}, β_{t}] such that p_{t}(I_{t}) = I_{t} and p_{t}(−β_{t}) = −β_{t}, so p_{t} can be seen as a unimodal map of I_{t} (which depends on t). It is also not difficult to see that Sp_{t}(x) < 0 if x ≠ 0.

By an affine reparametrization of the parameter t and of each interval I_{t}, we obtain a canonical one-parameter family of S-unimodal maps in the interval I, which we denote p_{t}, t ∈ B^{1}, which will be called the quadratic family as well.

2.4. Quasisymmetric maps. Let γ ≥ 1 be given. We say that a homeomorphism f : ℝ → ℝ is γ-quasisymmetric (γ-qs) if for all h > 0

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

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

3. Statement of the results

We would like to classify the typical behavior in generic families of unimodal maps. This classification should reveal refined information on the stochastic description of the dynamics of those typical parameter.

We will therefore consider a smooth enough family of unimodal maps Γ. The techniques of the present paper will need the fact that Γ is a family of S-unimodal maps. This includes two main restrictions: the negative Schwarzian and the quadratic critical point. The first one is serious, since this condition is not dense, but can be removed with more refined techniques (see [?]). The second one is no serious loss of generality, since quadratic critical point is certainly typical among unimodal maps.

We first consider the analytic case. In this case, the generic families we will consider will be families satisfying a non-degeneracy condition: an analytic family of S-unimodal maps is called standard if it contains a regular map. This condition is easy to verify, both in practice (if the family is non-degenerate this can be verified in finite time) as in theory (they are indeed generic for meaningful concepts). Such nondegenerate family will be called standard, and our main result is a dichotomy on the typical dynamical behavior in any such family.

Theorem A. Let Γ be a standard k-parameter family of S-unimodal maps. Then for almost every p ∈ B^{k}, f_{p} is either regular or satisfy CE, P.

Our second result about standard families is the robustness of a slightly weaker dichotomy under C^{2} perturbations of the family.

Theorem B. Let Γ be a standard k-parameter family of S-unimodal maps. Let Γ_{n} be a sequence of C^{2} families such that Γ^{n} → Γ in the C^{2} topology. Let X_{n} be the set of parameters p ∈ B^{k} such that Γ_{n} is either regular or satisfy BCE, CE, SE and WR. Then |X_{n}| → 1.

As a consequence, we can use a Baire argument to conclude that a dichotomy is still valid among topologically generic (residual set) smooth families, so the following is just an easy Corollary of Theorems A and B.

Smooth Dichotomy. In topologically generic k-parameter C^{r}, r = 3, 4, ..., ∞ families of S-unimodal maps, almost every parameter is either regular or satisfy BCE, CE, SE and WR.

It is good to recall that both types of behavior stablished on the dichothomy are indeed observable for open sets of families of unimodal maps ([?], [?]).

3.1. Ergodic meaning. The importance of the above dichotomy is the fact that each of the two possibilities has very well defined stochastic properties. We quickly recall those (we assume that maps are S-unimodal).

Regular maps have a periodic attractor whose basin is big both topologically (open and dense set) as in the measure-theoretical sense (full measure). Moreover the attractor and its basin are stable under C^{1} perturbations. The dynamics of such maps can be described in deterministic terms.

Maps satisfying CE and SE have non-deterministic dynamics. They can however describe them through their stochastic properties, and it turns out that such maps have the main good properties usually found in hyperbolic maps. First, there is a physical measure, that is an invariant probability which describes assymptotic behavior of orbits: for almost every x and for every continuous ϕ : I → ℝ,

lim \frac{1}{n} ∑_{k=0}^{n−1} ϕ(f^{k}(x)) = ∫ ϕdμ.

This physical measure has a positive Lyapunov exponent and is indeed absolutely continuous and supported on a cycle of intervals, so the assymptotic behavior is non-deterministic. The convergence to the assymptotic stochastic model is exponential, see the results on decay of correlations ([?], [?]) and convergence to equilibrium. Those properties are beautifully related to a spectral gap of a transfer operator and to zeta functions, see [?]. Notice finally that exponential decay of correlations is actually equivalent to the Collet-Eckmann condition (see [?]).

While the dynamics is highly unstable under deterministic perturbations (nearby maps can be regular for instance), the stochastic description given by the physical measure μ is robust under stochastic perturbations: the perturbed system has a stationary measure which is close to μ in the sense of the L^{1} distance between their densities ([?]). For studies of decay of correlations for the perturbed systems, see [?].

4. Analytic families

4.1. Hybrid conjugacy and holonomy maps. A hybrid conjugacy between two S-unimodal maps f and f̃ is a topologic conjugacy which, if there exists an attracting periodic orbit, is locally smooth in the basin of attraction of this periodic orbit.

Two S-unimodal maps f, f̃ are said to be hybrid equivalent if there exists a hybrid conjugacy between them. The set of all maps which are hybrid equivalent to some f is called hybrid class of f.

It follows from a result of Guckenheimer (see [?]) that any S-unimodal map f is hybrid conjugate to some quadratic map. The problem of uniqueness is much harder. The following result is due to Lyubich [?] and Graczyk-Swiatek [?], and is a consequence of (the proof of) the equivalent rigidity result for quadratic maps:

Theorem 4.1. Let h be a hybrid conjugacy between two analytic Sunimodal maps f and f̃. Then h is quasisymmetric.

This rigidity result (in the case of the quadratic family) implies that any S-unimodal map f is hybrid conjugate to a unique quadratic map χ(f). The map χ is called the straightening.

Lemma 4.2. If f is an analytic S-unimodal map and χ(f) is either regular or CE and P then f is too.

Proof. Since regularity is clearly preserved by hybrid conjugacy, we only have to analyze invariance of the conditions CE and P. By Theorem ??, we just have to prove invariance under quasisymmetric conjugacy.

By [?] the Collet-Eckmann condition is invariant under quasisymmetric conjugacy (and even under topological conjugacy). Since quasisymmetric maps are Hölder, it is easy to see that P is also invariant.

4.2. Hybrid laminations. It is natural to study the hybrid class of some map f. This is what is done in Theorem A of [?] in the analytic setting, where it is shown that in U_{a}, every hybrid class is a codimension-one analytic submanifold. Moreover, different hybrid class fit together in some nice structure, called hybrid lamination.

If Γ is a one-parameter family of S-unimodal maps, it can be naturally thought of as an analytic curve in some U_{a}.

A consequence of the nice structure of the hybrid lamination is the following

Lemma 4.3. If Γ is not contained in a hybrid class then there is an open set of parameters, with countable complement, where Γ is transverse to the hybrid lamination.

From Milnor-Thurston kneading theory, we know that any family that is not contained in a hybrid class must contain a regular map.

Standard families are strongly generic among analytic families An analytic family of S-unimodal maps which is not standard must be contained in some hybrid class. In particular, standard families are dense in the space of C^{n} families of S-unimodal maps, n = 3, ..., ∞.

Definine the map χ_{Γ} on B^{1} defined by χ_{Γ}(t) = χ(γ_{t}). In [?] the map χ_{Γ} is considered as the holonomy map from Γ to the quadratic family along the hybrid lamination in some U_{a}. Using this interpretation, they obtain

Theorem 4.4. Let Γ be a standard one-parameter family of unimodal maps. If Γ contains a non-regular map, then there is an open set U ⊂ B^{1} with countable complement such that the straightening χ_{Γ} is quasisymmetric in any compact interval J ⊂ U.

4.3. Dichotomy in the quadratic family. The main result of [?] is that almost every parameter in the quadratic family is either regular or Collet-Eckmann with a polynomial recurrence of the critical orbit. To obtain the same result for a standard family using Theorem ??, we will need a stronger estimate, since quasisymmetric maps are not in general absolutely continuous.

Let’s say that a set X ⊂ B^{1} has total qs-probability if the image of B^{1} \ X by any quasisymmetric map h : B^{1} → B^{1} has 0-Lebesgue measure.

By a slight modification of the proofs in [?] (see [?]), it is possible to obtain

Theorem 4.5. The set of quadratic maps which are either regular or CE and P has total qs-probability.

Remark 4.1. In [?] it is obtained a better result than polynomial recurrence, namely it is shown that the assymptotyc exponent of the recurrence

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

is exactly 1 for almost every non regular map. However, for a set of total qs-probability, we are only able to show that the assymptotic exponent is bounded.

4.4. Proof of Theorem A. Let Γ be a standard family. If all parameters are regular, there is nothing to prove, so assume that there is a non-regular parameter.

First assume Γ is one-parameter. By Theorems ?? and ??, for almost every t ∈ B^{1}, χ_{Γ}(t) is either regular or satisfies CE and P. By Lemma ??, this implies that γ_{t} is either regular or CE and P.

Assume now that Γ is a k-parameter family. Let p ∈ B^{k} be a regular parameter. Let L : B^{1} → B^{k} be an affine map such p ∈ L(B^{1}). Let Γ^{L} be the one-parameter family defined by γ_{t}^{L} = γ_{L(t)}. Then Γ^{L} is a standard one-parameter family and for almost every t, γ_{t}^{L} is either regular or CE and P. The result follows by application of Fubini’s Theorem.

5. Robustness of the dichotomy

To obtain the robustness claimed on Theorem B our approach will be to exploit an important result of Tsujii, whose core is a strong generalization of Benedicks-Carleson result and techniques. This result establishes that the CE and SE conditions are infinitesimally persistent in one-parameter families unfolding generically: they are density points of CE and SE parameters. The connection with our robustness result, which has a global nature, is done using an abstract (and technical) scheme.

5.1. Tsujii’s theorem. Let Γ be a C^{2} k-parameter family of unimodal maps. Assume that p_{0} is a parameter such that γ_{p_{0}} satisfies CE, BCE, SE and has a quadratic critical point. Tsujii’s Theorem considers the case where Γ is a generic unfolding at p_{0}. For one-parameter families, generic unfolding means precisely

∑_{k=0}^{∞} \frac{∂_{t}γ_{p}(γ_{p}^{j}(0))}{Dγ_{p}^{j}(f^{j}(0))} ≠ 0,

where ∂_{t} stands for the parameter derivative. This transversality condition will be called Tsujii transversality.

If Γ is a one-parameter family, we will say that (Γ, p_{0}) satisfies the Tsujii conditions if all above requirements are satisfied.

The following is an immediate consequence of the main theorem of Tsujii in [?].

Theorem 5.1. Let Γ be a C^{2} one-parameter family of unimodal maps. Assume (Γ, t_{0}) satisfies the Tsujii conditions. Then t_{0} is a density point of parameters t for which (Γ, t) satisfies the Tsujii conditions and for which γ_{t} is WR.

5.2. A higher dimensional version. To pass from one-parameter to k-parameters, we need the following easy proposition. Let’s say that p ∈ B^{k} is a density point of X along a line l through p if p is a density point of l ∩ X in l endowed with the linear Lebesgue measure.

Proposition 5.2. If p is a density point of X along almost every line, then p is a density point of X in B^{k}.

Proof. To compute d(B_{ϵ}(p), X), use polar coordinates, and integrate first along the rays, obtaining a density function ρ_{ϵ}(θ), θ ∈ S^{k−1}. To conclude, use Fatou’s Lemma

lim inf_{ϵ→0} ∫ ρ_{ϵ}(θ)dθ ≥ ∫ lim inf_{ϵ→0} ρ_{ϵ}(θ)dθ.

The k-parameter analog of the Tsujii transversality is the existence of a line along which the one-parameter transversality condition is satisfied. In other words, there exists a an affine map L : B^{1} → B^{k} passing through p_{0} such that the induced one-parameter family Γ ∘ L is Tsujii transverse at the parameter t_{0} such that L(t_{0}) = p_{0}.

Notice that from linearity, if (Γ, p_{0}) is Tsujii transverse, then all lines passing through p_{0} are Tsujii transverse except the lines parallel to a certain codimension-one space of ℝ^{k}.

Lemma 5.3. Let Γ be a C^{2} k-parameter family of unimodal maps. Assume (Γ, p_{0}) satisfies the Tsujii conditions. Then p_{0} is a density point of parameters p for which (Γ, p) satisfies the Tsujii conditions and for which γ_{p} is WR.

Proof. If Γ is Tsujii transverse at p_{0} then it is Tsujii transverse along almost every line through p_{0}. Along such a line it is a density point of parameters satisfying the Tsujii conditions and which are WR. The result follows from proposition ??.

5.2.1. Tsujii transversality and hybrid lamination. Let’s take a closer look at the Tsujii transversality for an analytic Γ. Let γ_{p} = f.

Assuming

∑_{k=0}^{∞} \frac{1}{Df^{k}(f(0))} < ∞ (5.1)

(in particular if f is CE), let

ν_{f}(v) = ∑_{k=0}^{∞} v(f^{k}(0))/Df^{k}(f(0))

be a functional defined on continuous vectorfields v on the interval. It is easy to see (under the summability condition) that ν_{f} does not vanish on the set of symmetric polynomials.

Lemma 5.4. The kernel of ν_{f} intersected with A_{a}^{ℝ} is the tangent space to the hybrid class of f.

Proof. Since ν is non-trivial over symmetric polynomials the above intersection is a closed codimension-one subspace of A_{a}^{ℝ}. It is enough to show that if v is tangent then ν_{f}(v) = 0. It is remarked in [?] that

Df^{n}(f(0)) ∑_{k=0}^{n−1} v(f^{k}(0))/Df^{k}(f(0))

is precisely ∂_{t}(γ_{t}^{v})^{n+1}(0), and this sequence is bounded independent of n if v is tangent. Since the summability condition (??) |Df^{n}(f(0))| → ∞, we have necessarily ν_{f}(v) = 0.

So Tsujii transversality can be interpreted for such a map (satisfying the summability condition (??)) as transversality of the family to the hybrid class of γ_{p}.

Since for maps with negative Schwarzian derivative CE implies the BCE, we can conclude from Theorem A, Lemma ?? and this discussion the following

Lemma 5.5. If Γ is a standard k-parameter family then almost every parameter is regular or satisfies the Tsujii conditions.

5.3. Generalities about density points.

5.3.1. Density points in abstract. Let U^{2} be the space of C^{2} unimodal maps (without, naturally, the hypothesis of negative Schwarzian derivative), and K be the space of C^{2} k-parameter families of unimodal maps.

Let X ⊂ K × B^{k} be the set of (Γ, p) such that either γ_{p} is regular or satisfies the Tsujii conditions and WR. For Γ ∈ K, let X_{Γ} = {p ∈ B^{k}|(Γ, p) ∈ X}.

Let Y ⊂ B^{k} be measurable with |Y | > 0. We define the density of X along Γ on Y as

d(Γ, Y) = \frac{|Y ∩ X_{Γ}|}{|Y |}.

Instead of defining the classical infinitesimal density:

lim inf_{ϵ→0} d(Γ, B_{ϵ}(p))

we will need to consider the stability of the density with respect to perturbations of Γ. With this in mind we introduce two parameters. Let

D^{−}(Γ^{0}, p) = lim inf_{{Γ→Γ^{0} \atop γ_{p}=γ_{p}^{0}}} lim inf_{ϵ→0} d(Γ, B_{ϵ}(p)),

D^{+}(Γ^{0}, p) = lim inf_{ϵ→0} lim inf_{Γ→Γ^{0}} d(Γ, B_{ϵ}(p)).

Lemma ?? and Tsujii’s result gives a direct way to estimate D^{−}

Lemma 5.6. Let Γ be a standard family. Then for almost every p ∈ B^{k}, D^{−}(Γ, p) = 1.

However, for computations of stability D^{+} is more relevant. We proceed to discuss the effect of the interchange of limits in the definition of D^{−} and D^{+}.

5.3.2. Approximation condition. Let’s say that (Γ^{0}, p_{0}) is well approximable if for any Γ^{n} → Γ^{0}, ϵ_{n} → 0 and any neighborhood V of Γ^{0}, there exists sequences k_{n} and l_{n} such that for any sequence m_{n} ≥ l_{n} we can find a family Γ̃ ∈ V such that

lim_{n→∞} \frac{|\{p ∈ B_{ϵ_{kn}}(p_{0})|γ̃_{p} ≠ γ_{p}^{m_{n}}\}|}{|B_{ϵ_{kn}}|} = 0

and furthermore γ̃_{p_{0}} = γ_{p_{0}}^{0} (this actually follows from the last property).

Lemma 5.7. For any Γ ∈ K and p ∈ B^{k}, (Γ, p) is well approximable.

Proof. We use the following easy interpolation result, which is a consequence of existence of C^{∞} partitions of unity (this interpolation obviously fails for analytic maps):

Let U ⊂ B^{k} be open. For each compact K ⊂ U, for all ϵ > 0 there exists a δ > 0 such that if Γ′ is δ-close to Γ then there exists Γ′′ ϵ-close to Γ coinciding with Γ out of U and coinciding with Γ′ in K.

We can now obtain Γ̃ in the following way. Consider k_{n} such that ϵ_{k_{n+1}}/ϵ_{k_{n}} → 0, and let U_{n} = int(B_{ϵ_{kn}} \ B_{ϵ_{k_{n+1}}}), K_{n} ⊂ U_{n} compact such that |K_{n}|/|U_{n}| → 1.

If we now choose l_{n} growing fast enough, for m_{n} ≥ l_{n} then Γ_{m_{n}} is very close to Γ and the above interpolation result shows that we can interpolate Γ and Γ_{m_{n}} inside each U_{n}, obtaining a family Γ̃, ϵ-close to Γ, such that for each n, Γ̃ and Γ_{m_{n}} coincide in K_{n}.

Lemma 5.8. In this setting,

D^{+}(Γ^{0}, p_{0}) ≥ D^{−}(Γ^{0}, p_{0}).

Proof. Take sequences Γ^{n} and ϵ_{j} satisfying

lim_{j→∞} lim inf_{n→∞} d(Γ^{n}, B_{ϵ_{j}}(p_{0})) = D^{+}(Γ^{0}, p_{0}).

and fix a neighborhood V of Γ^{0}. Let k_{n}, l_{n} be the sequences from the approximation condition. Let m_{n} > l_{n} be a sequence such that

lim_{n→∞} d(Γ^{m_{n}}, B_{ϵ_{kn}}(p_{0})) = D^{+}(Γ^{0}, p_{0}).

and let Γ̃ be as in the approximation condition. Then it is clear that

lim_{n→∞} d(Γ̃, B_{ϵ_{kn}}(p_{0})) = D^{+}(Γ^{0}, p_{0}).

Since we can get Γ̃ to approximate Γ^{0} by shrinking V, we get D^{+} ≥ D^{−}.

5.3.3. Proof of Theorem B. Let Γ be a standard analytic family. Then almost every parameter satisfy D^{−}(Γ, p) = 1. In particular, for almost every p we have D^{+}(Γ, p) = 1.

Fix ϵ > 0. Let p ∈ B^{k} be such that D^{+}(Γ, p) = 1. By definition of D^{+} there exists a sequence of balls U^{n}(p) centered on p and converging to p, and neighborhoods V^{n}(p) ⊂ K of Γ such that if Γ̃ ∈ V^{n}(p) then

d(Γ̃, U^{n}(p)) > 1 − ϵ/2.

By Vitali’s Lemma, there exists sequences p_{j}, n_{j} such that U^{n_{j}}(p_{j}) are disjoint and |∪U^{n_{j}}(p_{j})| = 1. Let m such that ∪_{j=1}^{m}U^{n_{j}}(p_{j}) > 1−ϵ/2. Let V = ∩_{j=1}^{m}V^{n_{j}}(p_{j}). Then if Γ ∈ V, d(Γ̃, B^{k}) > 1 − ϵ.

5.3.4. Proof of the Smooth Dichotomy. Standard families are dense among C^{n} families of S-unimodal maps, n = 3, ..., ∞, so Theorem B implies that for all ϵ the set of C^{n} families of S-unimodal maps such that the set of bad parameters (not regular or BCE, CE, SE and WR) has measure less then ϵ contains an open and dense set. The result follows by a Baire argument.

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

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