Statistical properties of generalized Viana maps
We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana [Vi97]. In particular we construct a class of multidimensional non-uniformly expanding attractors that exhibit both critical points and discontinuities and prove existence and uniqueness of an SRB measure with stretched-exponential decay of correlations, stretched-exponential large deviations and satisfying some limit laws. Moreover, generically such maps admit the coexistence of a dense subset of points with negative central Lyapunov exponent together with a full Lebesgue measure subset of points which have positive Lyapunov exponents in all directions. Finally, we discuss the existence of SRB measures for skew-products associated to hyperbolic parameters by the study of fibered hyperbolic maps.
Key words and phrases:Non-uniform hyperbolicity, Lyapunov exponents, SRB measure
Since the 1960’s, when the concept of uniform hyperbolicity was coined by Smale in [Sm67], a relevant question in dynamical systems is to construct examples that exhibit the hyperbolic features described by the theory. In fact, Hunt and Mackay [HM03] proved that uniformly hyperbolic dynamical systems, among which Smale’s horseshoe is a paradigmatic example, arise naturally in physical systems. On other direction, simple one-dimensional examples arising from populational dynamics led to consider the quadratic family , or equivalently , that despite the simple formulation presents very rich and complex dynamics. In fact, it follows from pioneering works by Jakobson, Benedicks and Carleson [Jak81, BC85] that there exists a positive Lebesgue measure set such that has an absolutely continuous ergodic probability measure with positive Lyapunov exponent for all , ie, is non-uniformly hyperbolic. Later, Graczyk and Świa̧tek [GS97] and Lyubich [Lyu97] proved that is hyperbolic for an open and dense set of parameters .
Much more recently, in a major breakthrough, in [KSvS07a] Kozlovski, Shen and van Strien proved that hyperbolicity is open and dense among maps of the interval or the circle . In particular this shows that although persistent, meaning a positive Lebesgue measure phenomenon for parametrized families, one-dimensional examples with robust non-uniform hyperbolicity do not exist.
In a higher dimensional setting the existence of robust non-uniform hyperbolicity was addressed by Viana [Vi97] that introduced a class of -maps of the cyclinder: they are any small -perturbations of the skew-product transformations
where is an expanding map of the unit circle with where and for some small and parameter such that the quadratic map is of Misiurewicz type. Despite the presence of a critical region, Viana proved that this class of transformations of the cylinder have positive Lyapunov exponents in every direction, that is
for Lebesgue almost every and all . Building over this, Alves [Al00] proved that there is a unique -invariant probability measure absolutely continuous with respect to Lebesgue. In other words, this class of -endomorphisms of the cylinder present a robust non-uniform hyperbolicity phenomenon. More recent contributions and extensions include the ones by Gouzel [Gou07] on the skew products with curve of neutral fixed points, by Buzzi, Sester and Tsujii [BST03] for -perturbations of the skew product with a weaker condition and later on by Schnellmann [Sc08] that considered -transformations and by Schnellmann, Gao, Shen [Sc09, GS12] considering Misiurewicz-Thurston quadratic maps as the base dynamics. In these results the proof that Lebesgue almost every point has only positive Lyapunov exponents exploits, in the spirit of [Vi97], some weak hyperbolicity condition, namely dominated decomposition.
An important challenge in dynamics is to construct multidimensional attractors with critical behaviour without dominated splittings but persistence of positive Lyapunov exponents in parameter space. It was proposed by Bonatti, Díaz and Viana in [BDV05] that such phenomena might occur in a parametrized family . As mentioned above, important contributions were given in [Sc09, GS12] where the authors proved non-uniform expansion for skew-product of quadratic maps over a Misiurewicz-Thurston quadratic map. Since parameters corresponding to Misiurewicz-Thurston quadratic maps have zero Lebesgue measure in the parameter space then the previous question remains open.
Our purpose in this paper is to study a class of quadratic skew-products over a Markov expanding map of the interval with at most countably many inverse branches: skew-products with piecewise expanding Markov map of the unit interval and for some small and parameter . One important motivation to consider skew products of quadratic maps over expanding dynamics with infinitely many branches is to understand if the technique of inducing can be an useful approach to the previous question since non-uniform expansion is well known to be related with inducing schemes leading to piecewise expanding maps with infinitely many branches. We prove that the behaviour of this class of transformations has different behaviours depending on the parameter .
In the one hand, if is a Misiurewicz quadratic map then we overcome the difficulty caused by the presence of critical points and infinitely many invertibility branches to prove the existence of two positive Lyapunov exponents at Lebesgue almost every point, that there exists a unique absolutely continuous invariant measure and that it satisfies good statistical properties. Moreover, we also prove that generically such transformations admit the coexistence of the full Lebesgue measure set of points which have only positive Lyapunov exponents together with a dense set of points with one positive and one negative Lyapunov exponents, a fact that was unkown even in the context of Viana maps in [Vi97]. On the other hand, if the quadratic map is hyperbolic then the skew product admits an hyperbolic SRB measure, provided that is small. In fact this will be consequence of a more result for fibered hyperbolic polynomials.
The difficulties in proving our results are substantially different depending if one is considers perturbations of a Misiurewicz or a hyperbolic quadratic map. In the first case, the strategy of [Vi97] for the positivity of Lyapunov exponents Lebesgue almost everywhere is to consider the iteration of admissible curves (a class of curves preserved under iteration) and to prove that Lebesgue almost every point in each admissible curve has a slow recurrence condition to the critical region and thus has exponential growth of the derivative. The key combinatorial argument in [Vi97] does not hold in our setting since any admissible curve is mapped onto countably many of them. In our Proposition 4.1 we overcome this difficulty and obtain a statistical argument showing that, up to consider a finite iteration of the skew-product , the proportion of points in any admissible curve whose iterate intersects a small neighborhood of the critical region is bounded by the Lebesgue measure of the corresponding neighborhood. Then, a large deviations argument shows that the measure of points in admissible curves that exhibit fast recurrence to a neighborhood of the critical region decrease subexponentially and, consequently, Lebesgue almost every point has two positive Lyapunov exponents. In the second case, corresponding to perturbation of hyperbolic quadratic maps, the problem can be understood as a random composition of nearby quadratic maps and so we consider a more general setting of fibered nearby hyperbolic interval maps. The hyperbolicity of the Julia set for fibered polynomial maps in the complex variable setting has been studied by Jonsson [Jo97] and Sester [Se99]. Here we use a small neighborhood of the periodic atracting orbit to prove that there exists a trapping region and an attractor. The transversality of admissible curves implies that the attractor does not coincide with the continuation of the periodic attracting orbits, but using a uniforme contraction property one proves that it still supports a unique hyperbolic SRB measure. Roughly, the complement of the basin of attraction is formed by points whose random iteration of quadratic maps that does not intersect the trapping region. This paper is organized as follows. In Section 2 we recall some definitions and state our main results. Some preliminary results on one-dimensional dynamics and admissible curves are given along Section 3. Finally, the proofs of the main results are given on Sections 4, 5 and 6.
2. Statement of Main Results
In this section we present the necessary definitions to state our main results.
Let be an at most countable partition of the unit interval by subintervals and be a piecewise differentiable map. We will say that is a Markov expanding map if and the restriction is a diffeomorphism with a extension to the closure for any , for all , and there exists so that . The later is the so called Rènyi condition, which is a sufficient condition to obtain the bounded distortion property in Subsection 3.1. Throughout we assume that . This is a natural assumption to obtain finite positive Lyapunov exponent for and related with the size of smaller intervals of . Now we introduce the family of skew-products of the space with countably many inverse branches.
We say that a piecewise map is a generalized Viana map if it is a skew-product given by
where is a piecewise linear Markov expanding map on and for some and parameter .
The assumptions on the parameter will be crucial. Recall that the quadratic map is Misiurewicz provided that the critical point is pre-periodic repelling. It is not hard to check that there exists an interval such that for every small enough. Then we define the attractor for by
and consider the restriction . Let us mention that although it seems reasonable that some other classes of infinitely branched interval expanding maps can be considered as base dynamics without the Rènyi assumption some condition on the decay of the size of the partition elements should be necessary (e.g. otherwise could exist SRB measures without finite positive Lyapunov exponent).
Observe that we defined generalized Viana maps as a class of skew-products. We shall consider in this space the topology that we now describe. Assume, without loss of generality, that . Given we say that is --close to the skew-product above if is a piecewise map and satisfies:
, the map is a Markov expanding map on ;
and is a diffeomorphism that admits a extension to the boundary and is a Markov partition for
the renormalized maps and given respectively by
satisfy and , where denotes the Lebesgue measure of the interval with boundary points and .
Let us make some comments on our assumptions. We will assume for notational simplicity that the partition is preserved under perturbations, in which case perturbation coincides with the usual notion for interval maps. Condition (i) implies that the perturbed map is a skew-product with countably many domains of invertibility over a Markov expanding map. These are natural assumptions if one assumes the base dynamics to be induced map from some one-dimensional nonuniformly expanding map. Condition (ii) implies the domains of invertibility of to be close to those of and that the dynamics in each domain is -close to the original one. So, these assumptions require the map to be close to from both the topological and the differentiable viewpoints. Clearly, this class of maps include the skew-products considered in [Vi97].
2.2. Statement of results
We are now in a position to state our main results.
Consider the skew-product given by and such that is Misiurewicz. Then there exists such that for any small it holds
for Lebesgue almost every and every . Moreover, there exists such that the same property holds for every that is --close to .
As a byproduct of the proof we obtain some estimates on the decay of the first time at which some hyperbolicity is obtained. These are known as hyperbolic times (see [Al00] for some details). In consequence, one can use the works of Araújo, Solano [ArS11] or Pinheiro [Pi11] to build an inducing scheme and deduce the existence of an SRB measure with good statistical properties. Recall that a -invariant and ergodic probability measure is an SRB measure if its basin of attraction
has positive Lebesgue measure. Here we establish not only uniqueness of the SRB measure as we obtain several important statistical properties. Let be denote the space of -Hölder continuous observables. We obtain the following:
Let be a generalized Viana map such that is Misiurewicz. Then, for any small :
is topologically mixing;
There exists a unique -invariant measure that is absolutely continuous with respect to Lebesgue on the attractor ;
has stretched-exponential decay of correlations, that is, there exists and such that
for all large and observables and ;
has stretched-exponential large deviations, meaning that there exists such that for all and there exists satisfying
satisfies the central limit theorem, the almost sure invariance principle, the local limit theorem and the Berry-Esseen theorem for Hölder observables
Furthermore, all these properties hold for every that is -close enough to .
Our strategy to deduce the later ergodic properties is to use recent contributions to the study of stretched-exponential large deviations and limit theorems using Markov induced maps e.g. by Melbourne and Nicol [MN08], Alves, Luzzatto, Freitas, Vaienti [ALFV11] or Alves and Schnelmann [AS13]. Our next main result concerns the coexistence of a dense set of points with a negative and a positive Lyapunov exponents together with a full Lebesgue measure set of points with only positive Lyapunov exponents.
Let be a generalized Viana map such that is Misiurewicz and let be a -open set of generalized Viana maps. Then there exists an open and dense subset such that for every
there is a dense set of points with a negative Lyapunov exponent, that is,
Lebesgue almost every point in has two positive Lyapunov exponents.
In view of the previous theorem an interesting question is to understand if, at least generically, all Lyapunov exponents are bounded away from zero. The later goes in the direction of understanding possible phase transitions for the topological pressure of the generalized Viana-map with respect to the family of potentials , that is, parameters such that has none or more than one equilibrium state. For that purpose it would be important to characterize the range of the entropy among invariant and ergodic measures with one negative Lyapunov exponent.
This final part of the section is devoted to a better understanding of the dynamics of quadratic skew-products where the parameters are driven among hyperbolic parameters. In general, to obtain hiperbolicity for the composition of fibered polynomials is a hard question. Jonsson [Jo97] and Sester [Se99] studied this topic in the complex variable setting and, in particular, obtained conditions equivalent to hyperbolicity. With this in mind we state our last main result. For simplicity, we will say that an ergodic measure is hyperbolic if is has only nonzero Lyapunov exponents and not all of the same sign.
Let denote a closed interval. Assume that is a compact Riemannian manifold, that a continuous map admits a unique SRB measure , that is a hyperbolic polynomial with negative Schwarzian derivative and is a -smooth function. There exists such that if then the skew product
is such that has an ergodic SRB measure whose basin of attraction contains Lebesgue almost every point in some open and proper subset of . If, in addition, is hyperbolic then is also hyperbolic.
Let us mention that it is not clear to us wether the transversality condition of admissible curves can be used to prove that the hyperbolic SRB measure is absolutely continuous with respect to Lebesgue. Related results were obtained by Tsujii [Tsu01] and Volk [Vo11]. Finally, the previous result applies for different coupling functions as rotations, -Maneville-Pommeau maps and, in particular, directly for Viana maps whose parameters are driven along hyperbolic one as follows.
Consider the skew-product given by , where and for so that is hyperbolic. Then for every small the map has an SRB measure , it is hyperbolic and supported in a proper attractor. Moreover, the same property holds for every that is -close enough to .
Let us mention that the SRB measure above is supported on a topological atractor as described in detail in Section 7. Finally, some interesting questions are to understand if the complement of the attractor is such that the intersection with the fiber over is a totally disconnected, compact set of zero Lebesgue measure. For that it would be necessary to prove that is an expanding set for . Moreover, since almost every parameter is regular or stochastic for the quadratic map it would be interesting to understand wether either of Theorem A or Corollary A hold for quadratic skew-products and Lebesgue almost every parameter .
2.3. Some applications
Let us finish this section with some examples.
Example 2.2 (Viana maps).
The class of maps considered in [Vi97] fit in the previous setting. In fact, assume and take the Markov expanding map on given by (where stands for the integer part). Then admits a -extension to the boundary elements and one can identify the boundary points of to obtain the expanding map on the circle given by , thus recovering the previous setting. In particular, in this context Theorems A and B are consequences of [Vi97, Al00].
In this context, it follows from Theorems C that -generic transformations in the neighborhood of Viana maps exhibit coexistence of a dense set of points with one negative Lyapunov exponent while Lebesgue almost every point has only positive Lyapunov exponents. Finally, it follows from Theorem A that quadratic skew-products with parameters driven among hyperbolic ones admit a unique SRB measure, it is hyperbolic and the complement of its basin of attraction is an expanding Cantor set of lines.
In the next class of examples we present a robust class of Markov expanding maps with discontinuities and infinitely many invertibility domains.
Example 2.3 (Quadratic skew-products over piecewise linear expanding maps).
Let be an arbitrary countable partition of the unit interval in subintervals with size smaller or equal to and let be piecewise linear satisfying . Since and for all then it is clear that is a piecewise Markov expanding map and satisfies the Rènyi condition. See Figure 1.
Moreover, if is a Markov expanding map that is --close enough to then it follows from Lemma 3.1 that it also satisfies the Rènyi condition thus satisfying all the hypothesis to be used as base dynamics.
In this section we recall some definitions and preliminaries that will be used in the proof of the main results.
3.1. One dimensional dynamics
Combinatorial description of Markov expanding maps
Here we describe the Markov expanding maps from the combinatorial point of view. Let be the Markov partition for . Then there is a semi-conjugacy between the dynamics of and the full shift given by
where the semi-conjugation is given by the itinerary map defined as and is the only point in satisfying for all . Set and for any partition element define as its -th itinerary. For simplicity we will denote by the element of whose itinerary is . This will be helpful to give a precise description of points that visit to definite regions of the phase space.
Here we show that Rényi condition is an open property among Markov expanding maps and relate this with the bounded distortion property.
Assume that is a -Markov expanding map satisfying the Rènyi condition . If for small then satisfies with .
Assume that . Using that is expanding it follows that and consequently
This proves that also satisfies the Rènyi condition and proves the lemma. ∎
In the second lemma we collect some bounded distortion estimates.
Let be a -Markov expanding map satisfying and the Rènyi condition . Then, for all and
Let for some be given. We may assume, without loss of generality, that is increasing. Then
which is clearly bounded from above by . Since were arbitrary then the lower bound also holds. This finishes the proof of the lemma. ∎
Finally let us recall that it is well known that if is a Markov expanding map with the Rènyi condition then there exists a unique absolutely continuous invariant probability measure. We will use the following strong Gibbs property for Lebesgue.
Let be a -Markov expanding map satisfying and the Rènyi condition . Then, for all and
The proof is a simple application of the usual change of coordinates to the diffeomorphism by means that
together with the previous bounded distortion estimates. ∎
Hyperbolicity in dimension one
In [Sm00], Smale proposed the density of hyperbolicity in dimension one as one of the problems for the 21st century. Recall that a endomorphism of a compact interval is hyperbolic if it has finitely many hyperbolic attracting periodic points and the complement of the basins of attraction is a hyperbolic set. Such major achievement was obtained by Kozlovski, Shen and van Strien.
(Theorem 2 in [KSvS07a]) Hyperbolic maps are dense in the space of maps of the compact interval or the circle for .
Now we collect some results on a mechanism to obtain expansion in the fiber direction. Roughly, expansion for random composition of perturbations of a Misiurewicz quadratic map is obtained if orbits avoid the critical region and the loss of expansion in each return to the critical region is proportional to the return depth. More precisely,
[Vi97, Lemmas 2.4 and 2.5] Given , take , and . There are constants and (depending only on ) and , such that, for every small there exists satisfying:
for some uniform constants ;
Given an interval , for every with the iterates satisfy for every ;
for all with ;
For every with there exists so that
for every with for every ; and
for all such that with and .
Taking the previous proposition into account it is important to estimate how close typical points return close to the critical region.
3.2. Partial hyperbolicity and admissible curves
Under our assumptions on the generalized Viana maps of Definition 2.1, we obtain that the map is indeed partially hyperbolic in the sense that the dynamics along the horizontal direction dominates the dynamics along the vertical fibers. This will be made precise in terms of admissible curves as we now describe.
A curve with is an admissible curve if it is differentiable, and for every .
The strong expansion assumption on the Markov map yields a domination property as we now describe.
If is an admissible curve then for every it follows that is an admissible curve. In particular is an at most countable collection of admissible curves.
This lemma follows from [Vi97, Lemma 2.1], whose argument we reproduce here for completeness and the reader’s convenience. Since it is enough to prove the lemma for and use the argument recursively, let be an admissible curve and take . Then, for every
where, by the chain rule and definition of ,
Analogously it is not hard to check that
which is smaller than since the partial derivatives of are smaller compared with the term . This proves that is an admissible curve. Since is at most countable then is the union of at most countable admissible curves. This finishes the proof of the lemma. ∎
The crucial property of admissible curves is that their images by are non-flat.
Let be an admissible curve and set . Then or for every and
for any interval .
The proof follows the same ideas of [Vi97, Lemma 2.2] even with the presence of discontinuities for . In fact, set and notice that for all
and . If then and it follows from (3.1) that . Otherwise, for it follows that . This proves the first statement of the lemma.
Now, notice that has three connected components and for all . Moreover, since the map is -differentiable then applying the Mean Value Theorem applied to on each connected component of it follows that Using that on the two connected components of a similar argument shows that The proof of the lemma is now complete. ∎
We obtain the following very useful consequence.
Let be an admissible curve and set . Then there exists (depending only on ) such that for any subinterval satisfying it holds
First note that the case corresponds to the previous lemma. Hence, let be arbitrary and fixed. If then and is an admissible curve. Therefore, for any subinterval satisfying . Then one can use the bounded distortion property to get
where and depends only on . The proof of the corollary is now complete. ∎
Let us mention that the bound in the right hand side of the expression in Corollary 3.9 depends on and it increases when approaches zero. In fact, as will be required to be very small for the large deviations argument in Subsection 4.2, it is necessary to obtain similar estimates where the right hand side above does not depend on .
4. Positive Lyapunov exponents for the skew-product
In this section we study the recurrence of typical points near the critical region and deduce existence of positive Lyapunov exponents Lebesgue almost everywhere.
4.1. Recurrence estimates
For notational simplicity set for every , with . In the next proposition we show that for a sufficiently large iterate of the measure of the set of points which have exponential deep returns decay exponentially fast with a rate that does not involve . More precisely, if is as given in Propositon 3.5 we obtain the following.
There exists small so that every that is --close to satisfies the following property: there exists and for any given there is a positive integer such that if is an admissible curve, then
for every .
Let be a fixed arbitrary admissible curve and set . On the one hand using Corollary 3.9 we deduce that
which satisfies the assertion in the corollary with and , provided that . So, through the remaining we assume . Let be maximal such that , and note that . One can write
where is the itinerary for the elements of . The strategy is to subdivide itineraries