Stefano Luzzatto

The Lorenz Attractor is Mixing

We study a class of geometric Lorenz flows, introduced independently by Afraimovič, Bykov & Sil′nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.

STATISTICAL PROPERTIES OF ONE-DIMENSIONAL MAPS WITH CRITICAL POINTS AND SINGULARITIES

We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Holder observations.

The boundary of hyperbolicity for Hénon-like families

We consider two-dimensional Henon-like families of difieomorphisms Ha;b depending on parameters a;b. We study the hyperbolicity of the horse- shoe which exists for large values of the parameter a and show that this hy- perbolicity is preserved uniformly until a flrst tangency takes place.

Lyapunov exponents and rates of mixing for one-dimensional maps

We show that one dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive some power of f is mixing and in particular the correlation of Holder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, to the average rate at which typical points start to exhibit exponential growth of the derivative.

Chapter 3 – Stochastic-Like Behaviour in Nonuniformly Expanding Maps

This chapter discusses the stochastic-like behaviour in nonuniformly expanding maps. Nonuniform expansivity is a special case of nonuniform hyperbolicity. This concept was first formulated and studied by Pesin and has become one of the main areas of research in dynamical systems for extensive and in-depth surveys. The formal definition is in terms of nonzero a Lyapunov exponents that means that the tangent bundle can be decomposed into subbundles in which vectors either contract or expand at an asymptotically exponential rate. Nonuniform expansivity corresponds to the case in which all the Lyapunov exponents are positive and, therefore, all the vectors expand asymptotically at an exponential rate. The natural setting for this situation is that of (noninvertible) local diffeomorphisms, whereas the theory of nonuniform hyperbolicity has been developed mainly for diffeomorphisms. For greater generality, and because this has great importance for applications one shall allow various kinds of critical and/or singular points for f or its derivative.

Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

We formulate and prove a Jakobson?Benedicks?Carleson-type theorem on the occurrence of non-uniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family fa(x) = x2 ? a which have an absolutely continuous invariant probability measure is at least 10?5000.

Quantitative hyperbolicity estimates in one-dimensional dynamics

We develop a rigorous computational method for estimating the Lyapunov exponents in uniformly expanding regions of the phase space for one-dimensional maps. Our method uses rigorous numerics and graph algorithms to provide results that are mathematically meaningful and can be achieved in an efficient way.

Uniform hyperbolicity for random maps with positive Lyapunov exponents

We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.

Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents

We show that for any C^1+alpha diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Omega_n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets, such that for any sequence mu_n of f-invariant ergodic probability measures with supp (mu_n) in Omega_n we have mu_n -> mu in the weak-* topology.

Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the central limit theorem and a vector-valued almost sure invariance principle. We also obtain results on decay of correlations and large deviations.