Vilton Pinheiro

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.

On the volume of singular-hyperbolic sets

An attractor Λ for a 3-vector field X is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that C 1 + α singular-hyperbolic attractors, for any  α > 0, always have zero volume, extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular attractors. Moreover, we prove that if Λ is a singular-hyperbolic attractor for X then either it has zero volume or X is an Anosov flow. We also present examples of C 1 singular-hyperbolic attractors with positive volume. In addition, we show that C 1 generically we have volume zero for C 1 robust classes of singular-hyperbolic attractors.

Sinai?Ruelle?Bowen measures for weakly expanding maps

We construct Sinai–Ruelle–Bowen measures for endomorphisms satisfying conditions far weaker than the usual non-uniform expansion. As a consequence, the definition of a non-uniformly expanding map can be weakened. We also prove the existence of an absolutely continuous invariant measure for local diffeomorphisms, only assuming the existence of hyperbolic times for Lebesgue at almost every point in the manifold.

Shadowing by non-uniformly hyperbolic periodic points and uniform hyperbolicity

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question asked by A Katok, in a related context.

Topological structure of (partially) hyperbolic sets with positive volume

We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is larger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also give a description of the limit set of almost every point belonging to a hyperbolic set or a partially hyperbolic set with positive volume.

Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the tail of the return time function can be controlled in terms of the time generic points needed to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem. 1. Dynamical and geometrical assumptions Let M be a compact Riemannian manifold of dimension d ≥ 1 with a normalized Riemannian volume | · |, which we call Lebesgue measure. Let f : M → M be a C local diffeomorphism for all x ∈ M \ C, where C is some critical set, which may include points at which the derivative Dfx is degenerate, as well as points of discontinuity and points at which the derivative is infinite. We assume the following natural non-degeneracy condition on C, which generalizes the notion of non-flat critical points for smooth one-dimensional maps. Definition 1. The critical set C ⊂ M is non-degenerate if |C| = 0 and there is a constant β > 0 such that for every x ∈M \ C we have dist(x, C) . ‖Dfxv‖/‖v‖ . dist(x, C)−β for all v ∈ TxM , and the functions log detDf and log ‖Df−1‖ are locally Lipschitz with Lipschitz constant . dist(x, C)−β . We now state our two dynamical assumptions: the first is on the growth of the derivative and the second is on the approach rate of orbit to the critical set. Notice that for a linear map A, the condition ‖A‖ > 1 only provides information about the existence of some expanded direction, whereas the condition ‖A−1‖ 0) implies that every direction is expanded. Received by the editors November 5, 2002. 2000 Mathematics Subject Classification. Primary 37D20, 37D50, 37C40. Work carried out at the Federal University of Bahia, University of Porto and Imperial College, London. Partially supported by CMUP, PRODYN, SAPIENS and UFBA. c ©2003 American Mathematical Society