José F. Alves

SRB measures for non-hyperbolic systems with multidimensional expansion

Abstract We construct ergodic absolutely continuous invariant probability measures for an open class of non-hyperbolic surface maps introduced by Viana (1997), who showed that they exhibit two positive Lyapunov exponents at almost every point. Our approach involves an inducing procedure, based on the notion of hyperbolic time that we introduce here, and contains a theorem of existence of absolutely continuous invariant measures for multidimensional piecewise expanding maps with countably many domains of smoothness.

Statistical stability for robust classes of maps with non-uniform expansion

We consider open sets of transformations in a manifold M, exhibiting nonuniformly expanding behaviour in some forward invariant domain U ‰ M. Assuming that each transformation has a unique SRB measure in U, and some general uniformity conditions, we prove that the SRB measure varies continuously with the dynamics in the L 1 -norm. As an application we show that an open class of maps introduced in [V1] fits this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map.

On the uniform hyperbolicity of some nonuniformly hyperbolic systems

We give sufficient conditions for the uniform hyperbolicity o certain nonuni- formly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly ex- panding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.

Strong statistical stability of non-uniformly expanding maps

We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behaviour. We give sufficient conditions for the continuous variation (in the L1-norm) of the densities of absolutely continuous (with respect to the Lebesgue measure) invariant probability measures for those transformations.

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.

Hyperbolic times: frequency versus integrability

We consider dynamical systems on compact manifolds that are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that a (strong) integrability of the first hyperbolic time map implies a positive frequency of hyperbolic times. We also present an example of a map with a positive frequency of hyperbolic times at Lebesgue almost every point, but whose first hyperbolic time map is not integrable with respect to the Lebesgue measure.

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.

Ergodic properties of Viana-like maps with singularities in the base dynamics

We consider two examples of Viana maps for which the base dynamics has singularities (discontinuities or critical points) and show the existence of a unique absolutely continuous invariant probability measure and related ergodic properties such as stretched exponential decay of correlations and stretched exponential large deviations.

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.

Strong stochastic stability for non-uniformly expanding maps

We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araujo [Random perturbations of non-uniformly expanding maps. Asterisque 286 (2003), 25–62], where the stochastic stability in the