Thermodynamic formalism for Lorenz maps

For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any H\"older continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.Comment: 36 page

Ergodic Formalism for topological Attractors and historic behavior

We introduce the concept of Baire Ergodicity and Ergodic Formalism. We use them to study topological and statistical attractors, in particular to establish the existence and finiteness of such attractors. We give applications for maps of the interval, non uniformly expanding maps, partially hyperbolic systems, strongly transitive dynamics and skew-products. In dynamical systems with abundance of historic behavior (and this includes all systems with some hyperbolicity, in particular, Axiom A systems), we cannot use an invariant probability to control the asymptotic topological/statistical behavior of a generic orbit. However, the results presented here can also be applied in this context, contributing to the study of generic orbits of systems with abundance of historic behavior.Comment: 37 pages, 4 figure

Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction

We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem.Comment: 23 page