Ergodic Formalism for topological Attractors and historic behavior

Abstract

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