Genericity of historic behavior for maps and flows

Abstract

We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to nontrivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Mañé and Shub examples of robustly transitive diffeomorphisms. This way, our unifying approach recovers a collection of known deep theorems on the genericity of the irregular set, for both additive and sub-additive potentials, and also provides a number of new applications.

Ergodic Formalism for topological Attractors and historic behavior

“…That is, the presence of a nonperiodic transitive hyperbolic attractor implies the abundance of historic behavior! There exists an extensive bibliography about historic behavior (for instance, see [BKNRS,CV,FV,Ga,KS16,KS17,LR,Li,Ta95,Th,Ya]), in particular about the topological entropy and Hausdorff dimension of the set of points with historic behavior. Pesin and Pitskel [PP] showed that, in the full shift σ : Σ + 2 , the topological entropy of the set of with historic behavior is equal to the entropy of the whole system, i.e., h top (σ| HB(σ) ) = h top (σ) = log 2.…”