Genericity of historic behavior for maps and flows

TL;DR

This paper provides a unifying criterion for the genericity of historic behavior in continuous maps on compact metric spaces, covering various classes of dynamical systems and recovering known results while offering new applications.

Contribution

It introduces a broad sufficient condition ensuring a residual set of points with historic behavior across multiple dynamical systems, unifying and extending previous theorems.

Findings

Residual set of historic points in minimal, non-uniquely ergodic maps

Applicability to maps with multiple invariant measures and homoclinic classes

Extension to non-uniformly expanding and partially hyperbolic diffeomorphisms

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 non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Ma\~n\'e 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.