Sensitivity and historic behavior for continuous maps on Baire metric spaces

TL;DR

This paper introduces a new notion of sensitivity for continuous maps on Baire metric spaces, providing conditions for the generic presence of points with historic behavior and extending known results to new contexts like geodesic flows and semigroup actions.

Contribution

It presents a novel sensitivity concept that ensures the genericity of historic behavior, extending classical theorems and applying to diverse dynamical systems.

Findings

Established a criterion linking sensitivity to historic behavior

Extended the understanding of irregular sets in various dynamical systems

Provided new results on geodesic flows and semigroup actions

Abstract

We introduce a notion of sensitivity, with respect to a continuous bounded observable, which provides a sufficient condition for a continuous map, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior. The applications of this criterion recover, and extend, several known theorems on the genericity of the irregular set, besides yielding a number of new results, including information on the irregular set of geodesic flows, in both negative and non-positive curvature, and semigroup actions.