Generic Points for Dynamical Systems with Average Shadowing

TL;DR

This paper establishes a connection between average shadowing properties and the existence of generic points for invariant measures in dynamical systems, using Besicovitch pseudometric properties to generalize known results.

Contribution

It introduces a new approach linking average shadowing to generic points via Besicovitch pseudometric, extending the theory to broader classes of dynamical systems.

Findings

Asymptotic pseudo orbits can be associated with invariant measures.

The set of generic points for ergodic measures is closed in the Besicovitch pseudometric.

Weak specification implies average asymptotic shadowing.

Abstract

It is proved that to every invariant measure of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for the measure. It follows that the asymptotic average shadowing property implies that every invariant measure has a generic point. The proof is based on the properties of the Besicovitch pseudometric DB which are of independent interest. It is proved among the other things that the set of generic points of ergodic measures is a closed set with respect to DB. It is also showed that the weak specification property implies the average asymptotic shadowing property thus the theory presented generalizes most known results on the existence of generic points for arbitrary invariant measures.