Ergodic properties of invariant measures for systems with average shadowing property

TL;DR

This paper investigates the ergodic properties of invariant measures in systems with the average shadowing property, extending known results and characterizing the structure and density of certain measure-related sets.

Contribution

It extends Sigmund's results to systems with average shadowing, describing the structure of invariant measure sets and their relation to orbit averages.

Findings

Every non-empty, compact, connected subset of invariant measures equals the set of accumulation points of orbit averages.

The set of points with a given measure set is dense in the support union of that measure set.

Residuality of points with full invariant measure set when the support union is isolated or entire.

Abstract

In this paper, we explore a topological system $f:M\rightarrow M$ with average shadowing property. We extend Sigmund's results and show that every non-empty, compact and connected subset $V\subseteq\mathcal {M}_{inv}(f)$ coincides with $V_f(y)$, where $\mathcal {M}_{inv}(f)$ denotes the space of invariant Borel probability measures on M, and $V_f(y)$ denotes the accumulation set of time average of Dirac measures supported at the orbit of $y$. We also show that the set $M_{V}=\{y\in M\,\,|\,\,V_{f}(y)=V\}$ is dense in $\Delta_{V}=\bigcup_{\nu\in V}supp(\nu)$. In particular, if $\Delta_{max}=\bigcup_{\nu\in\mathcal {M}_{inv}(f)}supp(\nu)$ is isolated or coincides with $M$, then $M_{max}=\{y: V_{f}(y)=\mathcal {M}_{inv}(f)\}$ is residual in $\Delta_{max}$.