Ergodic properties of invariant measures for systems with average shadowing property

Yiwei Dong, Xueting Tian, Xiaoping Yuan

Published 2013-12-02, updated 2015-11-19Version 3

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 $\Delta_{max}$.