Ergodic properties of invariant measures for systems with average shadowing property

Abstract

In this paper, we explore a topological system $f:M\to M$ 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 ${\Delta }_{max}$.

average shadowing may be not enough for results in this paper, we replaced average shadowing by asymptotic average shadowing. see the paper " Ergodic properties of systems with asymptotic average shadowing property, Journal of Mathematical Analysis and Applications, 2015, Vol. 432(1), 53-73."