Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 360583, 6 pages
http://dx.doi.org/10.1155/2011/360583
Research Article

A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

Risong Li and Xiaoliang Zhou

School of Science, Guangdong Ocean University, Zhanjiang 524025, China

Received 23 July 2011; Revised 22 September 2011; Accepted 5 October 2011

Academic Editor: Victor S. Kozyakin

Copyright © 2011 Risong Li and Xiaoliang Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove that if a continuous, Lyapunov stable map 𝑓 from a compact metric space 𝑋 into itself is topologically transitive and has the asymptotic average shadowing property, then 𝑋 is consisting of one point. As an application, we prove that the identity map 𝑖 𝑋 ∶ 𝑋 → 𝑋 does not have the asymptotic average shadowing property, where 𝑋 is a compact metric space with at least two points.