On ergodicity of systems with the asymptotic average shadowing property (Q931743)

The author studies the asymptotic average shadowing property of a continuous map \(f\) of the compact metric space \((X,d)\) into itself, which he introduced in [Nonlinear Anal., Theory Methods Appl. 67, No.~6 (A), 1680--1689 (2007; Zbl 1121.37011)]. A sequence \(\{x_i\}_{i=0}^{\infty}\) of points in \(X\) is called an asymptotic average pseudo-orbit of \(f\) if \[ \lim_{n \to \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f(x_i),x_{i+1}) = 0. \] A map \(f\) is said to have the asymptotic average shadowing property, if every asymptotic average pseudo-orbit \(\{x_i\}_{i=0}^{\infty}\) of \(f\) can be asymptotically shadowed in average by some point \(z\) in \(X\), i.e., if \(\lim_{n \to \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f^{i}(z),x_{i}) = 0.\) The main result of the article is following: if the Lyapunov stable map \(f\) has the asymptotic average shadowing property then \(f\) is topologically ergodic.