an:05939792
Zbl 1225.37031
Niu, Yingxuan; Su, Shoubao
On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property
EN
Chaos Solitons Fractals 44, No. 6, 429-432 (2011).
00281905
2011
j
37C50 37B05 37A25
asymptotic average shadowing property (AASP); ergodicity; equicontinuity; distality; Li-Yorke chaos
Summary: Let \(X\) be a compact metric space and \(f: X \rightarrow X\) be a continuous map. In this paper, we investigate the relationships between the asymptotic average shadowing property (AASP) and other notions known from topological dynamics. We prove that if \(f\) has the AASP and the minimal points of \(f\) are dense in \(X\), then for any \(n \geqslant 1\), \(f \times f \times \cdots \times f\) (\(n\) times) is totally strongly ergodic. As a corollary, it is shown that if \(f\) is surjective and equicontinuous, then \(f\) does not have the AASP. Moreover, we prove that if \(f\) is point distal, then \(f\) does not have the AASP. For \(f: [0, 1] \rightarrow [0, 1]\) being surjective and continuous, it is obtained that if \(f\) has two periodic points and the AASP, then \(f\) is Li-Yorke chaotic.