The average shadowing property and chaos for continuous flows

Abstract

Let X be a compact metric space and ϕ : R × X → X be a continuous flow. In this paper, we prove that if ϕ has the average shadowing property and the almost periodic points of ϕ are dense in X, then ϕ × ϕ is topologically ergodic. As a corollary, we obtain that if a Lyapunov stable flow ϕ has the average-shadowing property, then X is a singleton.

AMS Classification:

• 54H20
• 37D30
• 37C50

Keywords:

• The average-shadowing property
• Lyapunov stable
• almost periodic point
• topological ergodicity
• Auslander-Yorke chaos