Canonical coordinates and the pseudo orbit tracing property

Abstract

S. Smale [ 131 defines a hyperbolic closed invariant set ,4 for a flow 4 on a compact manifold M. Smale’s Axiom A flows are those where the nonwandering set Q is hyperbolic and 0 is the closure of the set of periodic points. One of the most important properties of a hyperbolic set of a diffeomorphism or flow is the pseudo orbit tracing property (also known as the shadowing property); see Section 1 for the definition. The pseudo orbit tracing property (POTP) is the key of the analysis of such maps or flows [I, 5, 6, 14, 163. Bowen and Reddy [l, 2, 121 and others have tried to generalize these ideas to homeomorphisms and flows on compact metric spaces. In [lo], Ombach shows that an expansive homeomorphism has canonical coordinates if and only if it has the pseudo orbit tracing property. In this paper we investigate the same question for expansive oneparameter flows with canonical coordinates only, but free of any hyperbolic assumption. Let 4 be a continuous flow on a compact manifold A4 and X a closedinvariant subset of M. For xEX and E>O let