On an Ordering of Dynamics of Homeomorphisms

Abstract

As is well known, an important part of dynamics of a homeomorphism (or a diffeomorphism) is on an invariant set, which is certainly not a manifold, such as recurrent set, nonwandering set and chain recurrent set. In many cases, the restriction of the map to such an invariant set possesses expansivity (or sensitive dependence on initial conditions, see Devaney [D] for the definition). For instance, from a result of Shub [Sh] we see that a diffeomorphism of a closed manifold can be C o-approximated by an Axiom A diffeomorphism which is expansive on the chain recurrent set.