Nonlocally maximal and premaximal hyperbolic sets

Abstract

We prove that for any closed manifold of dimension 3 or greater that there is an open set of smooth flows that have a hyperbolic set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the shadowing closure of a hyperbolic set is an intrinsic property for premaximality. Lastly, we review some results due to Anosov that concern premaximality. 2000 Mathematics Subject Classification. 37D20, 37D05, 37C05.