Nonlocally maximal and premaximal hyperbolic sets

TL;DR

This paper demonstrates the existence of hyperbolic sets in smooth flows on manifolds of dimension 3 or higher that are not contained in locally maximal sets, and explores properties related to premaximality.

Contribution

It establishes the presence of nonlocally maximal hyperbolic sets in higher-dimensional manifolds and analyzes the intrinsic nature of premaximality through shadowing closure stabilization.

Findings

Existence of open sets of flows with nonlocally maximal hyperbolic sets in dimension ≥3

Premaximality is characterized by stabilization of shadowing closure

Review of classical results on premaximality by Anosov

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.