Characterizations of \omega-Limit Sets of Topologically Hyperbolic Systems

TL;DR

This paper generalizes characterizations of -limit sets from shifts of finite type to broader topologically hyperbolic systems, showing equivalences among various dynamical properties.

Contribution

It extends known results about -limit set characterizations to a wider class of hyperbolic systems and establishes equivalences among key properties.

Findings

Characterization of -limit sets in topologically hyperbolic systems.

Equivalence of internal chain transitivity and weak incompressibility in compact spaces.

Generalization of shift of finite type results to broader systems.

Abstract

It is well known that \omega-limit sets are internally chain transitive and have weak incompressibility; the converse is not generally true, in either case. However, it has been shown that a set is weakly incompressible if and only if it is an abstract \omega-limit set, and separately that in shifts of finite type, a set is internally chain transitive if and only if it is a (regular) \omega-limit set. In this paper we generalise these and other results, proving that the characterization for shifts of finite type holds in a variety of topologically hyperbolic systems (defined in terms of expansive and shadowing properties), and also show that the notions of internal chain transitivity and weak incompressibility coincide in compact metric spaces.