Characterizations of $\omega$-limit sets in      topologically hyperbolic systems

Andrew D. Barwell; Chris Good; Piotr Oprocha; Brian E. Raines

doi:10.3934/dcds.2013.33.1819

Article Contents

2013, Volume 33, Issue 5: 1819-1833. Doi: 10.3934/dcds.2013.33.1819

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Characterizations of $\omega$-limit sets in topologically hyperbolic systems

1.
Heilbronn Institute of Mathematical Research, University of Bristol, Howard House, Queens Avenue, Bristol, BS8 1SN
2.
School of Mathematics, University of Birmingham, Birmingham, B15 2TT
3.
Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków
4.
Department of Mathematics, Baylor University, Waco, TX 76798–7328

Received: November 30, 2011

Revised: April 30, 2012

Published: April 2013

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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.

Keywords:

Mathematics Subject Classification: 37D20, 37E05, 54H20.

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