On certain hyperbolic sets
type: Item
instance of: scholarly article
title: On certain hyperbolic sets
zbMATH Open document ID: 1316.37019
DOI: 10.1134/S0001434610050020
author: D. V. Anosov
published in: Mathematical Notes
publication date: 2013-09-18T00:00:00Z
review text: A hyperbolic invariant set \(F\) is called ``locally maximal'' under the action of a diffeomorphism \(S\) if there exists a neighborhood \(U\) of \(F\) such that the maximal \(S\)-invariant subset in \(U\) is \(F\). An invariant, not locally maximal set \(F\) is called ``locally premaximal'' if any neighborhood of \(F\) contains a locally maximal set which contains \(F\). There are two examples of hyperbolic sets which are not locally premaximal, due to \textit{S. Crovisier} [C. R., Math., Acad. Sci. Paris 334, No. 5, 401--404 (2002; Zbl 1042.37022)] and due to \textit{T. Fisher} [Ergodic Theory Dyn. Syst. 26, No. 5, 1491--1509 (2006; Zbl 1122.37022)]. The author shows that, for some hyperbolic set \(F'\) to be not locally premaximal, it is sufficient to be equihomeomorphic to the hyperbolic sets either from Crovisier's and from Fisher's example. This is an interesting assertion, since local premaximality (or its absence) is a property referring to the behavior of orbits off, but near \(F'\), yet equihomeomorphisms relate only the intrinsic dynamics.
Mathematics Subject Classification ID: 37D05
zbMATH DE Number: 6209065
zbMATH Keywords: locally maximal
zbMATH Keywords: hyperbolic sets
zbMATH Keywords: invariant sets
zbMATH Keywords: hyperbolicity
MaRDI profile type: Publication
cites work: A remark on locally maximal hyperbolic sets
cites work: Hyperbolic sets that are not locally maximal
cites work: Q4179562
cites work: Q4840149
description: scientific article; zbMATH DE number 6209065