Markov partitions for hyperbolic sets
type: Item
instance of: scholarly article
title: Markov partitions for hyperbolic sets
zbMATH Open document ID: 1214.37005
author: Todd Fisher
author: Himal Rathnakumara
published in: Involve
publication date: 2010-05-10T00:00:00Z
review text: Let \(M\) be a compact manifold without boundary and let \(\text{Diff}(M)\) denote the set of diffeomorphisms from \(M\) to itself. Let \(f \in \text{Diff}(M)\) be given and \(\Lambda \subset M\) be a hyperbolic set for \(f\). If \(V\) is a neighborhood of \(\Lambda\) then it was shown by the first author of this paper [Ergodic Theory Dyn. Syst. 26, No.~5, 1491--1509 (2006; Zbl 1122.37022)] that there exists a hyperbolic set \({\widetilde \Lambda}\) for \(f\) such that \(\Lambda \subset {\widetilde \Lambda}\subset V\) and \({\widetilde \Lambda}\) has a Markov partition. Then there is a canonically associated symbolic space called a subshift of finite type associated to the Markov partition. The dynamical systems of \(f\) on \({\widetilde \Lambda}\) can be studied by the method of symbolic dynamics. Recall that a dynamical system \((X, f)\) is called topologically mixing if for any open sets \(U\) and \(V\) there exists some positive integer \(N\) such that \(f^n(U)\cap V \neq \emptyset\) for all \(n\geq N\); the dynamical system is topologically transitive if there exists a point \(x\in X\) such that the forward orbit of \(x\) (with respect to \(f\)) is dense in \(X.\) The purpose of this paper under review is to strengthen the result on Markov partition in the paper mentioned above. If \(\Lambda\) is a topologically mixing hyperbolic set for \(f\in \text{Diff}(M)\) and \(V\) is a neighborhood of of \(\Lambda\), then the authors prove that there exists a hyperbolic set \({\widetilde \Lambda}\) for \(f\) containing \(\Lambda\) and contained in \(V\) such that \(({\widetilde \Lambda}, f)\) has a Markov partition coming from an associated mixing subshift of finite type. Furthermore, if \((\Lambda, f)\) is transitive, then \(({\widetilde \Lambda}, f)\) has a Markov partition coming from an associated transitive subshift of finite type. As a corollary of their main result they show that with the same datum \((\Lambda, f)\) and \(V\) as in the main theorem, there exists a…
reviewed by: Liang-Chung Hsia
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37D15
zbMATH DE Number: 5706079
zbMATH Keywords: Markov partitions
zbMATH Keywords: hyperbolic
zbMATH Keywords: entropy
zbMATH Keywords: specification
zbMATH Keywords: finitely presented
MaRDI profile type: Publication
full work available at URL: involve.2009.2.549
OpenAlex ID: W2098358376
DOI: 10.2140/INVOLVE.2009.2.549
description: scientific article; zbMATH DE number 5706079