Measures of maximal dimension for hyperbolic diffeomorphisms
type: Item
instance of: scholarly article
title: Measures of maximal dimension for hyperbolic diffeomorphisms
zbMATH Open document ID: 1024.37025
author: Luís Barreira
author: Christian Wolf
published in: Communications in Mathematical Physics
publication date: 2003-08-21T00:00:00Z
review text: Let \(f: M\to M\) be a \(C^{1+\varepsilon}\) diffeomorphism and \(\Lambda\subset M\) a compact locally maximal hyperbolic set. It is well known that there is a unique measure of maximal entropy on \(\Lambda\). Moreover, the function \(\nu\mapsto h_{\nu}(f)\) defined on \(f\)-invariant probability measures is upper semicontinuous and its maximum satisfies the variational principle \(\max\{h_{\nu}(f)\}=h_{\text{top}}(f)\). The authors prove that if \(\dim M=2\) and \(f|_{\Lambda}\) is topologically mixing, then there is an ergodic measure \(\mu\) of maximal (Hausdorff) dimension \(\dim_H(\mu)=\max\{\dim_H(\nu)\}\). The map \(\nu\mapsto\dim_H(\mu)\) is neither upper nor lower semicontinuous. The ergodic measure of maximal dimension is not always unique, but the authors identify classes of diffeomorphisms for which the number of such measures is finite. The proofs are based on thermodynamic formalism.
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 1969274
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: hyperbolic sets
zbMATH Keywords: ergodic measures
zbMATH Keywords: thermodynamic formalism
MaRDI profile type: Publication
full work available at URL: s00220-003-0858-9
OpenAlex ID: W2008382491
Wikidata QID: Q126027216
DOI: 10.1007/S00220-003-0858-9
description: scientific article; zbMATH DE number 1969274