The Lyapunov spectrum for conformal expanding maps and Axiom-A surface diffeomorphisms
type: Item
instance of: scholarly article
title: The Lyapunov spectrum for conformal expanding maps and Axiom-A surface diffeomorphisms
zbMATH Open document ID: 0948.37031
DOI: 10.1023/A:1004591209134
author: Howard Weiss
published in: Journal of Statistical Physics
publication date: 1999-11-22T00:00:00Z
review text: The Lyapunov spectrum of a map \(g\) is the map \(l(\beta)\) which assigns to any \(\beta\in \mathbb{R}\) the Hausdorff dimension of the \(\beta\)-level set of the Lyapunov exponent of \(g\). In the paper it is shown that if \(g\) is a conformal expanding map (examples of such maps include Markov maps of an interval, rational maps with hyperbolic Julia sets and conformal toral endomorphisms) and the measure of maximal dimension does not have maximal entropy then the Lyapunov spectrum \(l(\beta)\) is real analytic and strictly convex on an interval (i.e., the range of the Lyapunov exponent contains an interval) and the level sets corresponding to this interval are dense in the phase space. Similar results are proved for most Axiom-A surface diffeomorphisms. Also some rigidity results are obtained for rational maps and geodesic flows.
reviewed by: Ľubomír Snoha
Mathematics Subject Classification ID: 37F15
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37D25
zbMATH DE Number: 1366077
zbMATH Keywords: Axiom-A surface diffeomorphism
zbMATH Keywords: conformal repeller
zbMATH Keywords: dimension spectrum
zbMATH Keywords: expanding map
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: Lyapunov spectrum
zbMATH Keywords: multifractal analysis
zbMATH Keywords: pointwise dimension
zbMATH Keywords: Markov maps
MaRDI profile type: Publication
full work available at URL: a:1004591209134
OpenAlex ID: W2117328022
description: scientific article; zbMATH DE number 1366077