Contributions to a rigidity conjecture
type: Item
instance of: scholarly article
title: Contributions to a rigidity conjecture
zbMATH Open document ID: 0917.28015
DOI: 10.1023/A:1005816204645
published in: Acta Applicandae Mathematicae
publication date: 1999-07-20T00:00:00Z
review text: Let \(p\) and \(q\) be two relatively prime positive integers and \(\mu\) a Borel probability measure invariant and ergodic under the semigroup generated by the actions of \(z^p\) and \(z^q\) on the unit circle \(| z| =1\). An open conjecture by Furstenberg states that \(\mu\) is either atomic or the Lebesgue measure. This paper discusses known partial results in this direction and generalizes them to nonlinear expanding differentiable maps of the circle \(S^1\). Let \(f\) and \(g\) be two commuting \(C^1\) expanding maps of \(S^1\) whose degrees \(p\) and \(q\) are relatively prime, and \(\mu\) an \(f,g\)-invariant measure, ergodic for the semigroup generated by \(f\) and \(g\). Then either \(\mu\) is the (unique) measure of maximal entropy for \(f\) and \(g\) or its entropy is zero with respect to both \(f\) and \(g\). If the (common) conjugacy between \(f\), \(g\) and \(z^p\), \(z^q\), respectively, is differentiable, then the unique measure of maximal entropy for \(f\) and \(g\) is absolutely continuous. If \(\mu\) is nonatomic and mixing for either \(f\) or \(g\), then \(\mu\) is the measure of maximal entropy. Lastly, any closed invariant subset under both \(f\) and \(g\) is either finite or \(S^1\).
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 1230522
zbMATH Keywords: rigidity conjecture
zbMATH Keywords: expanding maps
zbMATH Keywords: ergodic measures
zbMATH Keywords: entropy
author: Maria Pires de Carvalho
MaRDI profile type: Publication
full work available at URL: a:1005816204645
OpenAlex ID: W17714851
description: scientific article; zbMATH DE number 1230522