Intrinsic ergodicity of smooth interval maps
type: Item
instance of: scholarly article
title: Intrinsic ergodicity of smooth interval maps
zbMATH Open document ID: 0889.28009
DOI: 10.1007/BF02773637
author: Jérôme Buzzi
published in: Israel Journal of Mathematics
publication date: 1998-06-23T00:00:00Z
review text: Let \(f\) be a \(C^\infty\) self-map of the interval \([0,1]\) that may have infinitely many critical points, i.e., points where the derivative changes its sign. The main result of this paper is the following Theorem: If \(f\) has strictly positive topological entropy, then the set of ergodic measures of maximal entropy for \(f\) is finite and non-empty, and each topologically transitive component of \(f\) carries at most one such measure. This result generalizes work of Hofbauer (who proved the same result for maps with finitely many critical points but without smoothness assumptions). It makes essential use of the topological spectral decomposition for continuous \(f\) by Blokh, of bounds on the upper semicontinuity defect of the entropy in terms of local entropy by Newhouse and by Yomdin, of a version of Hofbauer's Markov diagrams adapted to maps with infinitely many critical points, and of Gurevic's results on measures of maximal entropy for countable state topological Markov chains. In an appendix, the author constructs examples of \(C^r\) interval maps showing that the \(C^\infty\) assumption made on \(f\) is indispensible for both the existence as well as the finiteness result for the set of ergodic measures of maximal entropy for \(f\).
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 28D15
zbMATH DE Number: 1062359
zbMATH Keywords: intrinsic ergodicity
zbMATH Keywords: smooth interval map
zbMATH Keywords: local entropy
zbMATH Keywords: critical points
zbMATH Keywords: ergodic measures of maximal entropy
reviewed by: G. Keller
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773637
OpenAlex ID: W2078794687
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description: scientific article; zbMATH DE number 1062359