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Existence of measures of maximal entropy for $ \mathcal{C}^r$ interval maps


Author: David Burguet
Journal: Proc. Amer. Math. Soc. 142 (2014), 957-968
MSC (2010): Primary 37E05, 37A35
DOI: https://doi.org/10.1090/S0002-9939-2013-12067-8
Published electronically: December 23, 2013
MathSciNet review: 3148530
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Abstract: We show that a $ \mathcal {C}^{r}$ $ (r>1)$ map of the interval $ f:[0,1]\rightarrow [0,1]$ with topological entropy larger than $ \frac {\log \Vert f'\Vert _{\infty }}{r}$ admits at least one measure of maximal entropy. Moreover the number of measures of maximal entropy is finite. It is a sharp improvement of the 2006 paper of Buzzi and Ruette in the case of $ \mathcal {C}^r$ maps and solves a conjecture of J. Buzzi stated in his 1995 thesis. The proof uses a variation of a theorem of isomorphism due to J. Buzzi between the interval map and the Markovian shift associated to the Buzzi-Hofbauer diagram.


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David Burguet
Affiliation: LPMA - CNRS UMR 7599, Université Paris 6, 75252 Paris Cedex 05, France
Email: david.burguet@upmc.fr

DOI: https://doi.org/10.1090/S0002-9939-2013-12067-8
Received by editor(s): April 24, 2012
Published electronically: December 23, 2013
Communicated by: Nimish Shah
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.