Large entropy implies existence of a maximal entropy measure for interval maps

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October 2006, 14(4): 673-688. doi: 10.3934/dcds.2006.14.673

Large entropy implies existence of a maximal entropy measure for interval maps

1.	Centre de Mathématiques de l'Ecole Polytechnique, U.M.R. 7640 du C.N.R.S., Ecole Polytechnique, 91128 Palaiseau Cedex
2.	Laboratoire de Mathématiques, Topologie et Dynamique, Bât. 425, Université Paris-Sud, F-91405 Orsay cedex, France

Received October 2004 Revised September 2005 Published January 2006

Abstract
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We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that h_top(f) > 2log || f'||∞ / r possesses measures with maximal entropy.

Keywords: Maximal entropy measure, Markov shift., interval map.

Mathematics Subject Classification: 37E05, 37C40, 37B4.

Citation: Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673

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