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

Jérôme Buzzi; Sylvie Ruette

doi:10.3934/dcds.2006.14.673

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2006, Volume 14, Issue 4: 673-688. Doi: 10.3934/dcds.2006.14.673

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

Jérôme Buzzi^1, and
Sylvie Ruette^2,

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

Received: September 30, 2004

Revised: August 31, 2005

Published: September 2006

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Abstract

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.

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Mathematics Subject Classification: 37E05, 37C40, 37B40.

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