# American Institute of Mathematical Sciences

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

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 htop(f)  >  2log || f'||∞ / r possesses measures with maximal entropy.
Citation: Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
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