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${C}^{r} $ surface diffeomorphisms with no maximal entropy measure

Published online by Cambridge University Press:  05 July 2013

JÉRÔME BUZZI
JÉRÔME BUZZI*
Affiliation:
C.N.R.S. & Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France email jerome.buzzi@math.u-psud.fr
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Abstract

For any $1\leq r\lt \infty $, we build on the disk, and therefore on any manifold, a ${C}^{r} $-diffeomorphism with no measure of maximal entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 6 , December 2014 , pp. 1770 - 1793
Copyright
© Cambridge University Press, 2013 

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References

Béguin, F., Crovisier, S. and Le Roux, F.. Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique. Ann. Sci. Éc. Norm. Supér. 40 (2) (2007), 251–308.Google Scholar
Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences). Springer, New York, 2004.Google Scholar
Burguet, D.. Existence of maximal entropy measures on the interval. Proc. Amer. Math. Soc., to appear.Google Scholar
Buzzi, J.. Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100 (1) (1997), 125–161.Google Scholar
Buzzi, J.. A continuous, piecewise affine surface map with no measure of maximal entropy. Preprint, arXiv:0902.2624, 2009.Google Scholar
Buzzi, J.. Maximal entropy measures for piecewise affine surface homeomorphisms. Ergod. Th. & Dynam. Sys. 29 (6) (2009), 1723–1763.Google Scholar
Buzzi, J. and Ruette, S.. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete Contin. Dyn. Syst. 14 (4) (2006), 673–688.Google Scholar
Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, 1976.Google Scholar
Díaz, L. J., Fisher, T., Pacifico, M.-J. and Vieitez, J. L.. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete Contin. Dyn. Syst. 32 (12) (2012), 4195–4207.Google Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and Its Applications). Cambridge University Press, Cambridge, 1997.Google Scholar
Misiurewicz, M.. Diffeomorphism without any measure with maximal entropy. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 903–910.Google Scholar
Newhouse, S. E.. Continuity properties of entropy. Ann. of Math. (2) 129 (1) (1989), 215–235.Google Scholar
Ruette, S.. Mixing ${c}^{r} $ maps of the interval without maximal measure. Israel J. Math. 127 (1) (2002), 253–277.Google Scholar
Sarig, O.. Symbolic dynamics for surface diffeomorphisms. J. Amer. Math. Soc. 26 (2013), 341–426.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics). Springer, New York, 1982.CrossRefGoogle Scholar
Yomdin, Y.. Volume growth and entropy. Israel J. Math. 57 (3) (1987), 285–300.Google Scholar