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Maximal entropy measures for piecewise affine surface homeomorphisms

Published online by Cambridge University Press:  21 May 2009

JÉRÔME BUZZI
JÉRÔME BUZZI*
Affiliation:
Centre de Mathématiques Laurent Schwartz (UMR 7640), C.N.R.S. and Ecole polytechnique, 91128 Palaiseau Cedex, France (email: jerome.buzzi@math.u-psud.fr)
*
Current address: Laboratoire de Mathématique (UMR 8628), C.N.R.S. and Université Paris-Sud, 91405 Orsay Cedex, France.
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Abstract

We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected ‘good’ returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1723 - 1763
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Billingsley, P.. Ergodic Theory and Information. Robert E. Krieger Publishing Co., Huntington, NY, 1978 (Reprint of the 1965 original).Google Scholar
[2]Bonatti, C., Díaz, L. and Viana, M.. Beyond Uniform Hyperbolicity. Springer, Berlin, 2005.Google Scholar
[3]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
[4]Boyle, M., Buzzi, J. and Gomez, R.. Almost isomorphism of countable state Markov shifts. J. Reine Angew. Math. 592 (2006), 23–47.Google Scholar
[5]Buzzi, J.. Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100 (1997), 125–161.CrossRefGoogle Scholar
[6]Buzzi, J.. Intrinsic ergodicity of Affine Maps in [0,1]d. Monatsh. Math. 124(2) (1997), 97–118.CrossRefGoogle Scholar
[7]Buzzi, J.. Ergodicité intrinsèque d’applications chaotiques fibrées. Bull. Soc. Math. France 126(1) (1998), 51–77.CrossRefGoogle Scholar
[8]Buzzi, J.. Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112 (1999), 357–380.CrossRefGoogle Scholar
[9]Buzzi, J.. Piecewise isometries have zero topological entropy. Ergod. Th. & Dynam. Sys. 21(5) (2001), 1371–1377.CrossRefGoogle Scholar
[10]Buzzi, J.. On entropy-expanding maps. Preprint CMLS, 2000.Google Scholar
[11]Buzzi, J.. Subshifts of quasi-finite type. Invent. Math. 159 (2005), 369–406.CrossRefGoogle Scholar
[12]Buzzi, J.. Puzzles of quasi-finite type, zeta functions and symbolic dynamics for multi-dimensional maps. Ann. Institut Fourier to appear.Google Scholar
[13]Buzzi, J.. Hyperbolicity through entropies, Lecture at the International Congress of Mathematical Physics, Rio, August 2006.Google Scholar
[14]Buzzi, J. and Ruette, S.. Existence of measures with maximal entropy for interval maps. Discrete Contin. Dyn. Syst. 14 (2006), 673–688.CrossRefGoogle Scholar
[15]Godbillon, C.. Travaux de D. Anosov et S. Smale sur les difféomorphismes. Séminaire Bourbaki 11 (1968–1969), Exposé No. 348, p. 13.Google Scholar
[16] A. Greven, G. Keller, G. Warnecke. (eds) Entropy (Princeton Series in Applied Mathematics). Princeton University Press, Princeton, NJ, 2003.Google Scholar
[17]Gurevič, B.. Shift entropy and Markov measures in the space of paths of a countable graph. Dokl. Akad. Nauk SSSR 192 (1970), 963–965 (Russian); English transl. Sov. Math. Dokl. 11 (1970), 744–747.Google Scholar
[18]Gurevič, B. and Savchenko, S.. Thermodynamic formalism for symbolic Markov chains with a countable number of states. Uspekhi Mat. Nauk 53(2(320)) (1998), 3–106 (in Russian); translation in Russian Math. Surveys 53(2) (1998), 245–344.Google Scholar
[19]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34(3) (1980), 213–237; Israel J. Math. 38(1–2) (1981), 107–115.CrossRefGoogle Scholar
[20]Hofbauer, F.. Periodic points for piecewise monotonic transformations. Ergod. Th. & Dynam. Sys. 5(2) (1985), 237–256.CrossRefGoogle Scholar
[21]Ishii, Y. and Sands, D.. Lap number entropy formula for piecewise affine and projective maps in several dimensions. Nonlinearity 20 (2007), 2755–2772.CrossRefGoogle Scholar
[22]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Etudes Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
[23]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems, With a Supplementary Chapter by Katok and Leonardo Mendoza (Encyclopedia of Mathematics and Its Applications, 54). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[24]Keller, G.. Lifting measures to Markov extensions. Monatsh. Math. 108(2–3) (1989), 183–200.CrossRefGoogle Scholar
[25]Kruglikov, B. and Rypdal, M.. Entropy via multiplicity. Discrete Contin. Dyn. Syst. 16(2) (2006), 395–410.CrossRefGoogle Scholar
[26]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[27]Misiurewicz, M.. Diffeomorphism without any measure with maximal entropy. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 21 (1973), 903–910.Google Scholar
[28]Misiurewicz, M. and Szlenk, W.. Entropy of piecewise monotone mappings. Studia Math. 67(1) (1980), 45–63.CrossRefGoogle Scholar
[29]Newhouse, S.. Continuity properties of entropy. Ann. of Math. (2) 129(2) (1989), 215–235.CrossRefGoogle Scholar
[30]Petersen, K.. Ergodic Theory. Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
[31]Rudolph, D.. Fundamentals of Measurable Dynamics. Clarendon Press, Oxford, 1990.Google Scholar
[32]Ruette, S.. Mixing Cr maps of the interval without maximal measure. Israel J. Math. 127 (2002), 253–277.CrossRefGoogle Scholar
[33]Tsujii, M.. Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143(2) (2001), 349–373.CrossRefGoogle Scholar
[34]Vere-Jones, D.. Ergodic properties of nonnegative matrices. Pacific J. Math. 22 (1967), 361–386.CrossRefGoogle Scholar
[35]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar
[36]Zweimuller, R.. Invariant measures for general(ized) induced transformations. Proc. Amer. Math. Soc. 133(8) (2005), 2283–2295.CrossRefGoogle Scholar