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    BUZZI, JÉRÔME 2014. surface diffeomorphisms with no maximal entropy measure. Ergodic Theory and Dynamical Systems, Vol. 34, Issue. 06, p. 1770.

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    YILDIZ, IZZET BURAK 2012. Discontinuity of topological entropy for Lozi maps. Ergodic Theory and Dynamical Systems, Vol. 32, Issue. 05, p. 1783.

    Sarig, Omri M. 2011. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, Vol. 5, Issue. 3, p. 593.

    Sarig, Omri M. 2011. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, Vol. 5, Issue. 3, p. 593.

  • Ergodic Theory and Dynamical Systems, Volume 29, Issue 6
  • December 2009, pp. 1723-1763

Maximal entropy measures for piecewise affine surface homeomorphisms

  • DOI:
  • Published online: 01 May 2009

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.

Corresponding author
Current address: Laboratoire de Mathématique (UMR 8628), C.N.R.S. and Université Paris-Sud, 91405 Orsay Cedex, France.
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[3]R. Bowen . Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401414.

[5]J. Buzzi . Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100 (1997), 125161.

[6]J. Buzzi . Intrinsic ergodicity of Affine Maps in [0,1]d. Monatsh. Math. 124(2) (1997), 97118.

[8]J. Buzzi . Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112 (1999), 357380.

[9]J. Buzzi . Piecewise isometries have zero topological entropy. Ergod. Th. & Dynam. Sys. 21(5) (2001), 13711377.

[11]J. Buzzi . Subshifts of quasi-finite type. Invent. Math. 159 (2005), 369406.

[19]F. Hofbauer . On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34(3) (1980), 213237; Israel J. Math.38(1–2) (1981), 107–115.

[21]Y. Ishii and D. Sands . Lap number entropy formula for piecewise affine and projective maps in several dimensions. Nonlinearity 20 (2007), 27552772.

[22]A. Katok . Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Etudes Sci. 51 (1980), 137173.

[24]G. Keller . Lifting measures to Markov extensions. Monatsh. Math. 108(2–3) (1989), 183200.

[29]S. Newhouse . Continuity properties of entropy. Ann. of Math. (2) 129(2) (1989), 215235.

[32]S. Ruette . Mixing Cr maps of the interval without maximal measure. Israel J. Math. 127 (2002), 253277.

[33]M. Tsujii . Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143(2) (2001), 349373.

[34]D. Vere-Jones . Ergodic properties of nonnegative matrices. Pacific J. Math. 22 (1967), 361386.

[35]P. Walters . An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.

[36]R. Zweimuller . Invariant measures for general(ized) induced transformations. Proc. Amer. Math. Soc. 133(8) (2005), 22832295.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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