Skip to main content
    • Aa
    • Aa

Maximal entropy measures for piecewise affine surface homeomorphisms


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.
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 3 *
Loading metrics...

Abstract views

Total abstract views: 35 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd May 2017. This data will be updated every 24 hours.