Ergodic Theory and Dynamical Systems

Research Article

Maximal entropy measures for piecewise affine surface homeomorphisms


a1 Centre de Mathématiques Laurent Schwartz (UMR 7640), C.N.R.S. and Ecole polytechnique, 91128 Palaiseau Cedex, France (email:


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.

(Received September 24 2007)

(Revised October 11 2008)


c1 Current address: Laboratoire de Mathématique (UMR 8628), C.N.R.S. and Université Paris-Sud, 91405 Orsay Cedex, France.