Markov extensions for multi-dimensional dynamical systems

By a result of F. Hofbauer [11], piecewise monotonic maps of the interval c an be identified with topological Markov chains with respect to measures wi th large entropy. We generalize this to arbitrary piecewise invertible dyna mical systems under the following assumption: the total entropy of the syst em should be greater than the topological entropy of the boundary of some r easonable partition separating almost all orbits. We get a sufficient condi tion for these maps to have a finite number of invariant and ergodic probab ility measures with maximal entropy. We illustrate our results by quoting a n application to a class of multi-dimensional, non-linear, non-expansive sm ooth dynamical systems.