Markov diagrams for multi-dimensional dynamical systems
by
Jerome Buzzi
Institut de Mathematiques de Luminy

Markov diagrams have been introduced by F. Hofbauer for the study of piecewise monotonic maps of positive entropy. We give a generalization of this construction to a multi-dimensional setting, the main condition being typically that the entropy of codimension 1 submanifolds should be strictly smaller than the total entropy. In this way we get a countable Markov partition of the natural extension of our system with some control at infinity. We are able to study absolutely continuous invariant probability measures for almost all piecewise smooth, piecewise expanding multi-dimensional maps. We prove in particular that any acim is a convex combination of finitely many ergodic acim's and that the acim's are exactly the measures maximizing the pressure. We also study measures of maximal entropy for some non-expanding smooth maps: fibered perturbations of products of smooth interval maps with positive entropy. We hope that the interest of the method, as well as the main ideas behind it, will be made manifest.

Date received: March 2, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabf-17.