The coding of non-uniformly expanding maps with application to endomorphisms of $CP^k$

Abstract

A classical approach to the study of a dynamical system is to code it using a partition. This leads to the question of the quality of this coding and in particular to the problem of finding checkable conditions ensuring that the coding is essentially one-to-one. We prove that, with respect to an invariant measure with only positive Lyapunov exponents, it is basically enough that the map be one-to-one on each piece of the partition. This is a multi-dimensional generalization of a well-known fact in one-dimensional dynamics. We apply this result to the study of the measure of maximal entropy of endomorphisms of complex projective spaces.