On invariant measures for expanding differentiable mappings

Abstract

This note concerns expanding differentiable mappings first studied by M. Shub, see [5] and [6]. These mappings are closely connected with Anosov diffeomorphisms. But while it is not known whether there always exists a finite Lebesgue measure invariant with respect to an Anosov diffeomorphism (see [1] and [6]), it turns out that such a measure always exists for any expanding differentiable mapping. The purpose of this note is to prove this fact. It seems that this may be of some interest and that is why we publish the proof although the arguments used in it have some points of similarity with the proof of Theorem 1 in [3], p. 483.