Measures with maximum total exponent and generic properties of $C^{1}$ expanding maps

Abstract

We show that a generic $C^{1}$ expanding map on a compact Riemannian manifold has a unique measure of maximum total exponent which is fully supported and of zero entropy. We also show that for $r\ge 2$ a generic $C^{r}$ expanding map does not have fully supported measures of maximum total exponent.