## Most expanding maps have no absolutely continuous invariant measure

### Volume 134 / 1999

Studia Mathematica 134 (1999), 69-78
DOI: 10.4064/sm-134-1-69-78

#### Abstract

We consider the topological category of various subsets of the set of expanding maps from a manifold to itself, and show in particular that a generic $C^1$ expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for $C^2$ or $C^{1+ε}$ expanding maps, for which it is known that there is always a unique absolutely continuous invariant probability measure.