A generic $C^1$ map has no absolutely continuous invariant probability measure

TL;DR

This paper proves that for a generic $C^1$ map on a smooth compact manifold, there are no absolutely continuous invariant probability measures, extending the Rokhlin tower lemma to non-invariant measures.

Contribution

It demonstrates that the set of $C^1$ maps lacking absolutely continuous invariant measures is residual, and introduces a generalized Rokhlin tower lemma for non-invariant measures.

Findings

Residual set of $C^1$ maps with no absolutely continuous invariant measures

Generalization of Rokhlin tower lemma to non-invariant measures

Applicable to manifolds of any dimension

Abstract

Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual (dense $G_\delta) set in the $C^1$ topology. In the course of the proof, we need a generalization of the usual Rokhlin tower lemma to non-invariant measures. That result may be of independent interest.