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

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

2006-05-29

arxiv.org/abs/math/0605729v2

Nonlinearity 19 (2006) 2717-2725

Mathematics

Dynamical Systems

12 pages

Scientific paper

10.1088/0951-7715/19/11/001

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.

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