A generic C1 map has no absolutely continuous invariant probability measure
Artur Avila, Jairo Bochi
Abstract
Let M be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension d 1. We consider the set of C1 maps f:M M which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual (dense Gδ) set in the C1$ 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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