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
Nearby in the stack
which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual (dense
Gδ)setinthe
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.