  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.
