Not sure if it is related, but in Lemma 1 of this paper (A.Avila, J. Bochi, A C1 generic map has no invariant absolutely continuous probability measure) it is proved that if a map has no invariant absolutely continuous probability measure, then, there exists a compact set $K$ of measure abitrarily close to $1$ which has an iterate with arbitrarily small measure.
Sorry if this has nothing to do, but from how I understood the question, at least this should be useful.