Residually many BV homeomorphisms map a null set onto a set of full measure

Abstract

Let $Q=(0,1{)}^2$ be the unit square in ${\mathbb{R}}^2$. We prove that in a suitable complete metric space of $BV$ homeomorphisms $f:Q\to Q$ with $f_{|\partial Q}=Id$, the generical homeomorphism (in the sense of Baire categories) maps a null set in a set of full measure and vice versa. Moreover we observe that, for $1\le p<2$, in the most reasonable complete metric space for such problem, the family of $W^{1,p}$ homemomorphisms satisfying the above property is of first category, instead.