*Accepted Paper*

**Inserted:** 20 jul 2017

**Last Updated:** 8 nov 2018

**Journal:** Proc. Roy. Soc. Edinburgh Sect. A

**Year:** 2017

**Abstract:**

Let $Q$ be the open unit square in $\mathbb{R}^2$. We prove that in a natural complete metric space of $BV$ homeomorphisms $f:Q\rightarrow Q$ with $f_{

\partial Q}=Id$, residually many homeomorphisms
(in the sense of Baire categories) map a null set in a set of full measure, and vice versa. Moreover we observe that, for $1\leq p<2$, the family of $W^{1,p}$ homemomorphisms satisfying the above property is of first category.

**Keywords:**
Sobolev homeomorphism, Baire categories, piecewise affine homeomorphism

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