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

TL;DR

This paper demonstrates that generically, in a suitable space of BV homeomorphisms fixing the boundary, null sets can be mapped onto sets of full measure, revealing surprising measure-theoretic properties of these mappings.

Contribution

It establishes that in a complete metric space of BV homeomorphisms, the typical map can send null sets to full measure sets, contrasting with the behavior of W^{1,p} homeomorphisms for p<2.

Findings

Generic BV homeomorphisms map null sets onto full measure sets.

In the space of W^{1,p} homeomorphisms for p<2, this property is of first category.

The result highlights a stark difference between BV and Sobolev homeomorphisms regarding measure-theoretic behavior.

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 $f:Q\rightarrow Q$ 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 $1\leq p<2$, 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.