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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
34/2015

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

Andrea Marchese

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.

Received:
03.06.15
Published:
04.06.15
MSC Codes:
46B35, 26B35
Keywords:
Sobolev homeomorphism, baire categories, piecewise affine homeomorphism

Related publications

inJournal
2019 Repository Open Access
Andrea Marchese

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

In: Proceedings of the Royal Society of Edinburgh / A, 149 (2019) 4, pp. 1047-1059