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

Andrea Marchese

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Submission date: 03. Jun. 2015 (revised version: June 2015)
Pages: 15
published in: Proceedings of the Royal Society of Edinburgh / A, 149 (2019) 4, p. 1047-1059 
DOI number (of the published article): 10.1017/prm.2018.104
Bibtex
with the following different title: Residually many BV homeomorphisms map a null set onto a set of full measure
MSC-Numbers: 46B35, 26B35
Keywords and phrases: Sobolev homeomorphism, baire categories, piecewise affine homeomorphism
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Abstract:
Let Q = [0,1]² be the unit square in ℝ². We prove that in a suitable complete metric space of BV homeomorphisms f : Q → Q with f_|∂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 ≤ 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.