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

Abstract

Let Q be the open unit square in R 2 . We prove that in a natural complete metric space of BV homeomorphisms f : Q → Q with f |∂Q = Id, residually many homeomorphisms (in the sense of Baire categories) map a null set onto a set of full measure, and vice versa. Moreover we observe that, for 1 ≤ p < 2, the family of W 1,p homemomorphisms satisfying the above property is of first category.