A measure and orientation preserving homeomorphism with approximate Jacobian equal $-1$ almost everywhere

TL;DR

This paper constructs a homeomorphism of a cube that preserves measure and orientation, has Jacobian -1 almost everywhere, and can be approximated by smooth measure-preserving diffeomorphisms.

Contribution

It introduces a novel homeomorphism with measure and orientation preservation, and demonstrates its approximation by smooth diffeomorphisms, advancing understanding of Jacobian properties.

Findings

Homeomorphism with Jacobian -1 a.e.

Approximation by measure-preserving diffeomorphisms

Preserves orientation and measure almost everywhere

Abstract

We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit $n$-dimensional cube onto itself, whose Jacobian is equal to $-1$ a.e. Moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.