# Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian

## TL;DR

This paper constructs a homeomorphism of a unit cube that is approximately differentiable almost everywhere, orientation-preserving, with negative Jacobian almost everywhere, and satisfies a sub-Lipschitz condition, including bi-Hölder continuity.

## Contribution

It introduces a novel example of a homeomorphism with negative Jacobian almost everywhere that is also bi-Hölder continuous, expanding understanding of such mappings.

## Key findings

- Homeomorphism with negative Jacobian a.e.
- Satisfies sub-Lipschitz and bi-Hölder conditions
- Preserves orientation and has the Lusin property (N)

## Abstract

We construct an a.e. approximately differentiable homeomorphism of a unit $n$-dimensional cube onto itself which is orientation preserving, has the Lusin property (N) and has the Jacobian determinant negative a.e. Moreover, the homeomorphism together with its inverse satisfy a rather general sub-Lipschitz condition, in particular it can be bi-H\"older continuous with an arbitrary exponent less than $1$.