Mappings of finite distortion: compactness of the branch set

TL;DR

This paper investigates the properties of finite distortion mappings, demonstrating that under certain conditions, their branch sets cannot be compact, and provides a construction showing the sharpness of these bounds.

Contribution

It establishes a new growth condition preventing compact branch sets in finite distortion mappings and constructs an example illustrating the bound's optimality.

Findings

Finite distortion mappings cannot have compact branch sets under certain growth conditions.

Constructed an example with a branch set homeomorphic to an (n-2)-dimensional torus.

The example's distortion approaches the established asymptotic bound.

Abstract

We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, has a branch set homeomorphic to $(n-2)$-dimensional torus and distortion arbitrarily close to the asymptotic bound.