On the infinitesimal space of uqr mappings

TL;DR

This paper investigates the structure of infinitesimal spaces of uniformly quasiregular mappings near fixed points, revealing complex behaviors that contrast with simpler conjugate models like linear maps.

Contribution

It demonstrates that at certain fixed points, the infinitesimal space of a uniformly quasiregular mapping can be uncountably infinite, contrasting with linear conjugates.

Findings

Infinitesimal spaces can be uncountably infinite at fixed points.

Uniformly quasiregular mappings can be conjugated to maps with complex infinitesimal structures.

Contrasts with simple linear models like x/2 or 2x.

Abstract

Generalized derivatives and infinitesimal spaces generalize the idea of derivatives to mappings which need not be differentiable. It is particularly powerful in the context of quasiregular mappings, where normal family arguments imply generalized derivatives always exist. The main result of this paper is to show that if $f$ is any uniformly quasiregular mapping with $x_0$ a topologically attracting or repelling fixed point, at which $f$ is locally injective, then $f$ may be conjugated to a uniformly quasiregular mapping $g$ with fixed point $0$ and so that the infinitesimal space of $g$ at $0$ contains uncountably many elements. This should be contrasted with the fact that $f$ (and also $g$) is conjugate to $x\mapsto x/2$ or $x\mapsto 2x$ in the attracting or repelling cases respectively.