The Orbits of Generalized Derivatives

TL;DR

This paper investigates the structure of the infinitesimal space of quasiregular mappings, showing it is either trivial or uncountably large, and explores the properties of orbits within this space.

Contribution

It introduces the concept of orbits for infinitesimal spaces of quasiregular mappings and characterizes their topological properties, including realizability in dimension two.

Findings

Infinitesimal space is either singleton or uncountably infinite.

Orbits form compact, connected subsets of rac{rac{{R}^n ackslash \u00a0{0}}.

Every such orbit set in dimension two can be realized as an orbit space.

Abstract

The infinitesimal space of a quasiregular mapping was introduced by Gutlyanskii et al and generalized the idea of a derivative for this class of mappings which is only differentiable almost everywhere. In this paper, we show that the infinitesimal space is either simple, that is, it consists of only one mapping, or it contains uncountable many. To achieve this, we define the orbit of a given point as its image under all elements of the infinitesimal space. We prove that this orbit is a compact and connected subset of $\mathbb{R}^n \setminus \{ 0 \}$ and moreover, every such set in dimension two can be realized as an orbit space. We conclude with some examples exhibiting features of orbits.