A Complete Realization of the orbits of generalized derivatives of Quasiregular Mappings

TL;DR

This paper demonstrates that any compact connected subset of ^n0 can be realized as an orbit space of a quasiconformal map, introducing new tools like the Zorich transform for quasiregular dynamics.

Contribution

It constructs higher-dimensional analogues of logarithmic spiral maps and introduces the Zorich transform to realize all such orbit spaces, advancing quasiregular dynamics.

Findings

Any compact connected subset of ^n0 is realizable as an orbit space.

Introduces the Zorich transform as a new tool in quasiregular dynamics.

Develops higher-dimensional analogues of logarithmic spiral maps.

Abstract

Quasiregular maps are differentiable almost everywhere maps which are analogous to holomorphic maps in the plane for higher real dimensions. Introduced by Gutlyanskii et al, the infinitesimal space is a generalization of the notion of derivatives for quasiregular maps. Evaluation of all elements in the infinitesimal space at a particular point is called the orbit space. We prove that any compact connected subset of $\R^n\setminus\{0\}$ can be realized as an orbit space of a quasiconformal map. To that end, we construct analogues of logarithmic spiral maps and interpolation between radial stretch maps in higher dimensions. For the construction of such maps, we need to implement a new tool called the Zorich transform, which is a direct analogue of the logarithmic transform. The Zorich transform could have further applications in quasiregular dynamics.