A Complete Realization of the Orbits of Generalized Derivatives of Quasiregular Mappings

Abstract

Quasiregular maps are differentiable almost everywhere and are analogous to holomorphic maps in the plane for higher real dimensions. Introduced by Gutlyanskii et al. (Ann Acad Sci Fenn 25:101–130, 2000), 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 $$\mathbb {R}^n\setminus \{0\}$$ 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.