No 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 [7], 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 \ {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.

We study the mean radius growth function for quasiconformal mappings. We give a new sub-class of quasiconformal mappings in R n , for n ≥ 2, called bounded integrable parameterization mappings, or BIP maps for short. These have the property that the restriction of the Zorich transform to each slice has uniformly bounded derivative in L n/(n−1) . For BIP maps, the logarithmic transform of the mean radius function is bi-Lipschitz. We then apply our result to BIP maps with simple infinitesimal spaces to show that the asymptotic representation is indeed quasiconformal by showing that its Zorich transform is a bi-Lipschitz map.