2019/08/29 by Alastair Fletcher, Fletcher, Alastair, Jacob Pratscher +1
Mathematics · #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1908.11320
openalex publication_date 2019/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Quasiregular maps are differentiable almost everywhere maps which are\nanalogous to holomorphic maps in the plane for higher real dimensions.\nIntroduced by Gutlyanskii et al, the infinitesimal space is a generalization of\nthe notion of derivatives for quasiregular maps. Evaluation of all elements in\nthe infinitesimal space at a particular point is called the orbit space. We\nprove that any compact connected subset of Rn\∖ 0 can be realized\nas an orbit space of a quasiconformal map. To that end, we construct analogues\nof logarithmic spiral maps and interpolation between radial stretch maps in\nhigher dimensions. For the construction of such maps, we need to implement a\nnew tool called the Zorich transform, which is a direct analogue of the\nlogarithmic transform. The Zorich transform could have further applications in\nquasiregular dynamics.\n