Analytic and quasiregular distortion of Nagata dimension

TL;DR

This paper investigates how various classes of functions, including analytic, quasiregular, and rational maps, distort the Nagata dimension of sets in the plane, revealing both preservation and change under different conditions.

Contribution

It establishes that quasiconformal and analytic functions preserve Nagata dimension on certain domains, while polynomials and rational maps preserve it more generally, and provides examples of dimension change by entire functions.

Findings

Quasiconformal mappings preserve Nagata dimension of compact subsets.

Polynomials and rational maps preserve Nagata dimension of their domain subsets.

Entire functions can increase or decrease Nagata dimension without compactness constraints.

Abstract

We study how analytic functions, and more generally quasiregular mappings, distort Nagata dimension. Quasiconformal mappings of domains preserve the Nagata dimension of compact subsets, in view of a result of Lang and Schlichenmaier. We establish the same conclusion for analytic functions defined on general planar domains. On the other hand, polynomials (and more generally, rational maps) preserve the Nagata dimension of arbitrary subsets of their domain. In the absence of the compactness assumption, we provide examples to show that an entire function can increase or decrease the Nagata dimension of subsets of the domain. Some of these results generalize to meromorphic functions, and separately to planar quasiregular maps in view of Stoilow factorization. We also show that conformal mappings can change the porosity behavior of noncompact subsets of their domain; this yields examples of planar conformal maps which take sets of Nagata dimension strictly less than two onto set of Nagata dimension two. We conclude with open questions and potential future work related to the distortion of Nagata dimension by higher-dimensional quasiregular maps.