Analytic and quasiregular distortion of Nagata dimension

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.

Figures (3)

• Figure 1: $\varepsilon s_n$ thickening of the elements $B_{i,n}$
• Figure 2: Elements of an admissible subfamily $\mathcal{I}_n$ shown in green.
• Figure 3: The spiral domain $\Omega_p$

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2: Nagata dimension increase by entire functions
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5: Nagata dimension decrease under entire functions
• Theorem 1.6: Conformal mappings can destroy porosity
• Corollary 1.7
• Definition 2.1
• Theorem 2.2: Finite stability, Theorem 2.7 in LangSchlichenmaier2005
• Theorem 2.3: Invariance under completion, LangSchlichenmaier2005