Iteration of quasiregular tangent functions in three dimensions
type: Item
instance of: scholarly article
title: Iteration of quasiregular tangent functions in three dimensions
zbMATH Open document ID: 1244.37029
DOI: 10.1090/S1088-4173-2012-00236-6
author: Alastair Fletcher
author: D. A. Nicks
publication date: 2012-06-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.3589
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 6040472
zbMATH Keywords: dynamics
zbMATH Keywords: Julia set
zbMATH Keywords: quasiregular map
zbMATH Keywords: Zorich map
MaRDI profile type: Publication
arXiv ID: 1112.3589
cites work: Iteration of quasiregular mappings
cites work: Dynamics of a higher dimensional analog of the trigonometric functions
cites work: The escaping set of a quasiregular mapping
cites work: Dynamics of meromorphic functions with direct or logarithmic singularities
cites work: Dynamics of meromorphic maps : maps with polynomial schwarzian derivative
cites work: Q4354140
cites work: On a method of Holopainen and Rickman
cites work: Quasiregular dynamics on the n-sphere
cites work: Julia sets of uniformly quasiregular mappings are uniformly perfect
cites work: Local dynamics of uniformly quasiregular mappings
cites work: Geometric function theory and nonlinear analysis
cites work: Dynamics of the family 𝜆tan𝑧
cites work: Uniformly quasiregular mappings of Lattès type
cites work: Quasiregular analogues of critically finite rational functions with parabolic orbifold
cites work: Rigidity of escaping dynamics for transcendental entire functions
cites work: Q3141918
cites work: On questions of Fatou and Eremenko
cites work: Fast escaping points of entire functions
cites work: Dynamic rays of bounded-type entire functions
cites work: Fixed points and normal families of quasiregular mappings
published in: Conformal Geometry and Dynamics
review text: Zorich constructed a quasiregular self-map of \(\mathbb{R}^3\) which can be considered as an analogue of the exponential function. The reviewer and \textit{A. Eremenko} [Ann. Acad. Sci. Fenn., Math. 36, No. 1, 165--175 (2011; Zbl 1234.37015)] used a similar method to construct a quasiregular analogue of the sine and cosine function. It was shown in [the reviewer, Duke Math. J. 154, No. 3, 599--630 (2010; Zbl 1218.37057)] that the dynamics of Zorich maps are in many ways similar to those of exponential functions, and in [the reviewer and Eremenko, loc. cit.] it was showed that this similarity is also present for trigonometric functions.NEWLINENEWLINEHere the authors construct a quasiregular analogue of the tangent function. Essentially, this is achieved by composing the Zorich map with a Möbius transformation. Then they show that certain results of \textit{L. Keen} and \textit{J. Kotus} [Conform. Geom. Dyn. 1, No. 4, 28--57 (1997; Zbl 0884.30019)] on the dynamics of \(\lambda \tan z\) have analogues for these maps. For example, they show that for suitable choices of the parameters the escaping set is totally disconnected while its closure is connected.
reviewed by: Walter Bergweiler
description: scientific article; zbMATH DE number 6040472