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Iteration of quasiregular tangent functions in three dimensions. (English) Zbl 1244.37029
Zorich constructed a quasiregular self-map of \(\mathbb{R}^3\) which can be considered as an analogue of the exponential function. The reviewer and 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.
Here 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 L. Keen and 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.

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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