zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] Walter Bergweiler, Iteration of quasiregular mappings, Comput. Methods Funct. Theory 10 (2010), no. 2, 455 – 481. · Zbl 1243.30057 · doi:10.1007/BF03321776 [2] W. Bergweiler, Fatou-Julia theory for non-uniformly quasiregular maps, to appear in Ergodic Theory Dynam. Systems, arxiv:1102.1910. [3] Walter Bergweiler and Alexandre Eremenko, Dynamics of a higher dimensional analog of the trigonometric functions, Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 1, 165 – 175. · Zbl 1234.37015 · doi:10.5186/aasfm.2011.3610 [4] Walter Bergweiler, Alastair Fletcher, Jim Langley, and Janis Meyer, The escaping set of a quasiregular mapping, Proc. Amer. Math. Soc. 137 (2009), no. 2, 641 – 651. · Zbl 1161.30008 · doi:10.1090/S0002-9939-08-09609-3 [5] Walter Bergweiler, Philip J. Rippon, and Gwyneth M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 368 – 400. · Zbl 1174.37011 · doi:10.1112/plms/pdn007 [6] Robert L. Devaney and Linda Keen, Dynamics of meromorphic maps: maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 55 – 79. · Zbl 0666.30017 [7] P. Domínguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 225 – 250. · Zbl 0892.30025 [8] David Drasin, On a method of Holopainen and Rickman, Israel J. Math. 101 (1997), 73 – 84. · Zbl 0891.30012 · doi:10.1007/BF02760922 [9] A. È. Erëmenko, On the iteration of entire functions, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 339 – 345. [10] Alastair N. Fletcher and Daniel A. Nicks, Quasiregular dynamics on the \?-sphere, Ergodic Theory Dynam. Systems 31 (2011), no. 1, 23 – 31. · Zbl 1209.37051 · doi:10.1017/S0143385709001072 [11] A. Fletcher and D. A. Nicks, Julia sets of uniformly quasiregular mappings are uniformly perfect, Math. Proc. Cambridge Philos. Soc. 151 (2011), 541-550. · Zbl 1235.37014 [12] Aimo Hinkkanen, Gaven J. Martin, and Volker Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand. 95 (2004), no. 1, 80 – 100. · Zbl 1067.30043 [13] Tadeusz Iwaniec and Gaven Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. · Zbl 1045.30011 [14] Linda Keen and Janina Kotus, Dynamics of the family \?tan\?, Conform. Geom. Dyn. 1 (1997), 28 – 57 (electronic). · Zbl 0884.30019 · doi:10.1090/S1088-4173-97-00017-9 [15] Volker Mayer, Uniformly quasiregular mappings of Lattès type, Conform. Geom. Dyn. 1 (1997), 104 – 111 (electronic). · Zbl 0897.30008 · doi:10.1090/S1088-4173-97-00013-1 [16] Volker Mayer, Quasiregular analogues of critically finite rational functions with parabolic orbifold, J. Anal. Math. 75 (1998), 105 – 119. · Zbl 0911.30018 · doi:10.1007/BF02788694 [17] D. A. Nicks, Wandering domains in quasiregular dynamics, pre-print, arXiv:1101.1483. [18] Lasse Rempe, Rigidity of escaping dynamics for transcendental entire functions, Acta Math. 203 (2009), no. 2, 235 – 267. · Zbl 1226.37027 · doi:10.1007/s11511-009-0042-y [19] Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. · Zbl 0816.30017 [20] P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1119 – 1126 (electronic). · Zbl 1058.37033 · doi:10.1090/S0002-9939-04-07805-0 [21] P. J. Rippon and G. M. Stallard, Fast escaping points of entire functions, to appear in Proc. Lond. Math. Soc. arXiv:1009.5081. · Zbl 1291.30160 [22] Günter Rottenfusser, Johannes Rückert, Lasse Rempe, and Dierk Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2) 173 (2011), no. 1, 77 – 125. · Zbl 1232.37025 · doi:10.4007/annals.2011.173.1.3 [23] Heike Siebert, Fixed points and normal families of quasiregular mappings, J. Anal. Math. 98 (2006), 145 – 168. · Zbl 1133.30322 · doi:10.1007/BF02790273 [24] Daochun Sun and Lo Yang, Iteration of quasi-rational mapping, Progr. Natur. Sci. 11 (2001), no. 1, 16 – 25. [25] V. A. Zorič, M. A. Lavrent$$^{\prime}$$ev’s theorem on quasiconformal space maps, Mat. Sb. (N.S.) 74 (116) (1967), 417 – 433 (Russian).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.