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Periodic domains of quasiregular maps

  • DANIEL A. NICKS (a1) and DAVID J. SIXSMITH (a1)
Abstract

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$ . We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is the best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

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[1] Baker I. N.. Wandering domains in the iteration of entire functions. Proc. Lond. Math. Soc. (3) 49(3) (1984), 563576.
[2] Baker I. N.. Infinite limits in the iteration of entire functions. Ergod. Th. & Dynam. Sys. 8(4) (1988), 503507.
[3] Barański K. and Fagella N.. Univalent Baker domains. Nonlinearity 14(3) (2001), 411429.
[4] Bergweiler W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29(2) (1993), 151188.
[5] Bergweiler W.. Singularities in Baker domains. Comput. Methods Funct. Theory 1(1) (2001), 4149.
[6] Bergweiler W.. Fixed points of composite entire and quasiregular maps. Ann. Acad. Sci. Fenn. Math. 31(2) (2006), 523540.
[7] Bergweiler W.. Karpińska’s paradox in dimension 3. Duke Math. J. 154(3) (2010), 599630.
[8] Bergweiler W.. Fatou–Julia theory for non-uniformly quasiregular maps. Ergod. Th. & Dynam. Sys. 33(1) (2013), 123.
[9] Bergweiler W., Drasin D. and Fletcher A.. The fast escaping set for quasiregular mappings. Anal. Math. Phys. 4(1–2) (2014), 8398.
[10] Bergweiler W. and Eremenko A.. Dynamics of a higher dimensional analog of the trigonometric functions. Ann. Acad. Sci. Fenn. Math. 36(1) (2011), 165175.
[11] Bergweiler W., Fletcher A. and Nicks D. A.. The Julia set and the fast escaping set of a quasiregular mapping. Comput. Methods Funct. Theory 14(2–3) (2014), 209218.
[12] Bergweiler W. and Hinkkanen A.. On semiconjugation of entire functions. Math. Proc. Cambridge Philos. Soc. 126(3) (1999), 565574.
[13] Bergweiler W. and Nicks D. A.. Foundations for an iteration theory of entire quasiregular maps. Israel J. Math. 201(1) (2014), 147184.
[14] Daneri S. and Pratelli A.. A planar bi-Lipschitz extension theorem. Adv. Calc. Var. 8(3) (2015), 221266.
[15] Evdoridou V.. Fatou’s web. Proc. Amer. Math. Soc. 144 (2016), 52275524.
[16] Fatou P.. Sur l’itération des fonctions transcendantes entières. Acta Math. 47(4) (1926), 337370.
[17] Fletcher A. N. and Nicks D. A.. Chaotic dynamics of a quasiregular sine mapping. J. Difference Equ. Appl. 19(8) (2013), 13531360.
[18] Fletcher A. N. and Nicks D. A.. Superattracting fixed points of quasiregular mappings. Ergod. Th. & Dynam. Sys. 36 (2016), 781793.
[19] García-Máynez A. and Illanes A.. A survey on unicoherence and related properties. An. Inst. Mat. Univ. Nac. Autónoma México 29 (1990), 1767.
[20] Herring M. E.. Mapping properties of Fatou components. Ann. Acad. Sci. Fenn. Math. 23(2) (1998), 263274.
[21] Iwaniec T. and Martin G.. Quasiregular semigroups. Ann. Acad. Sci. Fenn. Math. 21(2) (1996), 241254.
[22] Iwaniec T. and Martin G.. Geometric Function Theory and Non-linear Analysis (Oxford Mathematical Monographs) . The Clarendon Press, Oxford University Press, New York, 2001.
[23] Järvi P.. On the zeros and growth of quasiregular mappings. J. Anal. Math. 82 (2000), 347362.
[24] Kalaj D.. Radial extension of a bi-Lipschitz parametrization of a starlike Jordan curve. Complex Var. Elliptic Equ. 59(6) (2014), 809825.
[25] König H.. Conformal conjugacies in Baker domains. J. Lond. Math. Soc. (2) 59(1) (1999), 153170.
[26] Kuratowski K.. Topology Vol. II. New edition, revised and augmented. Translated from the French by A. Kirkor. Academic Press, New York; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968.
[27] Kuusalo T.. Generalized conformal capacity and quasiconformal metrics. Proceedings of the Romanian–Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings (Braşov, 1969). Publishing House of the Academy of the Socialist Republic of Romania, Bucharest, 1971, pp. 193202.
[28] Mohri M.. Quasiconformal metric and its application to quasiregular mappings. Osaka J. Math. 21(2) (1984), 225237.
[29] Nicks D. A. and Sixsmith D.. Hollow quasi-Fatou components of quasiregular maps. Math. Proc. Cambridge Philos. Soc. (2015), published online doi:10.1017/S0305004116000840.
[30] Rickman S.. Quasiregular Mappings (Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 26) . Springer, Berlin, 1993.
[31] Rippon P. J.. Baker domains of meromorphic functions. Ergod. Th. & Dynam. Sys. 26(4) (2006), 12251233.
[32] Rippon P. J.. Baker domains. Transcendental Dynamics and Complex Analysis (London Mathematical Society Lecture Note Series, 348) . Cambridge University Press, Cambridge, 2008, pp. 371395.
[33] Rippon P. J. and Stallard G. M.. On sets where iterates of a meromorphic function zip towards infinity. Bull. Lond. Math. Soc. 32(5) (2000), 528536.
[34] Rippon P. J. and Stallard G. M.. Fast escaping points of entire functions. Proc. Lond. Math. Soc. (3) 105(4) (2012), 787820.
[35] Rippon P. J. and Stallard G. M.. A sharp growth condition for a fast escaping spider’s web. Adv. Math. 244 (2013), 337353.
[36] Tukia P.. The planar Schönflies theorem for Lipschitz maps. Ann. Acad. Sci. Fenn. Ser. A I Math. 5(1) (1980), 4972.
[37] Vuorinen M.. Conformal Geometry and Quasiregular Mappings (Lecture Notes in Mathematics, 1319) . Springer, Berlin, 1988.
[38] Zorich V. A.. A theorem of M. A. Lavrent’ev on quasiconformal space maps. Mat. Sb. (N. S.) 74(116) (1967), 417433.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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