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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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I. N. Baker . Infinite limits in the iteration of entire functions. Ergod. Th. & Dynam. Sys. 8(4) (1988), 503507.

K. Barański and N. Fagella . Univalent Baker domains. Nonlinearity 14(3) (2001), 411429.

W. Bergweiler . Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29(2) (1993), 151188.

W. Bergweiler . Singularities in Baker domains. Comput. Methods Funct. Theory 1(1) (2001), 4149.

W. Bergweiler . Karpińska’s paradox in dimension 3. Duke Math. J. 154(3) (2010), 599630.

W. Bergweiler . Fatou–Julia theory for non-uniformly quasiregular maps. Ergod. Th. & Dynam. Sys. 33(1) (2013), 123.

W. Bergweiler , D. Drasin and A. Fletcher . The fast escaping set for quasiregular mappings. Anal. Math. Phys. 4(1–2) (2014), 8398.

W. Bergweiler and A. Eremenko . Dynamics of a higher dimensional analog of the trigonometric functions. Ann. Acad. Sci. Fenn. Math. 36(1) (2011), 165175.

W. Bergweiler , A. Fletcher and D. A. Nicks . The Julia set and the fast escaping set of a quasiregular mapping. Comput. Methods Funct. Theory 14(2–3) (2014), 209218.

W. Bergweiler and A. Hinkkanen . On semiconjugation of entire functions. Math. Proc. Cambridge Philos. Soc. 126(3) (1999), 565574.

W. Bergweiler and D. A. Nicks . Foundations for an iteration theory of entire quasiregular maps. Israel J. Math. 201(1) (2014), 147184.

S. Daneri and A. Pratelli . A planar bi-Lipschitz extension theorem. Adv. Calc. Var. 8(3) (2015), 221266.

V. Evdoridou . Fatou’s web. Proc. Amer. Math. Soc. 144 (2016), 52275524.

P. Fatou . Sur l’itération des fonctions transcendantes entières. Acta Math. 47(4) (1926), 337370.

A. N. Fletcher and D. A. Nicks . Chaotic dynamics of a quasiregular sine mapping. J. Difference Equ. Appl. 19(8) (2013), 13531360.

A. N. Fletcher and D. A. Nicks . Superattracting fixed points of quasiregular mappings. Ergod. Th. & Dynam. Sys. 36 (2016), 781793.

P. Järvi . On the zeros and growth of quasiregular mappings. J. Anal. Math. 82 (2000), 347362.

D. Kalaj . Radial extension of a bi-Lipschitz parametrization of a starlike Jordan curve. Complex Var. Elliptic Equ. 59(6) (2014), 809825.

H. König . Conformal conjugacies in Baker domains. J. Lond. Math. Soc. (2) 59(1) (1999), 153170.

S. Rickman . Quasiregular Mappings (Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 26) . Springer, Berlin, 1993.

P. J. Rippon . Baker domains of meromorphic functions. Ergod. Th. & Dynam. Sys. 26(4) (2006), 12251233.

P. J. Rippon . Baker domains. Transcendental Dynamics and Complex Analysis (London Mathematical Society Lecture Note Series, 348) . Cambridge University Press, Cambridge, 2008, pp. 371395.

P. J. Rippon and G. M. Stallard . On sets where iterates of a meromorphic function zip towards infinity. Bull. Lond. Math. Soc. 32(5) (2000), 528536.

P. J. Rippon and G. M. Stallard . Fast escaping points of entire functions. Proc. Lond. Math. Soc. (3) 105(4) (2012), 787820.

P. J. Rippon and G. M. Stallard . A sharp growth condition for a fast escaping spider’s web. Adv. Math. 244 (2013), 337353.

P. Tukia . The planar Schönflies theorem for Lipschitz maps. Ann. Acad. Sci. Fenn. Ser. A I Math. 5(1) (1980), 4972.

M. Vuorinen . Conformal Geometry and Quasiregular Mappings (Lecture Notes in Mathematics, 1319) . Springer, Berlin, 1988.

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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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