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Conformal Geometry and Dynamics

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Iteration of quasiregular tangent functions in three dimensions


Authors: A. N. Fletcher and D. A. Nicks
Journal: Conform. Geom. Dyn. 16 (2012), 1-21
MSC (2010): Primary 30C65; Secondary 30D05, 37F10
DOI: https://doi.org/10.1090/S1088-4173-2012-00236-6
Published electronically: February 7, 2012
MathSciNet review: 2888171
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Abstract: We define a new quasiregular mapping $ T:\mathbb{R}^3\to \mathbb{R}^3 \cup \{\infty \}$ that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family $ \{\lambda T:\lambda >0\}$, establishing results analogous to those of Devaney and Keen for the meromorphic family $ \{z\mapsto \lambda \tan z:\lambda >0\}$, although the methods used are necessarily original.


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  • 1. W. Bergweiler, Iteration of quasiregular mappings, Comput. Methods Funct. Theory 10 (2010), 455-481. MR 2791320
  • 2. W. Bergweiler, Fatou-Julia theory for non-uniformly quasiregular maps, to appear in Ergodic Theory Dynam. Systems, arxiv:1102.1910.
  • 3. W. Bergweiler and A. Eremenko, Dynamics of a higher dimensional analog of the trigonometric functions, Ann. Acad. Sci. Fenn. Math. 36 (2011), 165-175. MR 2797689 (2012b:37128)
  • 4. W. Bergweiler, A. Fletcher, J. K. Langley and J. Meyer, The escaping set of a quasiregular mapping, Proc. Amer. Math. Soc. 137 (2009), 641-651. MR 2448586 (2010f:30045)
  • 5. W. Bergweiler, P. J. Rippon and G. M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. Lond. Math. Soc. 97 (2008), 368-400. MR 2439666 (2010b:37122)
  • 6. R. L. Devaney and L. Keen, Dynamics of meromorphic maps: maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. 22 (1989), 55-79. MR 985854 (90e:58071)
  • 7. P. Domínguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn. Math. 23 (1998), 225-250. MR 1601879 (99b:30031)
  • 8. D. Drasin, On a method of Holopainen and Rickman, Israel J. Math. 101 (1997), 73-84. MR 1484869 (99f:30034)
  • 9. A. Eremenko, On the iteration of entire functions, Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center Publ., 23, PWN, Warsaw, (1989), 339-345. MR 1102727 (92c:30027)
  • 10. A. Fletcher and D. A. Nicks, Quasiregular dynamics on the $ n$-sphere, Ergodic Theory Dynam. Systems 31 (2011), 23-31. MR 2755919
  • 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.
  • 12. A. Hinkkanen, G. Martin and V. Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand. 95 (2004), 80-100. MR 2091483 (2005f:37094)
  • 13. T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, Oxford University Press, New York, 2001. MR 1859913 (2003c:30001)
  • 14. L. Keen and J. Kotus, Dynamics of the family $ \lambda \tan z$, Conform. Geom. Dyn. 1 (1997), 28-57. MR 1463839 (98h:58159)
  • 15. V. Mayer, Uniformly quasiregular mappings of Lattès type, Conform. Geom. Dyn. 1 (1997), 104-111. MR 1482944 (98j:30017)
  • 16. V. Mayer, Quasiregular analogues of critically finite rational functions with parabolic orbifold, J. Anal. Math. 75 (1998), 105-119. MR 1655826 (2000a:30043)
  • 17. D. A. Nicks, Wandering domains in quasiregular dynamics, pre-print, arXiv:1101.1483.
  • 18. L. Rempe, Rigidity of escaping dynamics for transcendental entire functions, Acta Math. 203 (2009), 235-267. MR 2570071 (2011b:37084)
  • 19. S. Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 26, Springer-Verlag, Berlin, 1993. MR 1238941 (95g:30026)
  • 20. P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc. 133 (2005), 1119-1126. MR 2117213 (2005j:37069)
  • 21. P. J. Rippon and G. M. Stallard, Fast escaping points of entire functions, to appear in Proc. Lond. Math. Soc. arXiv:1009.5081.
  • 22. G. Rottenfusser, J. Rückert, L. Rempe, and D. Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. 173 (2011), 77-125. MR 2753600 (2012b:37121)
  • 23. H. Siebert, Fixed points and normal families of quasiregular mappings, J. Anal. Math. 98 (2006), 145-168. MR 2254483 (2007e:30023)
  • 24. D. Sun and L. Yang, Iteration of quasi-rational mapping, Progr. Natur. Sci. (English Ed.) 11 (2001), 16-25. MR 1831577 (2002b:37059)
  • 25. V. A. Zorich, A theorem of M. A. Lavrent'ev on quasiconformal space maps, Math. USSR Sb. 3 (1967), 389-403; Transl. of Mat. Sb. (N.S.) 74 (1967), 417-433 (in Russian). MR 0223569 (36:6617)

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Additional Information

A. N. Fletcher
Affiliation: University of Warwick, Mathematics Institute, Coventry, England CV4 7AL

D. A. Nicks
Affiliation: Open University, Department of Mathematics and Statistics, Milton Keynes, England MK7 6AA
Address at time of publication: School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom

DOI: https://doi.org/10.1090/S1088-4173-2012-00236-6
Received by editor(s): December 15, 2011
Published electronically: February 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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