Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-04T21:06:22.234Z Has data issue: false hasContentIssue false
  • English
  • Français

On the iteration of quasimeromorphic mappings

Published online by Cambridge University Press:  06 July 2018

LUKE WARREN
LUKE WARREN*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 1 - 11
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Baker, I. N., Kotus, J. and LÜ, Y. Iterates of meromorphic functions I. Ergodic Theory Dynam. Systems. (2) 11 (1991), 241–248.Google Scholar
[2] Beardon, A. F. Iteration of Rational Functions. Grad. Texts in Math. vol.132 (Springer-Verlag, 1991).Google Scholar
[3] Bergweiler, W. Iteration of meromorphic functions. Bull. Amer. Math. Soc. (2) 29 (1993), 151–188.Google Scholar
[4] Bergweiler, W. Iteration of quasiregular mappings. Comput. Methods Funct. Theory. 10 (2010), 455–481.Google Scholar
[5] Bergweiler, W. Fatou–Julia theory for non-uniformly quasiregular maps. Ergodic Theory Dynam. Systems. (1) 33 (2013), 1–23.Google Scholar
[6] Bergweiler, W. and Nicks, D. A. Foundations for an iteration theory of entire quasiregular maps. Israel J. Math. (1) 201 (2014), 147–184.Google Scholar
[7] Domínguez, P. Dynamics of transcendental meromorphic functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 225–250.Google Scholar
[8] Fletcher, A. N. and Nicks, D. A. Quasiregular dynamics on the n-sphere. Ergodic Theory Dynam. Systems. 31 (2011), 23–31.Google Scholar
[9] Fletcher, A. N. and Nicks, D. A. Iteration of quasiregular tangent functions in three dimensions. Conform. Geom. Dyn. 16 (2012), 1–21.Google Scholar
[10] Iwaniec, T. and Martin, G. Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs (Oxford University Press, 2001).Google Scholar
[11] Martio, O. and Srebro, U. Periodic quasimeromorphic mappings in ℝn. Jour. d'An. Math. (1) 28 (1975), 20–40.Google Scholar
[12] Milnor, J. Dynamics in One Complex Variable, 3rd Edition. Ann. of Math. Stud. no.160 (Princeton University Press, 2006).Google Scholar
[13] Nicks, D. A. and Sixsmith, D. J. The dynamics of quasiregular maps of punctured space. Indiana Univ. Math. J. (to appear).Google Scholar
[14] Reshetnyak, Y. Space mappings with bounded distortion. (Russian). Sibirsk. Mat. Zh. 8 (1967), 629–659.Google Scholar
[15] Reshetnyak, Y. On the condition of the boundedness of index for mappings with bounded distortion. (Russian). Sibirsk. Mat. Zh. (2) 9 (1968), 368–374.Google Scholar
[16] Rickman, S. On the number of omitted values of entire quasiregular mappings. J. Anal. Math. 37 (1980), 100–117.Google Scholar
[17] Rickman, S. The analogue of Picard's theorem for quasiregular mappings in dimension three. Acta Math. 154 (1985), 195–242.Google Scholar
[18] Rickman, S. Quasiregular mappings. Ergeb. Math. Grenzgeb. vol.3, p. 26 (Springer-Verlag, 1993).Google Scholar
[19] Siebert, H. Fixpuntke und normale Familien quasiregulärer Abbildungen. Dissertation, University of Kiel (2004).Google Scholar
[20] Sun, D. and Yang, L. Iteration of quasi-rational mapping. Prog. Natur. Sci. (English Ed.). (1) 11 (2001), 16–25.Google Scholar
[21] Wallin, H. Metrical characterisation of conformal capacity zero. J. Math. Anal. Appl. (2) 58 (1977), 298–311.Google Scholar