Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-14T21:48:36.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Superattracting fixed points of quasiregular mappings

Published online by Cambridge University Press:  10 November 2014

ALASTAIR FLETCHER and
DANIEL A. NICKS
ALASTAIR FLETCHER
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115-2888, USA email fletcher@math.niu.edu
DANIEL A. NICKS
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK email Dan.Nicks@nottingham.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We investigate the rate of convergence of the iterates of an $n$-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 3 , May 2016, pp. 781 - 793
Copyright
© Cambridge University Press, 2014 

References

Bergweiler, W.. Fixed points of composite entire and quasiregular maps. Ann. Acad. Sci. Fenn. 31 (2006), 523–540.Google Scholar
Bergweiler, W.. Fatou-Julia theory for non-uniformly quasiregular maps. Ergod. Th. & Dynam. Sys. 33 (2013), 1–23.Google Scholar
Bergweiler, W., Drasin, D. and Fletcher, A.. The fast escaping set of a quasiregular mapping. Anal. Math. Phys. 4(1–2) (2014), 83–98.CrossRefGoogle Scholar
Bergweiler, W., Fletcher, A., Langley, J. and Meyer, J.. The escaping set of a quasiregular mapping. Proc. Amer. Math. Soc. 137 (2009), 641–651.Google Scholar
Bergweiler, W., Fletcher, A. and Nicks, D. A.. The Julia set and the fast escaping set of a quasiregular mapping. Comput. Methods Funct. Theory, to appear, doi:10.1007/s40315-014-0051-5, published online.Google Scholar
Fletcher, A.. Poincaré linearizers in higher dimensions. Proc. Amer. Math. Soc., to appear.Google Scholar
Fletcher, A. and Fryer, R.. On Böttcher coordinates and quasiregular maps. Contemp. Math. 575 (2012), 53–76 ; Volume title: Quasiconformal Mappings, Riemann Surfaces, and Teichmüller Spaces.Google Scholar
Fletcher, A. and Nicks, D. A.. Quasiregular dynamics on the n-sphere. Ergod. Th. & Dynam. Sys. 31 (2011), 23–31.Google Scholar
Gutlyanskiǐ, V. Ya., Martio, O., Ryazanov, V. I. and Vuorinen, M.. On local injectivity and asymptotic linearity of quasiregular mappings. Studia Math. 128(3) (1998), 243–271.Google Scholar
Gutlyanskiǐ, V. Ya., Martio, O., Ryazanov, V. I. and Vuorinen, M.. Infinitesimal geometry of quasiregular mappings. Ann. Acad. Sci. Fenn. Math. 25(1) (2000), 101–130.Google Scholar
Hinkkanen, A., Martin, G. and Mayer, V.. Local dynamics of uniformly quasiregular mappings. Math. Scand. 95(1) (2004), 80–100.CrossRefGoogle Scholar
Järvi, P.. On the zeros and growth of quasiregular mappings. J. Anal. Math. 82 (2000), 347–362.Google Scholar
Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160), 3rd edn. Princeton University Press, NJ, 2006.Google Scholar
Rickman, S.. Quasiregular Mappings (Ergebnisse der Mathematik und ihrer Grenzgebiete, 26). Springer, Berlin, 1993.Google Scholar
Rippon, P. and Stallard, G.. Fast escaping points of entire functions. Proc. Lond. Math. Soc. 105 (2012), 787–820.CrossRefGoogle Scholar
Sun, D. and Yang, L.. Quasirational dynamic system. Chinese Sci. Bull. 45 (2000), 1277–1279.CrossRefGoogle Scholar