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Julia sets of uniformly quasiregular mappings are uniformly perfect

Published online by Cambridge University Press:  18 July 2011

ALASTAIR N. FLETCHER  and
DANIEL A. NICKS
ALASTAIR N. FLETCHER
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL. e-mail: alastair.fletcher@warwick.ac.uk
DANIEL A. NICKS
Affiliation:
Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA. e-mail: d.nicks@open.ac.uk
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Abstract

It is well known that the Julia set J(f) of a rational map f: ℂ → ℂ is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: ℝn → ℝn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 541 - 550
Copyright
Copyright © Cambridge Philosophical Society 2011

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