Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Julia sets of uniformly quasiregular mappings are uniformly perfect

ALASTAIR N. FLETCHERa1 and DANIEL A. NICKSa2

a1 Mathematics Institute, University of Warwick, Coventry, CV4 7AL. e-mail: alastair.fletcher@warwick.ac.uk

a2 Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA. e-mail: d.nicks@open.ac.uk

Abstract

It is well known that the Julia set J(f) of a rational map f: 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: nn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

(Received December 14 2010)

(Revised May 05 2011)

(Online publication July 18 2011)