Julia sets of uniformly quasiregular mappings are uniformly perfect

被引：7

作者：

Fletcher, Alastair N. ^{[1
]}

Nicks, Daniel A. ^{[2
]}

机构：

[1] Univ Warwick, Inst Math, Coventry CV4 7AL, W Midlands, England

[2] Open Univ, Dept Math & Stat, Milton Keynes MK7 6AA, Bucks, England

来源：

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY | 2011年 / 151卷

关键词：

RATIONAL FUNCTIONS; DYNAMICS; DOMAINS;

D O I：

10.1017/S0305004111000478

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

It is well known that the Julia set J(f) of a rational map f : (C)over bar -> (C)over bar 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 : (R-n)over bar -> (R-n)over bar is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

引用

页码：541 / 550

页数：10

共 24 条

[1]

[Anonymous], 1993, QUASIREGULAR MAPPING

[2]

Baker IN., 1963, MATH Z, V81, P206

[3]

BEARDON AF, 1978, J LOND MATH SOC, V18, P475

[4]

BERGWEILER W, UNIFORM PERFECTNESS

[5]

Bergweiler W, 2009, P AM MATH SOC, V137, P641

[6]

Bergweiler W, 2010, COMPUT METH FUNCT TH, V10, P455

[7]

Carleson L., 1991, Complex Dynamics

[8]

EREMENKO A, 1992, JULIA SETS ARE UNIFO

[9] Quasiregular dynamics on the n-sphere [J].

Fletcher, Alastair N. ;

Nicks, Daniel A. .

ERGODIC THEORY AND DYNAMICAL SYSTEMS, 2011, 31 :23-31

[10] JULIA SETS OF RATIONAL FUNCTIONS ARE UNIFORMLY PERFECT [J].

HINKKANEN, A .

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 1993, 113 :543-559

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