Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T20:29:20.472Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions

Published online by Cambridge University Press:  01 September 1999

GWYNETH M. STALLARD
GWYNETH M. STALLARD
Affiliation:
Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f be a hyperbolic transcendental meromorphic function such that the finite singularities of f−1 are in a bounded set. We show that there exists 0<s(f)[les ]2 such that

formula here

for each point a in the Julia set of f, where

formula here

We then show that s(f)[les ]dimJ(f), the Hausdorff dimension of the Julia set, and give examples of such functions for which dimJ(f)>s(f). This contrasts with the situation for a hyperbolic rational function f where it is known that dimJ(f) = s(f).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 127 , Issue 2 , September 1999 , pp. 271 - 288
Copyright
The Cambridge Philosophical Society 1999

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.)