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The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

Published online by Cambridge University Press:  01 June 2000

GWYNETH M. STALLARD
GWYNETH M. STALLARD
Affiliation:
Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK
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Abstract

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 20 , Issue 3 , June 2000 , pp. 895 - 910
Copyright
2000 Cambridge University Press

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