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On semiconjugation of entire functions

Published online by Cambridge University Press:  01 May 1999

WALTER BERGWEILER
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany; e-mail: bergweiler@math.uni-kiel.de
A. HINKKANEN
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA; e-mail: aimo@math.uiuc.edu

Abstract

Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with gf=hg. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that gf=hg.

Type
Research Article
Copyright
The Cambridge Philosophical Society 1999

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