Mathematical Proceedings of the Cambridge Philosophical Society



On semiconjugation of entire functions


WALTER BERGWEILER a1 and A. HINKKANEN a2 1
a1 Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany; e-mail: bergweiler@math.uni-kiel.de
a2 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 g[circ B: composite function (small circle)]f=h[circ B: composite function (small circle)]g. 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 g[circ B: composite function (small circle)]f=h[circ B: composite function (small circle)]g.

(Received December 8 1997)
(Revised April 3 1998)



Footnotes

1 A.H. was partially supported by the U.S. National Science Foundation grant DMS 94-00999.