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Journal of the London Mathematical Society 1999 59(1):153-170; doi:10.1112/S002461079800698X
© 1999 by London Mathematical Society
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© The London Mathematical Society

Conformal Conjugacies in Baker Domains

Harald König

Institut für Mathematik, Universität Hannover Postfach 6009, D-30060 Hannover, Germany

Received 6 March 1996. Revision received 25 July 1996.

Let F be a meromorphic function in the complex plane. We investigate the behaviour of the iterates of F in a Baker domain B. In particular, we describe the dynamics of the orbits with the help of conformal conjugacies; that is, we determine a function {varphi} which is univalent in a large simply connected subdomain of B such that {varphi}(F(z))=T({varphi}(z)) holds throughout B. Here T is either a parabolic or hyperbolic Möbius transformation mapping either a half plane or C onto itself. This functional equation is always solvable in a Baker domain if F has only finitely many poles. Moreover, there is an example of a function with infinitely many poles where one cannot find an appropriate conformal conjugacy in an invariant Baker domain.


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