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Random iteration of analytic maps

Published online by Cambridge University Press:  04 May 2004

A. F. BEARDON ,
T. K. CARNE ,
D. MINDA  and
T. W. NG
A. F. BEARDON
Affiliation:
Centre for Mathematical Studies, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: [email protected], [email protected])
T. K. CARNE
Affiliation:
Centre for Mathematical Studies, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: [email protected], [email protected])
D. MINDA
Affiliation:
Mathematical Sciences Department, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA (e-mail: [email protected])
T. W. NG
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong (e-mail: [email protected])
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Abstract

We consider analytic maps $f_j:D\to D$ of a domain D into itself and ask when does the sequence $f_1\circ\dotsb\circ f_n$ converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence $\{w_1,w_2,\dotsc\}$ in D whose values are not taken by any fj in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 3 , June 2004 , pp. 659 - 675
Copyright
2004 Cambridge University Press

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