Ergodic Theory and Dynamical Systems

Random iteration of analytic maps

A. F. BEARDON a1, T. K. CARNE a1, D. MINDA a2 and T. W. NG a3
a1 Centre for Mathematical Studies, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail:,
a2 Mathematical Sciences Department, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA (e-mail:
a3 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong (e-mail:

Article author query
beardon a   [Google Scholar] 
carne t   [Google Scholar] 
minda d   [Google Scholar] 
ng t   [Google Scholar] 


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.

(Received August 2 2000)
(Accepted January 15 2004)