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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Random holomorphic iterations and degenerate subdomains of the unit disk


Authors: Linda Keen and Nikola Lakic
Journal: Proc. Amer. Math. Soc. 134 (2006), 371-378
MSC (2000): Primary 32G15; Secondary 30C60, 30C70, 30C75
Published electronically: August 25, 2005
MathSciNet review: 2176004
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Abstract: Given a random sequence of holomorphic maps $f_1,f_2,f_3,\ldots$ of the unit disk $\Delta$to a subdomain $X$, we consider the compositions

\begin{displaymath}F_n=f_1 \circ f_{2} \circ \ldots f_{n-1} \circ f_n.\end{displaymath}

The sequence $\{F_n\}$ is called the iterated function system coming from the sequence $f_1,f_2,f_3,\ldots.$ We prove that a sufficient condition on the domain $X$ for all limit functions of any $\{F_n\}$ to be constant is also necessary. We prove that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.


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Additional Information

Linda Keen
Affiliation: Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York 10468
Email: linda.keen@lehman.cuny.edu

Nikola Lakic
Affiliation: Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York 10468
Email: nikola.lakic@lehman.cuny.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08280-8
PII: S 0002-9939(05)08280-8
Received by editor(s): March 8, 2004
Published electronically: August 25, 2005
Additional Notes: The first author was partially supported by a PSC-CUNY Grant
The second author was partially supported by NSF grant DMS 0200733
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society