Random holomorphic iterations and degenerate subdomains of the unit disk
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- by Linda Keen and Nikola Lakic PDF
- Proc. Amer. Math. Soc. 134 (2006), 371-378 Request permission
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 \[ F_n=f_1 \circ f_{2} \circ \ldots f_{n-1} \circ f_n.\] 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.References
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Additional Information
- Linda Keen
- Affiliation: Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York 10468
- MR Author ID: 99725
- 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
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 371-378
- MSC (2000): Primary 32G15; Secondary 30C60, 30C70, 30C75
- DOI: https://doi.org/10.1090/S0002-9939-05-08280-8
- MathSciNet review: 2176004