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Zbl 0868.30027
Beardon, A.F.
Iteration of analytic Euclidean contractions. (English)
[A] Ali, Rosihan M. (ed.) et al., Computational methods and function theory 1994. Proceedings of the conference, Penang, Malaysia, March 21--25, 1994. Singapore: World Scientific. Ser. Approx. Decompos. 5, 57-74 (1995). ISBN 981-02-2129-0/hbk

Summary: Let $D$ be a bounded convex domain and suppose that $f$ is an analytic map of $D$ into itself with $|f'|<1$. Then the diameter of $f^n(D)$ converges to zero and we obtain upper and lower estimates on the rate of this convergence. The methods used also give results in other circumstances; for example, for contractions of a domain in the $n$-dimensional Euclidean space.

MSC 2000:: *30D05 Functional equations in the complex domain
37F99 Complex dynamical systems

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