Walter Bergweiler:

Quasinormal Families and Periodic Points

Let n >= 2 be an integer and K > 1. By fⁿ we denote the n-th iterate of a function f. Let F be the family of all functions f holomorphic in some domain such that ¦(fⁿ)'(ξ)¦ < = K whenever fⁿ(ξ)=ξ. We show that F is quasinormal of order 1. If K is sufficiently small, then F is normal. We also show that if f is a transcendental entire function, then f has a sequence (ξ_k) of periodic points of period n such that (fⁿ)'(ξ_k) -> ∞ as k -> ∞.

Bibliographical note: in "Complex Analysis and Dynamical Systems II (Nahariya, 2003)," edited by M. Agranovski, L. Karp and D. Shoikhet, Contemp. Math. 382, Amer. Math. Soc., Providence, 2005, 55-63

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[Thu Feb 19 18:56:36 2009]

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