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Invariant domains and singularities

Published online by Cambridge University Press:  24 October 2008

Walter Bergweiler
Walter Bergweiler
Affiliation:
Lehrstuhl II für Mathematik, RWTH Aachen, D-52056 Aachen, Germany*
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Abstract

Let U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U. Let P(f) be the closure of the set of the forward orbits of all critical and asymptotic values of f. We show that there exists a sequence pn∈P(f) such that dist(pn, U) = o(|pn|), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist (P(f), U) > 0. In this example, U is bounded by a Jordan curve.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 3 , May 1995 , pp. 525 - 532
Copyright
Copyright © Cambridge Philosophical Society 1995

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