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Complex asymptotics of Poincaré functions and properties of Julia sets

Published online by Cambridge University Press:  01 November 2008

GREGORY DERFEL ,
PETER J. GRABNER  and
FRITZ VOGL
GREGORY DERFEL
Affiliation:
Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer Sheva 84105, Israel. e-mail: derfel@math.bgu.ac.il
PETER J. GRABNER
Affiliation:
Institut für Analysis und Computational Number Theory (Math A), Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria. e-mail: peter.grabner@tugraz.at
FRITZ VOGL
Affiliation:
Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraβe 8–10, 1040 Wien, Austria. e-mail: F.Vogl@gmx.at
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Abstract

The asymptotic behaviour of the solutions of Poincaré's functional equation f(λz) = p(f(z)) (λ > 1) for p a real polynomial of degree ≥ 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(zρF(logλz)), if f(z) → ∞ for z ∞ and z ∈ W, where F denotes a periodic function of period 1 and ρ = logλ deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 699 - 718
Copyright
Copyright © Cambridge Philosophical Society 2008

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