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arXiv · 0704.3952

Complex asymptotics of Poincar\'e functions and properties of Julia sets

Gregory Derfel·Peter J. Grabner·Fritz Vogl
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Abstract

The asymptotic behaviour of the solutions of Poincar\'e's functional equation $f(\lambda z)=p(f(z))$ ($\lambda>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring periodic function 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 $p$.

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Gregory Derfel, Peter J. Grabner, Fritz Vogl. 2007-04-30. Complex asymptotics of Poincar\'e functions and properties of Julia sets. https://doi.org/10.1017/s0305004108001564

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