Complex asymptotics of Poincaré functions and properties of Julia sets

Details Complex asymptotics of Poincaré functions and properties of Julia sets Complex asymptotics of Poincaré functions and properties of Julia sets

2007-04-30

arxiv.org/abs/0704.3952v2

Mathematics

Complex Variables

Final version accepted for publication in Math. Proc. Camb. Philos. Soc

Scientific paper

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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