Complex asymptotics of Poincaré functions and properties of Julia sets

Abstract

The asymptotic behaviour of the solutions of Poincaré's functional equation $f(\lambda z)=p(f(z))$ ($\lambda >1$) for $p$ a real polynomial of degree $\ge 2$ 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 $p$.

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