Resumen de Complex asymptotics of Poincaré functions and properties of Julia sets

Gregory Derfel, Peter J. Grabner, Fritz Vogl

• The asymptotic behaviour of the solutions of Poincaré's functional equation f(?z) = p(f(z)) (? > 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(z? F(log?z)), if f(z) ? 8 for z 8 and z W, where F denotes a periodic function of period 1 and ? = log? 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.