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(zρ F(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.
Original language | English |
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Pages (from-to) | 699-718 |
Number of pages | 20 |
Journal | Mathematical Proceedings of the Cambridge Philosophical Society |
Volume | 145 |
Issue number | 3 |
DOIs | |
State | Published - 1 Nov 2008 |
ASJC Scopus subject areas
- General Mathematics