  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 $p$.
