Bernd O. Stratmann, Mariusz Urba\'nski
Jarn\'{\i}k and Julia; a Diophantine Analysis for Parabolic Rational Maps
Preprint series: Mathematica Gottingensis

MSC:

30C35 General theory of conformal mappings

Abstract: In this paper we derive a Diophantine analysis

for Julia sets of parabolic rational maps. We generalise two

theorems of Dirichlet and Jarn\'{\i}k in number theory to
the
theory of iterations of these maps.

On the basis of these results, we then derive
a
`weak multifractal analysis' of
the conformal measure
naturally
associated with a parabolic rational map. The
results in this paper
contribute to a further development of

Sullivan's famous dictionary translating between the theory
of Kleinian groups and the theory of rational maps.
Keywords: Rational maps, Julia sets, conformal measures, Diophantine analysis, multifractals