B.O. Stratmann, M. Urbanski
Metrical Diophantine Analysis For Tame Parabolic Iterated Function Systems
Preprint series: Mathematica Gottingensis

MSC:

51M15 Geometric constructions

Abstract: In this paper we study various aspects of tame parabolic
iterated function systems which satisfy a certain open
set condition. We give a detailed analysis of the conformal
measure associated to this type of system. In particular,
we derive a formula which describes in a uniform way the
scaling of this measure at arbitrary points of the associated
limit set. Furthermore, we obtain a metrical Diophantine
analysis for these parabolic limit sets in the spirit of
theorems of Jarnik and Khintchine in number theory.
Subsequently, we show that this gives rise to refinements of
the description of the conformal measure in terms of Hausdorff
and packing measures with respect to certain explicit gauge
functions.
Keywords: Fractal Geometry, Conformal Analysis, Diophantine Analysis