Metrical diophantine analysis for tame parabolic iterated function systems

Abstract

We study various aspects of tame finite parabolic iterated function systems that satisfy a certain open set condition. The first goal in our analysis of these systems is a detailed investigation of the conformal measure on the associated limit sets. We derive a formula that describes in a uniform way the scaling of this measure at arbitrary limit points. The second goal is to provide 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 Diophantine analysis gives rise to refinements of the description of the conformal measure in terms of Hausdorff and packing measures with respect to certain gauge functions.