Porosity in Conformal Infinite Iterated Function Systems Original Research Article

In this paper we deal with the problem of porosity of limit sets of conformal (infinite) iterated function systems. We provide a necessary and sufficient condition for the limit sets of these systems to be porous. We pay special attention to the systems generated by continued fractions with restricted entries and we give a complete description of the subsets I of positive integers such that the set JI of all numbers whose continued fraction expansion entries are contained in I, is porous. We then study such porous sets in greater detail examining their Hausdorff dimensions, Hausdorff measures, packing measures, and other geometric characteristics. We also show that the limit set generated by the complex continued fraction algorithm is not porous, the limit sets of all plane parabolic iterated function systems are porous, and of all real parabolic iterated function systems are not porous. We provide a very effective necessary and sufficient condition for the limit set of a finite conformal iterated function system to be porous.