Porosity in Conformal Infinite Iterated Function Systems

✍ Scribed by Mariusz Urbański

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 212 KB
• Volume: 88
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2645

✦ Synopsis

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 J I 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.
2001 Academic Press

1. INTRODUCTION, PRELIMINARIES

A bounded subset X of a Euclidean space is said to be porous if there exists a positive constant c>0 such that each open ball B centered at a point of X and of an arbitrary radius 00 such that each open ball B centered at a point of X and of an arbitrary radius 0<}r 1 contains an open ball of radius cr disjoint from X. Fixing }, c is called a porosity constant of X.
It is easy to see that each porous set has the box counting dimension less than the dimension of the Euclidean space it is contained in. Further