Complex Continued Fractions with Restricted Entries

We study special infinite iterated function systems derived from complex continued fraction expansions with restricted entries. We focus our attention on the corresponding limit set whose Hausdorff dimension will be denoted by h. Our primary goal is to determine whether the h-dimensional Hausdorff and packing measure of the limit set is positive and finite. 1. Preliminaries The theory of uniformly hyperbolic dynamical systems leads naturally to the study of finite Markov partitions and iterated function systems obtained from a finite number of contractions. (See [6], comp. [8] for further literature.) For non-uniformly-hyperbolic dynamical systems, a part of the corresponding theory has been developed. It goes back to the papers by Schweiger [12], and Thaler [14, 15] on interval maps with an indifferent fixed point. There the concept of jump transformation is introduced and explored. It has a natural Markov partition with infinitely many cells. Further development of this subject can be found for example in [1, 2, 3, 7, 13, 17, 18]. Here we discuss particular examples of conformal repellers, obtained as limit sets of iterated function systems. Each iterated function system is obtained from an infinite number of contractions. We show that under certain conditions the repellers possess zero Hausdorff measure and positive finite packing measure. Specifically, let (X, ρ) be a compact metric space, and let I be a countable set with at least two elements. Define S = {φi : X → X | i ∈ I }, a collection of injective contractions from X to X for which there exists 0 < s < 1 such that ρ(φi(x), φi(y)) ≤ sρ(x, y), for every i ∈ I and for every pair of points x, y ∈ X . Any such collection is called an iterated function system (abbreviated as i.f.s.). Set I∗ = ⋃ m≥1 I , and, for ω ∈ I,m ≥ 1, define φω = φω1 ◦ φω2 ◦ · · · ◦ φωm . 1991 Mathematics Subject Classification. 58F23, 58F03, 58F12.