Non-autonomous conformal iterated function systems and Moran-set constructions

Abstract

We study non-autonomous conformal iterated function systems with finite or countable infinite alphabet alike. These differ from the usual (autonomous) iterated function systems in that the contractions applied at each step in time are allowed to vary. (In the case where all maps are affine similarities, the resulting system is also called a “Moran set construction”.) We shall show that, given a suitable restriction on the growth of the number of contractions used at each step, the Hausdorff dimension of the limit set of such a system is determined by an equation known as Bowen’s formula. We also give examples that show the optimality of our results. In addition, we prove Bowen’s formula for a class of infinite alphabet-systems and deal with Hausdorff measures for finite systems, as well as continuity of topological pressure and Hausdorff dimension for both finite and infinite systems. In particular we strengthen the existing continuity results for infinite autonomous systems. As a simple application of our results, we show that, for a transcendental meromorphic function f f , the Hausdorff dimension of the set of transitive points (i.e., those points whose orbits are dense in the Julia set) is bounded from below by the hyperbolic dimension of f f .