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


Authors: Lasse Rempe-Gillen and Mariusz Urbański
Journal: Trans. Amer. Math. Soc. 368 (2016), 1979-2017
MSC (2010): Primary 28A80; Secondary 37C45, 37F10
DOI: https://doi.org/10.1090/tran/6490
Published electronically: May 22, 2015
MathSciNet review: 3449231
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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$, 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$.


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Additional Information

Lasse Rempe-Gillen
Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom
Email: l.rempe@liverpool.ac.uk

Mariusz Urbański
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
Email: urbanski@unt.edu

DOI: https://doi.org/10.1090/tran/6490
Received by editor(s): February 23, 2013
Received by editor(s) in revised form: February 2, 2014
Published electronically: May 22, 2015
Additional Notes: The first author was supported by EPSRC Fellowship EP/E052851/1 and a Philip Leverhulme Prize
The second author was supported in part by NSF Grant DMS 1001874.
Article copyright: © Copyright 2015 American Mathematical Society