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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Non-autonomous conformal iterated function systems and Moran-set constructions
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by Lasse Rempe-Gillen and Mariusz Urbański PDF
Trans. Amer. Math. Soc. 368 (2016), 1979-2017 Request permission

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 $f$.

References
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Additional Information
  • Lasse Rempe-Gillen
  • Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom
  • MR Author ID: 738017
  • ORCID: 0000-0001-8032-8580
  • 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
  • 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.
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 1979-2017
  • MSC (2010): Primary 28A80; Secondary 37C45, 37F10
  • DOI: https://doi.org/10.1090/tran/6490
  • MathSciNet review: 3449231