Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

Request Permissions   Purchase Content 


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
Published electronically: May 22, 2015
MathSciNet review: 3449231
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Rufus Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11-25. MR 556580 (81g:57023)
  • [2] Yuxia Dai, Zhixiong Wen, Lifeng Xi, and Ying Xiong, Quasisymmetrically minimal Moran sets and Hausdorff dimension, Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 1, 139-151. MR 2797687 (2012e:30022),
  • [3] M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), no. 2, 365-384. MR 1107011 (92f:58097)
  • [4] Doug Hensley, Continued fractions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2351741 (2009a:11019)
  • [5] Mark Holland and Yiwei Zhang, Dimension results for inhomogeneous Moran set constructions, Dyn. Syst. 28 (2013), no. 2, 222-250. MR 3170613,
  • [6] Kenneth Falconer, Fractal geometry, Mathematical foundations and applications, John Wiley & Sons, Ltd., Chichester, 1990. MR 1102677 (92j:28008)
  • [7] Thomas Jordan and Michał Rams, Increasing digit subsystems of infinite iterated function systems, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1267-1279. MR 2869111 (2012m:28010),
  • [8] Tomasz Łuczak, On the fractional dimension of sets of continued fractions, Mathematika 44 (1997), no. 1, 50-53. MR 1464375 (98i:11059a),
  • [9] R. Daniel Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105-154. MR 1387085 (97c:28020),
  • [10] R. Daniel Mauldin and Mariusz Urbański, Graph directed Markov systems, Geometry and dynamics of limit sets, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003. MR 2003772 (2006e:37036)
  • [11] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15-23. MR 0014397 (7,278f)
  • [12] Feliks Przytycki and Mariusz Urbański, Conformal fractals: ergodic theory methods, London Mathematical Society Lecture Note Series, vol. 371, Cambridge University Press, Cambridge, 2010. MR 2656475 (2011g:37002)
  • [13] Lasse Rempe, Hyperbolic dimension and radial Julia sets of transcendental functions, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1411-1420. MR 2465667 (2010h:37100),
  • [14] Mario Roy and Mariusz Urbański, Regularity properties of Hausdorff dimension in infinite conformal iterated function systems, Ergodic Theory Dynam. Systems 25 (2005), no. 6, 1961-1983. MR 2183304 (2008c:37042),
  • [15] Mario Roy, Hiroki Sumi, and Mariusz Urbański, Analytic families of holomorphic iterated function systems, Nonlinearity 21 (2008), no. 10, 2255-2279. MR 2439479 (2009h:37096),
  • [16] Mario Roy, Hiroki Sumi, and Mariusz Urbański, Lambda-topology versus pointwise topology, Ergodic Theory Dynam. Systems 29 (2009), no. 2, 685-713. MR 2486790 (2010c:37050),
  • [17] Mario Roy and Mariusz Urbański, Random graph directed Markov systems, Discrete Contin. Dyn. Syst. 30 (2011), no. 1, 261-298. MR 2773143 (2012m:37097),
  • [18] Mariusz Urbański and Anna Zdunik, The finer geometry and dynamics of the hyperbolic exponential family, Michigan Math. J. 51 (2003), no. 2, 227-250. MR 1992945 (2004d:37068),
  • [19] Zhiying Wen, Moran sets and Moran classes, Chinese Sci. Bull. 46 (2001), no. 22, 1849-1856. MR 1877244 (2002i:28015),

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 28A80, 37C45, 37F10

Retrieve articles in all journals with MSC (2010): 28A80, 37C45, 37F10

Additional Information

Lasse Rempe-Gillen
Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom

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

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

American Mathematical Society