Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T17:59:13.768Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Published online by Cambridge University Press:  24 November 2009

KEMAL ILGAR EROĞLU ,
STEFFEN ROHDE  and
BORIS SOLOMYAK
KEMAL ILGAR EROĞLU
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD) FI-40014, Finland (email: kieroglu@hotmail.com)
STEFFEN ROHDE
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle WA 98195, USA (email: rohde@math.washington.edu, solomyak@math.washington.edu)
BORIS SOLOMYAK
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle WA 98195, USA (email: rohde@math.washington.edu, solomyak@math.washington.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the ‘overlap set’ 𝒪 is finite, and which are ‘invertible’ on the attractor A, in the sense that there is a continuous surjection q:A→A whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that q is not a local homeomorphism precisely at 𝒪. We suppose also that there is a rational function p with the Julia set J such that (A,q) and (J,p) are conjugate. We prove that if A has bounded turning and p has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS {λz,λz+1} where λ is a complex parameter in the unit disk, such that its attractor Aλ is a dendrite, which happens whenever 𝒪 is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map qλ on Aλ. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map pc (z)=z2 +c, with the Julia set Jc such that (Aλ,qλ) and (Jc,pc) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1665 - 1684
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aikawa, H., Lundh, T. and Mizutani, T.. Martin boundary of a fractal domain. Potential Anal. 18(4) (2003), 311–357.CrossRefGoogle Scholar
[2]Bandt, C.. On the Mandelbrot set for pairs of linear maps. Nonlinearity 15 (2002), 1127–1147.CrossRefGoogle Scholar
[3]Bandt, C. and Keller, K.. Self-similar sets 2. A simple approach to the topological structure of fractals. Math. Nachr. 154 (1991), 27–39.CrossRefGoogle Scholar
[4]Bandt, C. and Rao, H.. Topology and separation of self-similar fractals in the plane. Nonlinearity 20 (2007), 1463–1474.CrossRefGoogle Scholar
[5]Barnsley, M. F. and Harrington, A. N.. A Mandelbrot set for pairs of linear maps. Phys. D 15 (1985), 421–432.CrossRefGoogle Scholar
[6]Beaucoup, F., Borwein, P., Boyd, D. W. and Pinner, C.. Multiple roots of [−1,1] power series. J. Lond. Math. Soc. 57 (1998), 135–147.CrossRefGoogle Scholar
[7]Bousch, T.. Connexité locale et par chemins hölderiens pour les systèmes itérés de fonctions. Preprint, 1993, http://topo.math.u-psud.fr/∼bousch.Google Scholar
[8]Bruin, H. and Schleicher, D.. Symbolic dynamics of quadratic polynomials. Institut Mittag–Leffler, Preprint, 2002.Google Scholar
[9]Bruin, H. and Schleicher, D.. Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials. Preprint, 2008, arXiv:0801.4662v1.CrossRefGoogle Scholar
[10]Carleson, L., Jones, P. and Yoccoz, J.-Ch.. Julia and John. Bol. Soc. Bras. Mat. 25 (1994), 1–30.CrossRefGoogle Scholar
[11]Devaney, R. L., Rocha, M. M. and Siegmund, S.. Rational maps with generalized Sierpiński gasket Julia sets. Topology Appl. 154 (2007), 11–27.CrossRefGoogle Scholar
[12]Douady, A. and Hubbard, J.. Études dynamiques des polynômes complexes. Publications Mathématiques d’Orsay 84-02 and 85-04, Département de Mathématiques d’Orsay, Université Paris-Sud 11, 1984 and 1985.Google Scholar
[13]Ghamsari, M., Näkki, R. and Väisälä, J.. John disks and extension of maps. Monatsh. Math. 117(1–2) (1994), 63–94.CrossRefGoogle Scholar
[14]Haïssinsky, P. and Pilgrim, K.. Coarse expanding conformal dynamics. Astérisque to appear, Preprint, 2006, arXiv: math.DS/0612617.Google Scholar
[15]Hata, M.. On the structure of self-similar sets. Japan J. Appl. Math. 2 (1985), 381–414.CrossRefGoogle Scholar
[16]Heinonen, J.. Lectures on Analysis on Metric Spaces. Springer, New York, 2001.CrossRefGoogle Scholar
[17]Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[18]Kaffl, A.. On the structure of abstract Hubbard trees and the space of abstract kneading sequences of degree two. Ergod. Th. & Dynam. Sys. 27 (2007), 1215–1238.CrossRefGoogle Scholar
[19]Kameyama, A.. Julia sets and self-similar sets. Topology Appl. 54 (1993), 241–251.CrossRefGoogle Scholar
[20]Kameyama, A.. Julia sets of post-critically finite rational maps and topologically self-similar sets. Nonlinearity 13 (2000), 165–188.CrossRefGoogle Scholar
[21]Keller, K.. Invariant factors. Julia Equivalences and the (Abstract) Mandelbrot Set (Lecture Notes in Mathematics, 1732). Springer, Berlin, 2000.CrossRefGoogle Scholar
[22]McMullen, C. and Sullivan, D.. Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system. Adv. Math. 135 (1998), 351–395.CrossRefGoogle Scholar
[23]Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160). Princeton University Press, Princeton, NJ, 2006.Google Scholar
[24]Mihalache, N.. Julia and John revisited. Preprint, 2008, arXiv:0803.3889.Google Scholar
[25]Näkki, R. and Väisälä, J.. John disks. Expo. Math. 9 (1991), 3–43.Google Scholar
[26]Przytycki, F. and Rohde, S.. Rigidity of holomorphic Collet–Eckmann repellers. Ark. Mat. 37 (1999), 357–371.CrossRefGoogle Scholar
[27]Solomyak, B.. On the ‘Mandelbrot set’ for pairs of linear maps: asymptotic self-similarity. Nonlinearity 18(5) (2005), 1927–1943.CrossRefGoogle Scholar
[28]Solomyak, B. and Xu, H.. On the ‘Mandelbrot set’ for a pair of linear maps and complex Bernoulli convolutions. Nonlinearity 16 (2003), 1733–1749.Google Scholar
[29]Tukia, P. and Väisälä, J.. Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Math. 5 (1980), 97–114.CrossRefGoogle Scholar