Hostname: page-component-5db58dd55d-pjp64 Total loading time: 0 Render date: 2026-05-27T21:15:47.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial models of expanding dynamical systems

Published online by Cambridge University Press:  24 January 2013

VOLODYMYR NEKRASHEVYCH
VOLODYMYR NEKRASHEVYCH*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX, USA (email: nekrash@math.tamu.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove homotopical rigidity of expanding dynamical systems, by showing that they are determined by a group-theoretic invariant. We use this to show that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules. Moreover, the cut-and-paste rules can be found algorithmically from the algebraic invariant.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 938 - 985
Copyright
©2013 Cambridge University Press 

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.)

Article purchase

Temporarily unavailable

References

[1]Aleshin, S. V.. A free group of finite automata. Moscow Univ. Math. Bull. 38(4) (1983), 10–13.Google Scholar
[2]Alexandroff, P.. Über Gestalt und Lage abgeschlossener Mengen beliebiger Dimension. Ann. of Math. (2) 30 (1928–1929), 101–187.CrossRefGoogle Scholar
[3]Bartholdi, L., Grigorchuk, R. and Nekrashevych, V.. From fractal groups to fractal sets. Fractals in Graz 2001. Analysis – Dynamics – Geometry – Stochastics. Eds. Grabner, P. and Woess, W.. Birkhäuser, Basel, 2003, pp. 25–118.Google Scholar
[4]Bass, H. and Lubotzky, A.. Tree Lattices (Progress in Mathematics, 176). Birkhäuser Boston, Boston, MA, 2001, with appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg and J. Tits.Google Scholar
[5]Bonk, M. and Meyer, D.. Expanding Thurston maps. Preprint, 2010, arXiv:1009.3647v1.Google Scholar
[6]Brady, T.. A partial order on the symmetric group and new K(π,1)’s for the braid groups. Adv. Math. 161(1) (2001), 20–40.Google Scholar
[7]Brauer, W.. Automates topologiques et ensembles reconnaissables. Séminaire M. P. Schützenberger, A. Lentin et M. Nivat, 1969/70: Problèmes Mathématiques de la Théorie des Automates. Secrétariat mathématique, Paris, 1970, Exp. 18, p. 24.Google Scholar
[8]Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-Positive Curvature (Grundlehren der Mathematischen Wissenschaften, 319). Springer, Berlin, 1999.Google Scholar
[9]Bruin, H. and Schleicher, D.. Symbolic dynamics of quadratic polynomials. Report No. 7, Institut Mittag-Leffler, 2001/2002.Google Scholar
[10]Bullett, S.. Dynamics of quadratic correspondences. Nonlinearity 1(1) (1988), 27–50.Google Scholar
[11]Bullett, S.. Dynamics of the arithmetic–geometric mean. Topology 30(2) (1991), 171–190.Google Scholar
[12]Bullett, S.. Critically finite correspondences and subgroups of the modular group. Proc. Lond. Math. Soc. (3) 65(2) (1992), 423–448.Google Scholar
[13]Bullett, S. and Penrose, C.. A gallery of iterated correspondences. Exp. Math. 3(2) (1994), 85–105.Google Scholar
[14]Cannon, J. W., Floyd, W. J. and Parry, W. R.. Finite subdivision rules. Conform. Geom. Dyn. 5 (2001), 153–196 (electronic).Google Scholar
[15]Cannon, J. W., Floyd, W. J. and Parry, W. R.. Constructing subdivision rules from rational maps. Conform. Geom. Dyn. 11 (2007), 128–136 (electronic).CrossRefGoogle Scholar
[16]Cox, D. A.. The arithmetic–geometric mean of Gauss. Enseign. Math. (2) 30(3–4) (1984), 275–330.Google Scholar
[17]Culler, M. and Vogtmann, K.. Moduli of graphs and automorphisms of free groups. Invent. Math. 84(1) (1986), 91–119.Google Scholar
[18]Douady, A. and Hubbard, J. H.. Étude Dynamiques des Polynômes Complexes (Première Partie) (Publications Mathematiques d’Orsay, 02). Université de Paris-Sud, 1984.Google Scholar
[19]Douady, A. and Hubbard, J. H.. Étude Dynamiques des Polynômes Complexes (Deuxième Partie) (Publications Mathematiques d’Orsay, 04) Université de Paris-Sud, 1985.Google Scholar
[20]Douady, A. and Hubbard, J. H.. A proof of Thurston’s topological characterization of rational functions. Acta Math. 171(2) (1993), 263–297.Google Scholar
[21]Eilenberg, S.. Automata, Languages and Machines. Vol. A. Academic Press, New York, 1974.Google Scholar
[22]Fried, D. L.. Finitely presented dynamical systems. Ergod. Th. & Dynam. Sys. 7 (1987), 489–507.Google Scholar
[23]Glasner, Y. and Mozes, S.. Automata and square complexes. Geom. Dedicata 111 (2005), 43–64.Google Scholar
[24]Grigorchuk, R. I., Nekrashevich, V. V. and Sushchanskii, V. I.. Automata, dynamical systems and groups. Proc. Steklov Inst. Math. 231 (2000), 128–203.Google Scholar
[25]Gromov, M.. Hyperbolic groups. Essays in Group Theory (Mathematical Sciences Research Institute Publications, 8). Ed. Gersten, S. M.. Springer, 1987, pp. 75–263.CrossRefGoogle Scholar
[26]Ishii, Y. and Smillie, J.. Homotopy shadowing. Amer. J. Math. 132(4) (2010), 987–1029.Google Scholar
[27]Jeandel, E.. Topological automata. Theory Comput. Syst. 40(4) (2007), 397–407.Google Scholar
[28]Kaimanovich, V. A. and Lyubich, M.. Conformal and harmonic measures on laminations associated with rational maps. Mem. Amer. Math. Soc. 173(820) (2005), vi+119.Google Scholar
[29]Katsura, T.. A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. I. Fundamental results. Trans. Amer. Math. Soc. 356(11) (2004), 4287–4322 (electronic).Google Scholar
[30]Katsura, T.. A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. II. Examples. Internat. J. Math. 17(7) (2006), 791–833.Google Scholar
[31]Katsura, T.. A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. III. Ideal structures. Ergod. Th. & Dynam. Sys. 26(6) (2006), 1805–1854.Google Scholar
[32]Katsura, T.. A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. IV. Pure infiniteness. J. Funct. Anal. 254(5) (2008), 1161–1187.Google Scholar
[33]Koch, S.. Teichmüller theory and endomorphisms of ℙn. PhD Thesis, Université de Provence, 2007.Google Scholar
[34]Lubotzky, A., Mozes, S. and Zimmer, R. J.. Superrigidity for the commensurability group of tree lattices. Comment. Math. Helv. 69 (1994), 523–548.Google Scholar
[35]Lyubich, M. and Minsky, Y.. Laminations in holomorphic dynamics. J. Differential Geom. 47(1) (1997), 17–94.Google Scholar
[36]Macedońska, O., Nekrashevych, V. V. and Sushchansky, V. I.. Commensurators of groups and reversible automata. Dopov. Nats. Akad. Nauk Ukr. Mat. Pryr. Tekh. Nauky 12 (2000), 36–39.Google Scholar
[37]Nekrashevych, V.. Self-similar Groups (Mathematical Surveys and Monographs, 117). American Mathematical Society, Providence, RI, 2005.Google Scholar
[38]Nekrashevych, V.. Symbolic dynamics and self-similar groups. Holomorphic Dynamics and Renormalization. A Volume in Honour of John Milnor’s 75th Birthday (Fields Institute Communications, 53). Eds. Lyubich, M. and Yampolsky, M.. American Mathematical Society, Providence, RI, 2008, pp. 25–73.Google Scholar
[39]Nekrashevych, V.. Iterated monodromy groups. Groups St. Andrews in Bath. Vol. 1 (London Mathematical Society Lecture Note Series, 387). Cambridge University Press, Cambridge, 2011, pp. 41–93.Google Scholar
[40]Shub, M.. Endomorphisms of compact differentiable manifolds. Amer. J. Math. 91 (1969), 175–199.Google Scholar
[41]Shub, M.. Expanding maps. Global Analysis (Proceedings of Symposia in Pure Mathematics, 14). American Mathematical Society, Providence, RI, 1970, pp. 273–276.Google Scholar
[42]Spanier, E. H.. Algebraic Topology. McGraw-Hill, New York, 1966.Google Scholar
[43]Vorobets, M. and Vorobets, Y.. On a free group of transformations defined by an automaton. Geom. Dedicata 124(1) (2007), 237–249.Google Scholar