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Complex dynamics of Möbius semigroups

Published online by Cambridge University Press:  07 February 2012

DAVID FRIED ,
SEBASTIAN M. MAROTTA  and
RICH STANKEWITZ
DAVID FRIED
Affiliation:
Department of Mathematics, Boston University, USA (email: [email protected])
SEBASTIAN M. MAROTTA
Affiliation:
Department of Mathematics, University of the Pacific, USA (email: [email protected])
RICH STANKEWITZ
Affiliation:
Department of Mathematical Sciences, Ball State University, USA (email: [email protected])
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Abstract

We study the dynamics of semigroups of Möbius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Möbius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Möbius semigroups, based on a random dynamics variant of the Fibonacci sequence.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 1889 - 1929
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Barnsley, M. F., Elton, J. H. and Hardin, D. P.. Recurrent iterated function systems. Constr. Approx. 5(1) (1989), 3–31.Google Scholar
[2]Beardon, A. F.. The Geometry of Discrete Groups (Graduate Texts in Mathematics, 91). Springer, New York, 1983.CrossRefGoogle Scholar
[3]Beardon, A. F.. Iteration of Rational Functions (Graduate Texts in Mathematics, 132). Springer, New York, 1991.CrossRefGoogle Scholar
[4]Brück, R.. Geometric properties of Julia sets of the composition of polynomials of the form z2+cn. Pacific J. Math. 198(2) (2001), 347–372.CrossRefGoogle Scholar
[5]Brück, R., Büger, M. and Reitz, S.. Random iterations of polynomials of the form z2+cn: connectedness of Julia sets. Ergod. Th. & Dynam. Sys. 19(5) (1999), 1221–1231.Google Scholar
[6]Büger, M.. Self-similarity of Julia sets of the composition of polynomials. Ergod. Th. & Dynam. Sys. 17(6) (1997), 1289–1297.Google Scholar
[7]Büger, M.. On the composition of polynomials of the form z2+cn. Math. Ann. 310(4) (1998), 661–683.Google Scholar
[8]Caruso, H. A. and Marotta, S. M.. Sequences of complex numbers resembling the fibonacci series. Revista Ciências Exatas e Naturais 2(1) (2000), 49–59.Google Scholar
[9]Elton, J. H.. An ergodic theorem for iterated maps. Ergod. Th. & Dynam. Sys. 7(4) (1987), 481–488.CrossRefGoogle Scholar
[10]Epstein, D. and Petronio, C.. An exposition of Poincaré’s polyhedron theorem. Enseign. Math. (2) 40 (1994), 113–170.Google Scholar
[11]Fornæss, J.E. and Sibony, N.. Random iterations of rational functions. Ergod. Th. & Dynam. Sys. 11(4) (1991), 687–708.CrossRefGoogle Scholar
[12]Greenberg, L.. Commensurable groups of Moebius transformations. Discontinuous Groups and Riemann Surfaces (Annals of Mathematics Studies, 79). Princeton University Press, Princeton, 1974, pp. 227–237.Google Scholar
[13]Harada, T.. The dynamics of nearly abelian polynomial semigroups at infinity. Proc. Japan Acad. Ser. A Math. Sci. 73(3) (1997), 36–37.Google Scholar
[14]Hayes, B.. The vibonacci numbers. Am. Sci. 87(4) (1999), 296–301.CrossRefGoogle Scholar
[15]Hecke, E.. Dirichlet Series (Planographed Lecture Notes, Princeton Institute for Advanced Study). Edwards Brothers, Ann Arbor, 1938.Google Scholar
[16]Hinkkanen, A. and Martin, G. J.. The dynamics of semigroups of rational functions I. Proc. Lond. Math. Soc. (3) 73(2) (1996), 358–384.Google Scholar
[17]Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30(5) (1981), 713–747.CrossRefGoogle Scholar
[18]Klimenko, E. and Kopteva, N.. Discrete groups with a parabolic generator. Sibirsk. Mat. Zh. 46(6) (2005), 1324–1334.Google Scholar
[19]Keen, L. and Series, C.. The Riley slice of Schottky space. Proc. Lond. Math. Soc. (3) 69(1) (1994), 72–90.Google Scholar
[20]Marden, A.. Outer Circles: An Introduction to Hyperbolic 3-manifolds. Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
[21]Maskit, B.. Kleinian Groups (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 287). Springer, Berlin, 1988.Google Scholar
[22]Milnor, J.. Dynamics in One Complex Variable, 3rd edn(Annals of Mathematics Studies, 160). Princeton University Press, Princeton, NJ, 2006.Google Scholar
[23]Mumford, D., Series, C. and Wright, D.. Indra’s Pearls: The Vision of Felix Klein. Cambridge University Press, New York, 2002.CrossRefGoogle Scholar
[24]Munkres, J. R.. Topology: A First Course. Prentice Hall, Englewood Cliffs, NJ, 1975.Google Scholar
[25]Parkkonen, J.. The outside of the Teichmuller space of punctured tori in Maskit’s embedding. Ann. Acad. Sci. Fenn. Math. 24(2) (1999), 305–342.Google Scholar
[26]Pennec, X.. Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vision 25(1) (2006), 127–154.Google Scholar
[27]Ratcliffe, J. G.. Foundations of Hyperbolic Manifolds, 2nd edn(Graduate Texts in Mathematics, 149). Springer, New York, 2006.Google Scholar
[28]Stankewitz, R.. Completely invariant Julia sets of rational semigroups. PhD Thesis, University of Illinois, 1998.Google Scholar
[29]Stankewitz, R.. Completely invariant Julia sets of polynomial semigroups. Proc. Amer. Math. Soc. 127(10) (1999), 2889–2898.Google Scholar
[30]Stankewitz, R.. Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s. Proc. Amer. Math. Soc. 128(9) (2000), 2569–2575.Google Scholar
[31]Stankewitz, R.. Density of repelling fixed points in the Julia set of a rational or entire semigroup. J. Difference Equ. Appl. 16(5) (2010), 763–771.Google Scholar
[32]Stankewitz, R.. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. To appear in Discrete and Continuous Dynamical Systems: Proceedings of the Conference Dynamical Systems II, Denton 2009. Preprint, Ball State University, 2010.Google Scholar
[33]Stankewitz, R. and Sumi, H.. Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups. Trans. Amer. Math. Soc. 363(10) (2011), 5293–5319.Google Scholar
[34]Sumi, H.. Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles. Ergod. Th. & Dynam. Sys. 30(6) (2011), 1869–1902.CrossRefGoogle Scholar
[35]Sumi, H.. On dynamics of hyperbolic rational semigroups. J. Math. Kyoto Univ. 37(4) (1997), 717–733.Google Scholar
[36]Sumi, H.. Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21(2) (2001), 563–603.CrossRefGoogle Scholar
[37]Sumi, H.. Random dynamics of polynomials and devil’s staircase-like functions in the complex plane. Appl. Math. Comput. 187 (2007), 489–500.Google Scholar
[38]Sumi, H.. Dynamics of postcritically bounded polynomials semigroups I: connected components of the Julia sets. Discrete Contin. Dyn. Syst. 29(3) (2011), 1205–1244.CrossRefGoogle Scholar
[39]Sun, Y. and Yang, C.-C.. On the connectivity of the Julia set of a finitely generated rational semigroup. Proc. Amer. Math. Soc. 130(1) (2001), 49–52 (electronic).Google Scholar
[40]Viswanath, D.. Lyapunov exponents from random fibonacci sequences to the Lorenz equations.PhD Thesis, Cornell University, 1998.Google Scholar
[41]Wright, D. J.. Searching for the cusp. Spaces of Kleinian Groups and Hyperbolic 3-manifolds (London Mathematical Society Lecture Notes, 329). Cambridge University Press, Cambridge, 2005, pp. 301–336.Google Scholar