Random backward iteration algorithm for Julia sets of rational semigroups

Rich Stankewitz; Hiroki Sumi

doi:10.3934/dcds.2015.35.2165

Article Contents

2015, Volume 35, Issue 5: 2165-2175. Doi: 10.3934/dcds.2015.35.2165

This issue Previous Article On the least energy sign-changing solutions for a nonlinear elliptic system Next Article Center of planar quintic quasi--homogeneous polynomial differential systems

Random backward iteration algorithm for Julia sets of rational semigroups

Rich Stankewitz^1, and
Hiroki Sumi^2,

1.
Department of Mathematical Sciences, Ball State University, Muncie, IN 47306
2.
Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043

Received: January 31, 2014

Revised: September 30, 2014

Published: May 2017

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Abstract

We provide proof that a random backward iteration algorithm for approximating Julia sets of rational semigroups, previously proven to work in the context of iteration of a rational function of degree two or more, extends to rational semigroups (of a certain type). We also provide some consequences of this result.

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Mathematics Subject Classification: Primary: 37F10, 30D05.

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References

[1]	M. F. Barnsley, J. H. Elton and D. P. Hardin, Recurrent iterated function systems, Constr. Approx., 5 (1989), 3-31.doi: 10.1007/BF01889596.
[2]	D. Boyd, An invariant measure for finitely generated rational semigroups, Complex Variables Theory Appl., 39 (1999), 229-254.doi: 10.1080/17476939908815193.
[3]	T. Butz, W. Conatser, B. Dean, K. Hart, Y. Li and R. Stankewitz, Julia 2.0 fractal drawing program, http://rstankewitz.iweb.bsu.edu/JuliaHelp2.0/Julia.html.
[4]	J. H. Elton, An ergodic theorem for iterated maps, Ergodic Theory Dynam. Systems, 7 (1987), 481-488.doi: 10.1017/S0143385700004168.
[5]	D. Fried, S. M. Marotta and R. Stankewitz, Complex dynamics of Möbius semigroups, Ergodic Theory Dynam. Systems, 32 (2012), 1889-1929.doi: 10.1017/S014338571100054X.
[6]	N. Fujishima, Chaotic dynamical systems and fractals, Bachelor thesis, Faculty of Integrated Human Studies, Kyoto University, under supervision of Shigehiro Ushiki, 2013.
[7]	H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32.doi: 10.1007/BF02760620.
[8]	Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions, Journal of Fudan University, 35 (1996), 387-392.
[9]	J. Hawkins and M. Taylor, Maximal entropy measure for rational maps and a random iteration algorithme, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1442-1447.doi: 10.1142/S021812740300731X.
[10]	A. Hinkkanen and G. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc., 73 (1996), 358-384.doi: 10.1112/plms/s3-73.2.358.
[11]	A. Hinkkanen and G. Martin, Julia sets of rational semigroups, Math. Z., 222 (1996), 161-169.doi: 10.1007/BF02621862.
[12]	J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.doi: 10.1512/iumj.1981.30.30055.
[13]	A. F. A. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Bras. Math., 14 (1983), 45-62.doi: 10.1007/BF02584744.
[14]	M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergod. Th. & Dynam. Sys., 3 (1983), 351-385.doi: 10.1017/S0143385700002030.
[15]	R. Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Math., 14 (1983), 27-43.doi: 10.1007/BF02584743.
[16]	R. Stankewitz, Completely Invariant Julia Sets of Rational Semigroups, Ph.D. Thesis. University of Illinois, 1998.
[17]	R. Stankewitz, Completely invariant Julia sets of polynomial semigroups, Proc. Amer. Math. Soc., 127 (1999), 2889-2898.doi: 10.1090/S0002-9939-99-04857-1.
[18]	R. Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl., 40 (2000), 199-210.doi: 10.1080/17476930008815219.
[19]	R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups, Ann. Acad. Sci. Fenn. Math., 29 (2004), 357-366.
[20]	D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2), 122 (1985), 401-418.doi: 10.2307/1971308.
[21]	H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.doi: 10.1088/0951-7715/13/4/302.
[22]	H. Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. Lond. Math. Soc. (3), 102 (2011), 50-112.doi: 10.1112/plms/pdq013.
[23]	H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, Adv. Math., 245 (2013), 137-181.doi: 10.1016/j.aim.2013.05.023.
[24]	W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Science Bulletin, 37 (1992), 969-971.