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May  2015, 35(5): 2165-2175. doi: 10.3934/dcds.2015.35.2165

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  February 2014 Revised  October 2014 Published  December 2014

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.
Keywords: random complex dynamics, Rational semigroups, Julia sets, random iteration, invariant measure., Markov process.
Mathematics Subject Classification: Primary: 37F10, 30D0.
Citation: Rich Stankewitz, Hiroki Sumi. Random backward iteration algorithm for Julia sets of rational semigroups. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2165-2175. doi: 10.3934/dcds.2015.35.2165
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.

show all references

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.

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