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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 & 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.  Google Scholar

[2]

D. Boyd, An invariant measure for finitely generated rational semigroups, Complex Variables Theory Appl., 39 (1999), 229-254. doi: 10.1080/17476939908815193.  Google Scholar

[3]

T. Butz, W. Conatser, B. Dean, K. Hart, Y. Li and R. Stankewitz, Julia 2.0 fractal drawing program,, , ().   Google Scholar

[4]

J. H. Elton, An ergodic theorem for iterated maps, Ergodic Theory Dynam. Systems, 7 (1987), 481-488. doi: 10.1017/S0143385700004168.  Google Scholar

[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.  Google Scholar

[6]

N. Fujishima, Chaotic dynamical systems and fractals, Bachelor thesis, Faculty of Integrated Human Studies, Kyoto University, under supervision of Shigehiro Ushiki, 2013. Google Scholar

[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.  Google Scholar

[8]

Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions, Journal of Fudan University, 35 (1996), 387-392.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Hinkkanen and G. Martin, Julia sets of rational semigroups, Math. Z., 222 (1996), 161-169. doi: 10.1007/BF02621862.  Google Scholar

[12]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[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.  Google Scholar

[14]

M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergod. Th. & Dynam. Sys., 3 (1983), 351-385. doi: 10.1017/S0143385700002030.  Google Scholar

[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.  Google Scholar

[16]

R. Stankewitz, Completely Invariant Julia Sets of Rational Semigroups, Ph.D. Thesis. University of Illinois, 1998. Google Scholar

[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.  Google Scholar

[18]

R. Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl., 40 (2000), 199-210. doi: 10.1080/17476930008815219.  Google Scholar

[19]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups, Ann. Acad. Sci. Fenn. Math., 29 (2004), 357-366.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Science Bulletin, 37 (1992), 969-971. Google Scholar

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.  Google Scholar

[2]

D. Boyd, An invariant measure for finitely generated rational semigroups, Complex Variables Theory Appl., 39 (1999), 229-254. doi: 10.1080/17476939908815193.  Google Scholar

[3]

T. Butz, W. Conatser, B. Dean, K. Hart, Y. Li and R. Stankewitz, Julia 2.0 fractal drawing program,, , ().   Google Scholar

[4]

J. H. Elton, An ergodic theorem for iterated maps, Ergodic Theory Dynam. Systems, 7 (1987), 481-488. doi: 10.1017/S0143385700004168.  Google Scholar

[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.  Google Scholar

[6]

N. Fujishima, Chaotic dynamical systems and fractals, Bachelor thesis, Faculty of Integrated Human Studies, Kyoto University, under supervision of Shigehiro Ushiki, 2013. Google Scholar

[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.  Google Scholar

[8]

Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions, Journal of Fudan University, 35 (1996), 387-392.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Hinkkanen and G. Martin, Julia sets of rational semigroups, Math. Z., 222 (1996), 161-169. doi: 10.1007/BF02621862.  Google Scholar

[12]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[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.  Google Scholar

[14]

M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergod. Th. & Dynam. Sys., 3 (1983), 351-385. doi: 10.1017/S0143385700002030.  Google Scholar

[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.  Google Scholar

[16]

R. Stankewitz, Completely Invariant Julia Sets of Rational Semigroups, Ph.D. Thesis. University of Illinois, 1998. Google Scholar

[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.  Google Scholar

[18]

R. Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl., 40 (2000), 199-210. doi: 10.1080/17476930008815219.  Google Scholar

[19]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups, Ann. Acad. Sci. Fenn. Math., 29 (2004), 357-366.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Science Bulletin, 37 (1992), 969-971. Google Scholar

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