Backward iteration algorithms for Julia sets of Möbius semigroups

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November 2016, 36(11): 6475-6485. doi: 10.3934/dcds.2016079

Backward iteration algorithms for Julia sets of Möbius semigroups

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 November 2015 Revised March 2016 Published August 2016

Abstract
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We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick attractor in the sense given in [8].

Keywords: random complex dynamics, Rational semigroups, random iteration, Julia sets, Markov process, Möbius maps, invariant measure..

Mathematics Subject Classification: Primary: 37F10, 30D0.

Citation: Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079

References:

[1]	M. Barnsley, Fractals Everywhere,, Academic Press, (1988). Google Scholar
[2]	M. F. Barnsley and S. G. Demko, Rational approximation of fractals,, in Rational approximation and interpolation (Tampa, (1983), 73. doi: 10.1007/BFb0072400. Google Scholar
[3]	M. F. Barnsley, J. H. Elton and D. P. Hardin, Recurrent iterated function systems,, Constr. Approx., 5 (1989), 3. doi: 10.1007/BF01889596. Google Scholar
[4]	A. F. Beardon, The Geometry of Discrete Groups,, vol. 91 of Graduate Texts in Mathematics, (1995). Google Scholar
[5]	D. Boyd, An invariant measure for finitely generated rational semigroups,, Complex Variables Theory Appl., 39 (1999), 229. doi: 10.1080/17476939908815193. Google Scholar
[6]	T. Butz, W. Conatser, B. Dean, K. Hart, Y. Li and R. Stankewitz, Julia 2.0 fractal drawing program,, URL , (). Google Scholar
[7]	J. H. Elton, An ergodic theorem for iterated maps,, Ergodic Theory Dynam. Systems, 7 (1987), 481. doi: 10.1017/S0143385700004168. Google Scholar
[8]	D. Fried, S. M. Marotta and R. Stankewitz, Complex dynamics of Möbius semigroups,, Ergodic Theory Dynam. Systems, 32 (2012), 1889. doi: 10.1017/S014338571100054X. Google Scholar
[9]	H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces,, Israel J. Math., 46 (1983), 12. doi: 10.1007/BF02760620. Google Scholar
[10]	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. doi: 10.1142/S021812740300731X. Google Scholar
[11]	A. Hinkkanen and G. Martin, The dynamics of semigroups of rational functions I,, Proc. London Math. Soc., 3 (1996), 358. doi: 10.1112/plms/s3-73.2.358. Google Scholar
[12]	A. Hinkkanen and G. Martin, Julia sets of rational semigroups,, Math. Z., 222 (1996), 161. doi: 10.1007/BF02621862. Google Scholar
[13]	J. E. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713. doi: 10.1512/iumj.1981.30.30055. Google Scholar
[14]	A. Freire, A. Lopes and R. Mañé, An invariant measure for rational maps,, Bol. Soc. Bras. Math., 14 (1983), 45. doi: 10.1007/BF02584744. Google Scholar
[15]	M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,, Ergod. Th. & Dynam. Sys., 3 (1983), 351. doi: 10.1017/S0143385700002030. Google Scholar
[16]	R. Mañé, On the uniqueness of the maximizing measure for rational maps,, Bol. Soc. Bras. Math., 14 (1983), 27. doi: 10.1007/BF02584743. Google Scholar
[17]	E. Mihailescu, Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets,, Discrete Contin. Dyn. Syst., 32 (2012), 961. doi: 10.3934/dcds.2012.32.961. Google Scholar
[18]	R. Stankewitz, Completely Invariant Julia Sets of Rational Semigroups,, Ph.D. Thesis. University of Illinois, (1998). Google Scholar
[19]	R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, II,, Discrete Contin. Dyn. Syst., 32 (2012), 2583. doi: 10.3934/dcds.2012.32.2583. Google Scholar
[20]	R. Stankewitz and H. Sumi, Random backward iteration algorithm for Julia sets of rational semigroups,, Discrete Contin. Dyn. Syst., 35 (2015), 2165. doi: 10.3934/dcds.2015.35.2165. Google Scholar
[21]	H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995. 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. doi: 10.1112/plms/pdq013. Google Scholar

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References:

[1]	M. Barnsley, Fractals Everywhere,, Academic Press, (1988). Google Scholar
[2]	M. F. Barnsley and S. G. Demko, Rational approximation of fractals,, in Rational approximation and interpolation (Tampa, (1983), 73. doi: 10.1007/BFb0072400. Google Scholar
[3]	M. F. Barnsley, J. H. Elton and D. P. Hardin, Recurrent iterated function systems,, Constr. Approx., 5 (1989), 3. doi: 10.1007/BF01889596. Google Scholar
[4]	A. F. Beardon, The Geometry of Discrete Groups,, vol. 91 of Graduate Texts in Mathematics, (1995). Google Scholar
[5]	D. Boyd, An invariant measure for finitely generated rational semigroups,, Complex Variables Theory Appl., 39 (1999), 229. doi: 10.1080/17476939908815193. Google Scholar
[6]	T. Butz, W. Conatser, B. Dean, K. Hart, Y. Li and R. Stankewitz, Julia 2.0 fractal drawing program,, URL , (). Google Scholar
[7]	J. H. Elton, An ergodic theorem for iterated maps,, Ergodic Theory Dynam. Systems, 7 (1987), 481. doi: 10.1017/S0143385700004168. Google Scholar
[8]	D. Fried, S. M. Marotta and R. Stankewitz, Complex dynamics of Möbius semigroups,, Ergodic Theory Dynam. Systems, 32 (2012), 1889. doi: 10.1017/S014338571100054X. Google Scholar
[9]	H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces,, Israel J. Math., 46 (1983), 12. doi: 10.1007/BF02760620. Google Scholar
[10]	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. doi: 10.1142/S021812740300731X. Google Scholar
[11]	A. Hinkkanen and G. Martin, The dynamics of semigroups of rational functions I,, Proc. London Math. Soc., 3 (1996), 358. doi: 10.1112/plms/s3-73.2.358. Google Scholar
[12]	A. Hinkkanen and G. Martin, Julia sets of rational semigroups,, Math. Z., 222 (1996), 161. doi: 10.1007/BF02621862. Google Scholar
[13]	J. E. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713. doi: 10.1512/iumj.1981.30.30055. Google Scholar
[14]	A. Freire, A. Lopes and R. Mañé, An invariant measure for rational maps,, Bol. Soc. Bras. Math., 14 (1983), 45. doi: 10.1007/BF02584744. Google Scholar
[15]	M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,, Ergod. Th. & Dynam. Sys., 3 (1983), 351. doi: 10.1017/S0143385700002030. Google Scholar
[16]	R. Mañé, On the uniqueness of the maximizing measure for rational maps,, Bol. Soc. Bras. Math., 14 (1983), 27. doi: 10.1007/BF02584743. Google Scholar
[17]	E. Mihailescu, Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets,, Discrete Contin. Dyn. Syst., 32 (2012), 961. doi: 10.3934/dcds.2012.32.961. Google Scholar
[18]	R. Stankewitz, Completely Invariant Julia Sets of Rational Semigroups,, Ph.D. Thesis. University of Illinois, (1998). Google Scholar
[19]	R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, II,, Discrete Contin. Dyn. Syst., 32 (2012), 2583. doi: 10.3934/dcds.2012.32.2583. Google Scholar
[20]	R. Stankewitz and H. Sumi, Random backward iteration algorithm for Julia sets of rational semigroups,, Discrete Contin. Dyn. Syst., 35 (2015), 2165. doi: 10.3934/dcds.2015.35.2165. Google Scholar
[21]	H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995. 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. doi: 10.1112/plms/pdq013. Google Scholar

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