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Random backward iteration algorithm for Julia sets of rational 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 |
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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