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

Backward iteration algorithms for Julia sets of Möbius 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  November 2015 Revised  March 2016 Published  August 2016

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 and Continuous Dynamical Systems, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079
References:
[1]

M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.

[2]

M. F. Barnsley and S. G. Demko, Rational approximation of fractals, in Rational approximation and interpolation (Tampa, Fla., 1983), vol. 1105 of Lecture Notes in Math., Springer, Berlin, 1984, 73-88. doi: 10.1007/BFb0072400.

[3]

M. F. Barnsley, J. H. Elton and D. P. Hardin, Recurrent iterated function systems, Constr. Approx., 5 (1989), 3-31. doi: 10.1007/BF01889596.

[4]

A. F. Beardon, The Geometry of Discrete Groups, vol. 91 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995, Corrected reprint of the 1983 original.

[5]

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

[6]

T. Butz, W. Conatser, B. Dean, K. Hart, Y. Li and R. Stankewitz, Julia 2.0 fractal drawing program, URL https://github.com/bsumath/julia/wiki.

[7]

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

[8]

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.

[9]

H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32. doi: 10.1007/BF02760620.

[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), 1442-1447. doi: 10.1142/S021812740300731X.

[11]

A. Hinkkanen and G. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc., 3 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358.

[12]

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

[13]

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

[14]

A. Freire, A. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Bras. Math., 14 (1983), 45-62. doi: 10.1007/BF02584744.

[15]

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

[16]

R. Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Math., 14 (1983), 27-43. doi: 10.1007/BF02584743.

[17]

E. Mihailescu, Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets, Discrete Contin. Dyn. Syst., 32 (2012), 961-975. doi: 10.3934/dcds.2012.32.961.

[18]

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

[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), 2583-2589. doi: 10.3934/dcds.2012.32.2583.

[20]

R. Stankewitz and H. Sumi, Random backward iteration algorithm for Julia sets of rational semigroups, Discrete Contin. Dyn. Syst., 35 (2015), 2165-2175. doi: 10.3934/dcds.2015.35.2165.

[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.

show all references

References:
[1]

M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.

[2]

M. F. Barnsley and S. G. Demko, Rational approximation of fractals, in Rational approximation and interpolation (Tampa, Fla., 1983), vol. 1105 of Lecture Notes in Math., Springer, Berlin, 1984, 73-88. doi: 10.1007/BFb0072400.

[3]

M. F. Barnsley, J. H. Elton and D. P. Hardin, Recurrent iterated function systems, Constr. Approx., 5 (1989), 3-31. doi: 10.1007/BF01889596.

[4]

A. F. Beardon, The Geometry of Discrete Groups, vol. 91 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995, Corrected reprint of the 1983 original.

[5]

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

[6]

T. Butz, W. Conatser, B. Dean, K. Hart, Y. Li and R. Stankewitz, Julia 2.0 fractal drawing program, URL https://github.com/bsumath/julia/wiki.

[7]

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

[8]

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.

[9]

H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32. doi: 10.1007/BF02760620.

[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), 1442-1447. doi: 10.1142/S021812740300731X.

[11]

A. Hinkkanen and G. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc., 3 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358.

[12]

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

[13]

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

[14]

A. Freire, A. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Bras. Math., 14 (1983), 45-62. doi: 10.1007/BF02584744.

[15]

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

[16]

R. Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Math., 14 (1983), 27-43. doi: 10.1007/BF02584743.

[17]

E. Mihailescu, Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets, Discrete Contin. Dyn. Syst., 32 (2012), 961-975. doi: 10.3934/dcds.2012.32.961.

[18]

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

[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), 2583-2589. doi: 10.3934/dcds.2012.32.2583.

[20]

R. Stankewitz and H. Sumi, Random backward iteration algorithm for Julia sets of rational semigroups, Discrete Contin. Dyn. Syst., 35 (2015), 2165-2175. doi: 10.3934/dcds.2015.35.2165.

[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.

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