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Random iteration for infinite nonexpansive iterated function systems. (English) Zbl 1374.37025

Summary: We prove that the random iteration algorithm works for strict attractors of infinite iterated function systems. The system is assumed to be compactly branching and nonexpansive. The orbit recovering an attractor is generated by a deterministic process and the algorithm is always convergent. We also formulate a version of the random iteration for uncountable equicontinuous systems.{
©2015 American Institute of Physics}

MSC:

37B25 Stability of topological dynamical systems
28A80 Fractals
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Barnsley, M. F.; Hutchinson, J. E.; Stenflo, Ö., V-variable fractals: Fractals with partial self similarity, Adv. Math., 218, 2051-2088 (2008) · Zbl 1169.28006 · doi:10.1016/j.aim.2008.04.011
[2] Barnsley, M. F.; Leśniak, K., The chaos game on a general iterated function system from a topological point of view, Int. J. Bifurcation Chaos Appl. Sci. Eng., 24, 1450139 (2014) · Zbl 1304.28003 · doi:10.1142/S0218127414501399
[3] Barnsley, M. F.; Leśniak, K.; Rypka, M., Chaos game for IFSs on topological spaces · Zbl 1341.37013
[4] Barnsley, M. F.; Vince, A., The chaos game on a general iterated function system, Ergodic Theor. Dyn. Syst., 31, 1073-1079 (2011) · Zbl 1221.37079 · doi:10.1017/S0143385710000428
[5] Barrientos, P. G.; Fakhari, A.; Sarizadeh, A., Density of fiberwise orbits in minimal iterated function systems on the circle, Discrete Contin. Dyn. Syst., 34, 3341-3352 (2014) · Zbl 1351.37167 · doi:10.3934/dcds.2014.34.3341
[6] Berstel, J.; Perrin, D., The origins of combinatorics on words, Eur. J. Combinatorics, 28, 996-1022 (2007) · Zbl 1111.68092 · doi:10.1016/j.ejc.2005.07.019
[7] Calude, C. S.; Priese, L.; Staiger, L., Disjunctive Sequences: An Overview (1997)
[8] Diaconis, P.; Freedman, D., Iterated random functions, SIAM Rev., 41, 45-76 (1999) · Zbl 0926.60056 · doi:10.1137/S0036144598338446
[9] Fernau, H., Infinite iterated function systems, Math. Nachr., 170, 79-91 (1994) · Zbl 0817.28006 · doi:10.1002/mana.19941700107
[10] Goodman, G. S.; Peitgen, H.-O.; Henriques, J. M.; Penedo, L. F., A probabilist looks at the chaos game, Fractals in the Fundamental and Applied Sciences, 159-168 (1991)
[11] Hoggar, S. G.; McFarlane, I., Faster fractal pictures by finite fields and far rings, Discrete Math., 138, 267-280 (1995) · Zbl 0816.68088 · doi:10.1016/0012-365X(94)00209-2
[12] Jadczyk, A., Quantum Fractals: From Heisenberg’s Uncertainty to Barnsley’s Fractality (2014) · Zbl 1309.81007
[13] Käenmäki, A.; Reeve, H. W. J., Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets, J. Fractal Geom., 1, 83-152 (2014) · Zbl 1292.28016 · doi:10.4171/JFG/3
[14] Kieninger, B., Iterated function systems on compact Hausdorff spaces (2002) · Zbl 1019.54020
[15] Kunze, H.; LaTorre, D.; Mendivil, F.; Vrscay, E. R., Fractal-Based Methods in Analysis (2012) · Zbl 1237.28002
[16] Lasota, A.; Myjak, J., Semifractals on Polish spaces, Bull. Pol. Acad. Sci., Math., 46, 179-196 (1998) · Zbl 0921.28007
[17] Leśniak, K., Random iteration and projection method
[18] Leśniak, K., On discrete stochastic processes with disjunctive outcomes, Bull. Aust. Math. Soc., 90, 149-159 (2014) · Zbl 1296.60188 · doi:10.1017/S0004972714000124
[19] Leśniak, K., Random iteration for nonexpansive iterated function systems: derandomized algorithm, Int. J. Appl. Nonlinear Sci., 1, 360-363 (2014) · Zbl 1344.37064 · doi:10.1504/IJANS.2014.068267
[20] Mantica, G., Direct and inverse computation of Jacobi matrices of infinite iterated function systems, Numer. Math., 125, 705-731 (2013) · Zbl 1282.65050 · doi:10.1007/s00211-013-0551-7
[21] Mauldin, R. D.; Urbański, M., Graph Directed Markov Systems. Geometry and Dynamics of Limit Sets (2003) · Zbl 1033.37025
[22] Miculescu, R.; Mihail, A., On a question of A. Kameyama concerning self-similar metrics, J. Math. Anal. Appl., 422, 265-271 (2015) · Zbl 1316.28004 · doi:10.1016/j.jmaa.2014.08.008
[23] Perry, J. C., Lipscomb”s universal space is the attractor of an infinite iterated function system, Proc. Am. Math. Soc., 124, 2479-2489 (1996) · Zbl 0879.54015 · doi:10.1090/S0002-9939-96-03554-X
[24] Reiter, C. A., Fractals, Visualization and J. (2007)
[25] Staiger, L., How large is the set of disjunctive sequences, J. UCS, 8, 348-362 (2002) · Zbl 1258.68091 · doi:10.3217/jucs-008-02-0348
[26] Stankewitz, R.; Sumi, H., Random backward iteration algorithm for Julia sets of rational semigroups, Discrete Contin. Dyn. Syst., 35, 2165-2175 (2015) · Zbl 1362.37093 · doi:10.3934/dcds.2015.35.2165
[27] Vince, A., Möbius iterated functions systems, Trans. Am. Math. Soc., 365, 491-509 (2013) · Zbl 1286.28009 · doi:10.1090/S0002-9947-2012-05624-8
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