Abstract
A discrete-time Markov process on a locally compact metric space obtained by randomly iterating a finite number of Lipschitz maps is considered. The probability of choosing a particular map at each step is allowed to depend on the current position. Conditions are given that guarantee a unique invariant, attractive measure for the process. An application to real Julia sets associated with polynomials is presented.
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References
M. F. Barnsley and S. Demko. Iterated Function Systems and the Global Construction of Fractals. Proc. R. Soc. London. Vol. A 399 (1985), 243–275.
J. Hutchinson. Fractals and Self Similarity. Indiana Univ. J. of Math. 30 (1981), 713–747.
B. Mandelbrot. The Fractal Geometry of Nature. 1982. W. H. Freeman. San Francisco.
O. Onicescu and G. Mihoc. Sur les chaines de variables statistique. Bull. Soc. Math, de France 59 (1935), 174–192.
S. Karlin. Some Random Walks Arising in Learning Models. Pac. J. Math. 3 (1953), 725–756.
L. Dubins and D. Freedman. Invariant Probabilities for Certain Markov Processes. Ann. Math. Stat. 37 (1966), 837–848.
J. Bellissard, D. Bessis and P. Moussa. Chaotic States of Almost Periodic Schrödinger Operators. Phys. Rev. Lett. 49 (1982), 701–704.
D. Bessis, J. S. Geronimo and P. Moussa. Function Weighted Measures and Orthogonal Polynomials on Julia Sets. Const. Approx. 4 (1988), 157–173.
P. Diaconis and M. Shahshahani. Products of Random Matrices and Computer Image Generation in Random Matrices and Their Applications. 50 Conf. Math. A.M.S., Providence, RI (1986).
S. Demko, L. Hodges and B. Naylor. Construction of Fractal Objects with Iterated Function Systems. SIGGRAPH ’85 Proceedings (1985), 271–278.
M. Berger and H. M. Soner. Random Walks Generated by Affine Mappings. J. Theor. Prob. 1 (1988), 239–254.
M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo. Invariant Measures for Markov Processes Arising from Iterated Function Systems with Place-Dependent Probabilities. Ann. Inst. H. Poincaré Prob, et Stat. 24 (1988), 367–394.
M. F. Barnsley and J. H. Elton. A New Class of Markov Processes for Image Encoding. Adv. Appi. Prob. 20 (1988), 14–32.
H. Brolin. Invariant Sets Under Iteration of Rational Functions. Ark. Mat. 6 (1965), 103–144.
M. F. Barnsley, J. S. Geronimo and A. N. Harrington. On the Invariant Sets of a Family of Quadratic Maps. Commun. Math. Phys. 88 (1983), 479–501.
P. Moussa. Iteration des polynomes et propriétés d’orthogonalite. Ann. Inst. H. Poincaré 44 (1986), 315–325.
M. F. Barnsley, J. S. Geronimo and A. N. Harrington. Almost Periodic Jacobi Matrices Associated with Julia Sets. Comm. Math. Phys. 99 (1985), 303–317.
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© 1990 Springer-Verlag Berlin Heidelberg
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Geronimo, J.S. (1990). Iterating Random Maps and Applications. In: Luck, JM., Moussa, P., Waldschmidt, M. (eds) Number Theory and Physics. Springer Proceedings in Physics, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75405-0_22
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DOI: https://doi.org/10.1007/978-3-642-75405-0_22
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