Abstract
The paper deals with a theoretical justification of the effect, observed in computer experiments, of convergence of orbits (without tending to any particular point) in random dynamical systems on the circle. The corresponding theorem is proved under certain assumptions satisfied, in particular, in some C 1-open domain in the space of random dynamical systems.
It follows from this theorem that the corresponding skew product has two invariant measurable sections, naturally called an attractor and a repeller. Moreover, it turns out that convergence of orbits and the uniqueness of a stationary measure, phenomena that are mutually exclusive in the case of a single map, typically coexist in random dynamical systems.
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References
T. Kaijser, “On stochastic perturbations of iterations of circle maps,” Phys. D, 68, 201–231 (1993).
A. N. Shiryaev, Probability, Graduate Texts in Mathematics, Vol. 95. Springer-Verlag, New York, 1996.
Y. Le Jan, “Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants,” Ann. Inst. H. Pincaré Probab. Statist., 23 No. 1, 111–120 (1987).
H. Furstenberg, “Noncommuting random matrices products,” Trans. Amer. Math. Soc., 108, 377–428 (1963).
H. Furstenberg, “Boundary theory and stochastic processes on homogeneous spaces,” In:Proc. Sympos. Pure Math., Vol. 26, 1973, pp. 193–229.
H. Furstenberg and H. Kesten, “Products of random matrices,” Ann. Math. Stat., 31, 457–469 (1960).
H. Furstenberg and Yu. Kifer, “Random matrix products and measures on projective spaces,” Israel J. Math., 46, No. 1–2, 12–32 (1983).
V. Kaimanovich and H. Mazur, “The Poisson boundary of the mapping class group,” Invent. Math, 125, 221–264 (1996).
H. Crauel, “Extremal exponents of random dynamical systems do not vanish,” J. Dynam. Differential Equations, 2, No. 3, 245–291 (1990).
A. S. Gorodetskii and Yu. S. Ilyashenko, “Some new robust properties of invariant sets and attractors of dynamical systems,” Funkts. Anal. Prilozhen., 33, No. 2, 16–30 (1999).
A. S. Gorodetskii and Yu. S. Ilyashenko, “Some properties of skew products over the horseshoe and solenoid,” Trudy MIAN, 231, pp. 96–118 (2000).
J. E. Hutchinson, “Fractals and self-similarity,” Indiana Univ. Math. J., 30, 271–280 (1981).
P. H. Baxendale, “Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms,” Probab. Theory Related Fields, 81, 521–554 (1989).
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 36–54, 2004
Original Russian Text Copyright © by V. A. Kleptsyn and M. B. Nalskii
Supported in part by RFBR grants 02-01-00482 and 02-01-22002 and CRDF grant RM1-2358-MO-02.
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Kleptsyn, V.A., Nalskii, M.B. Contraction of orbits in random dynamical systems on the circle. Funct Anal Its Appl 38, 267–282 (2004). https://doi.org/10.1007/s10688-005-0005-9
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DOI: https://doi.org/10.1007/s10688-005-0005-9