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Spectral decomposition of expanding probabilistic dynamical systems. (English) Zbl 0939.37003

Summary: We study probabilistic combinations of expanding dynamical systems, which we call expanding probabilistic dynamical systems, in one dimension. If the system is composed by exact endomorphisms we prove that the probabilistic dynamical system is an exact Markov semigroup, and we determine a generalized spectral decomposition of the associated Markov operator on densities for an example of the tent map coupled with the 2-Renyi map.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37A50 Dynamical systems and their relations with probability theory and stochastic processes
28D05 Measure-preserving transformations
39B12 Iteration theory, iterative and composite equations
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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[1] I. Antoniou, F. Bosco, Spectral decomposition of contracting probabilistic dynamical systems, Chaos Solitons and Fractals, submitted.; I. Antoniou, F. Bosco, Spectral decomposition of contracting probabilistic dynamical systems, Chaos Solitons and Fractals, submitted. · Zbl 0933.37003
[2] Lasota, A.; Mackey, M. C., (Chaos Fractals and Noise (1994), Springer: Springer New York) · Zbl 0784.58005
[3] Kifer, Yu., (Ergodic Theory of Random Transformations (1986), Birkhauser: Birkhauser Basel) · Zbl 0604.28014
[4] Pei-Dong Liu; Min Qian, (Lecture Notes in Mathematics, Vol. 1606 (1995), Springer: Springer New York)
[5] Ulam, S. M.; von Neumann, J., Bull. Am. Math. Soc., 51, 660 (1947)
[6] Krylov, N.; Bogolioubov, N., C.R. Acad. Sci. Paris, 204, 1386 (1937)
[7] Yosida, K.; Kakutani, S., Ann. Math., 42, 188 (1941)
[8] Koopman, B., (Proc. Nat. Acad. Sci. USA, 17 (1931)), 315
[9] Dynkin, E., (Markov Processes (1965), Springer: Springer Berlin) · Zbl 0132.37901
[10] Foguel, S., (The Ergodic Theory of Markov Processes (1969), Van Nostrand: Van Nostrand New York) · Zbl 0282.60037
[11] Morita, T., Osaka J. Math., 22, 489 (1985)
[12] Feller, W., (An Introduction to Probability Theory and its Applications, Vol. 2 (1968), Wiley: Wiley London) · Zbl 0155.23101
[13] Halmos, P., (A Hilbert Space Problem Book (1982), Springer: Springer New York)
[14] Hoffman, K., (Banach Spaces of Analytic Functions (1988), Dover: Dover New York) · Zbl 0734.46033
[15] Brown, J. R., (Ergodic Theory and Topological Dynamics (1976), Academic Press: Academic Press New York) · Zbl 0334.28011
[16] Rychlik, M., Studia Math., 76, 69 (1983)
[17] Hofbauer, F.; Keller, G., Math. Z., 180, 119 (1982)
[18] Antoniou, I.; Tasaki, S., Int. J. Qua. Chem., 46, 425 (1993)
[19] Treves, F. F., (Topological Vector Spaces, Distributions and Kernels (1967), Academic Press: Academic Press New York) · Zbl 0171.10402
[20] Bohm, A., (Quantum Mechanics, Foundations and Applications (1986), Springer: Springer Berlind) · Zbl 0994.81502
[21] Gantmacher, F. R., Matrix Theory (1960), Chelsea · Zbl 0085.01001
[22] Antoniou, I.; Bi Qiao, Chaos, Solitons and Fractals, 7, 1895 (1996)
[23] Ruelle, D., Publ. Math. IHES, 72, 175 (1990)
[24] Baladi, V., Commun. Math. Phys., 186, 671 (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.