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On a Gauss-Kuzmin-type problem for a generalized Gauss-Kuzmin operator. (English) Zbl 1237.37013
The author studies the operator defined by W. Fluch [Anz. Österr. Akad. Wiss. Math.-Naturwiss. Kl. 124 (1987), 73–76 (1988; Zbl 0709.11040)] in the context of random systems with complete connections (see [M. Iosifescu and S. Grigorescu, Dependence with complete connections and its applications. Cambridge Tracts in Mathematics. 96. Cambridge (UK): Cambridge University Press (1990; Zbl 0749.60067)]), and provides the corresponding Gauss-Kuzmin-type problem.
MSC:
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60J05 Discrete-time Markov processes on general state spaces
11K50 Metric theory of continued fractions
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[1] W. Fluch, “Ein verallgemeinerter Gauss-Kuzmin-operator,” Österreichische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Anzeiger, vol. 124, pp. 73-76, 1987. · Zbl 0709.11040
[2] Ch. Ganatsiou, “A random system with complete connections associated with a generalized Gauss-Kuzmin operator,” Revue Roumaine de Mathématiques Pures et Appliquées. Romanian Journal of Pure and Applied Mathematics, vol. 40, no. 2, pp. 85-89, 1995. · Zbl 0865.11055
[3] M. Iosifescu and S. Grigorescu, Dependence with Complete Connections and Its Applications, vol. 96 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0749.60067
[4] Ch. Ganatsiou, “On a Gauss-Kuzmin type problem for piecewise fractional linear maps with explicit invariant measure,” International Journal of Mathematics and Mathematical Sciences, vol. 24, no. 11, pp. 753-763, 2000. · Zbl 0971.60101 · doi:10.1155/S0161171200003872 · eudml:48937
[5] M. Iosifescu, “On the application of random systems with complete connections to the theory of f-expansions,” in Progress in Statistics (European Meeting Statisticians, Budapest, 1972), vol. 9 of Colloq. Math. Soc. János Bolyai, pp. 335-363, North-Holland, Amsterdam, The Netherlands, 1974. · Zbl 0299.60077
[6] M. Iosifescu, “Recent advances in the metric theory of continued fractions,” in Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Prague, 1978), Vol. A, pp. 27-40, Reidel, Dordrecht, The Netherlands, 1978. · Zbl 0406.10041
[7] S. Kalpazidou, “On a random system with complete connections associated with the continued fraction to the nearer integer expansion,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 30, no. 7, pp. 527-537, 1985. · Zbl 0576.60100
[8] S. Kalpazidou, “On the application of dependence with complete connections to the metrical theory of G-continued fractions. Dependence with complete connections,” Lietuvos Matematikos Rinkinys, vol. 27, no. 1, pp. 68-79, 1987, English translation: Lithuanian Mathematical Journal, vol. 27, no. 1, pp. 32-40, 1987. · Zbl 0644.10035 · doi:10.1007/BF00972019
[9] S. Kalpazidou, “On a problem of Gauss-Kuzmin type for continued fraction with odd partial quotients,” Pacific Journal of Mathematics, vol. 123, no. 1, pp. 103-114, 1986. · Zbl 0563.28012 · doi:10.2140/pjm.1986.123.103
[10] Ch. Ganatsiou, “On the ergodic behaviour of a random system with complete connections associated with a concrete piecewise fractional linear map,” Far East Journal of Dynamical Systems, vol. 10, no. 2, pp. 145-152, 2008. · Zbl 1161.37013 · pphmj.com
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