Convergence of iterates of Lasota-Mackey-Tyrcha operators

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Department of Mathematics, University of South Africa, P.O. Box 392, Pretoria 0001, South Africa

We provide sufficient and necessary conditions for asymptotic periodicity of iterates of strong Feller stochastic operators.

stochastic (Markov) operator, strong Feller kernel, stationary density, asymptotic periodicity

K. Baron and A. Lasota, Asymptotic properties of Markov operators defined by Volterra type integrals, Ann. Polon. Math. 57 (1993), 161-175.
W. Bartoszek, Asymptotic stability of the iterates of contractions on Banach lattices, in: Proc. Internat. Conf. on Function Spaces, Poznań 1986, J. Musielak (ed.), Teubner, Leipzig, 1987, 153-157.
W. Bartoszek, Asymptotic periodicity of the iterates of positive contractions on Banach lattices, Studia Math. 91 (1988), 179-188.
W. Bartoszek, Asymptotic properties of the iterates of stochastic operators on (AL) Banach lattices, Ann. Polon. Math. 52 (1990), 165-173.
A. Lasota, T. Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751-764.
A. Lasota and M. C. Mackey, Chaos, Fractals and Noise: Stochastic Aspects of Dynamics, Appl. Math. Sci. 97, Springer, New York, 1993.
A. Lasota, M. C. Mackey and J. Tyrcha, The statistical dynamics of recurrent biological events, J. Math. Biology 30 (1992), 775-800.
S. Łojasiewicz, An Introduction to the Theory of Real Functions, Wiley, Chichester, 1988.
F. Räbiger, Attractors and asymptotic periodicity of positive operators on Banach lattices, Tübingen Ber. Funktionanal. 3 (1993/94), 184-203.
D. Revuz, Markov Chains, Elsevier, Amsterdam, 1984.
H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York, 1974.