It is proved that a doubly stochastic operator P is weakly asymptotically cyclic if it almost overlaps supports. If moreover P is Frobenius-Perron or Harris then it is strongly asymptotically cyclic.
@article{bwmeta1.element.bwnjournal-article-apmv72z2p145bwm,
author = {Bartoszek, Wojciech},
title = {On asymptotic cyclicity of doubly stochastic operators},
journal = {Annales Polonici Mathematici},
volume = {72},
year = {1999},
pages = {145-152},
zbl = {0957.47011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv72z2p145bwm}
}
Bartoszek, Wojciech. On asymptotic cyclicity of doubly stochastic operators. Annales Polonici Mathematici, Tome 72 (1999) pp. 145-152. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv72z2p145bwm/
[000] [B1] W. Bartoszek, Asymptotic stability of iterates of positive contractions on Banach lattices, in: Proc. Int. Conf. Function Spaces (Poznań, 1986), Teubner Texte zur Math. 103, Teubner, 1986, 153-157.
[001] [B2] W. Bartoszek, Asymptotic periodicity of the iterates of positive contractions on Banach lattices, Studia Math. 91 (1988), 179-188. | Zbl 0675.47025
[002] [B3] W. Bartoszek, Asymptotic properties of the iterates of stochastic operators on (AL) Banach lattices, Ann. Polon. Math. 52 (1990), 165-173. | Zbl 0719.47022
[003] [BB] W. Bartoszek and T. Brown, On Frobenius-Perron operators which overlap supports, Bull. Polish Acad. Sci. Math. 45 (1997), 17-24. | Zbl 0891.47006
[004] [Br] J. R. Brown, Ergodic Theory and Topological Dynamics, Academic Press, New York, 1976.
[005] [F] S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Reinhold, New York, 1969.
[006] [K1] J. Komornik, Asymptotic periodicity of the iterates of Markov operators, Tôhoku Math. J. 38 (1986), 15-27. | Zbl 0578.47020
[007] [K2] J. Komornik, Asymptotic decomposition of Markov operators, Bull. Polish Acad. Sci. Math. 35 (1987), 321-327. | Zbl 0642.47026
[008] [KL] U. Krengel and M. Lin, On the deterministic and asymptotic σ-algebras of a Markov operator, Canad. Math. Bull. 32 (1989), 64-73. | Zbl 0638.60079
[009] [L] A. Lasota, Invariant principle for discrete time dynamical systems, Univ. Iagel. Acta Math. 31 (1994), 111-127. | Zbl 0830.58018
[010] [LM] A. Lasota and M. C. Mackey, Chaos, Fractals and Noise: Stochastic Aspects of Dynamics, Springer, New York, 1993.
[011] [R1] R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262. | Zbl 0838.47040
[012] [R2] R. Rudnicki, Asymptotic stability of Markov operators: a counter-example, ibid. 45 (1997), 1-5.
[013] [Z] R. Zaharapol, Strongly asymptotically stable Frobenius-Perron operators, preprint, 1997.