Previous |  Up |  Next

Article

Keywords:
hypercyclicity; transitivity; recurrence; the narrow topology; random dynamical system
Summary:
Let $(\Omega ,\mathcal {F},\mathbb {P})$ be a probability space, where $\mathcal {F}$ is countably generated, and $X$ be a Polish space. Let $\varphi $ be a random dynamical system with time $\mathbb {T}$ on $X$. The skew product flow $ \{\Theta _{t} , \ t\in \mathbb {T} \} $ induced by $\varphi $ is a family of continuous operators acting on $\Pr _{\Omega }(X)$, the set of all probability measures on $X\times \Omega $ with marginal $\mathbb {P}$, which is a Polish space equipped with the narrow topology. In this work, we introduce and study the notion of narrow recurrence of the flow $\{\Theta _{t},\ t\in \mathbb {T} \} $ on ${\rm Pr}_{\Omega }(X)$ and we give some results, which can be considered as an initiation of applications of properties of topological dynamics on stochastic process theory and random dynamical systems.
References:
[1] Amouch, M., Bachir, A., Benchiheb, O., Mecheri, S.: Weakly recurrent operators. Mediterr. J. Math. 20 (2023), Article ID 169, 16 pages. DOI 10.1007/s00009-023-02374-6 | MR 4565105 | Zbl 1525.47017
[2] Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin (1998). DOI 10.1007/978-3-662-12878-7 | MR 1723992 | Zbl 0906.34001
[3] Bès, J., Chan, K. C., Sanders, R.: Every weakly sequentially hypercyclic shift is norm hypercyclic. Math. Proc. R. Ir. Acad. 105A (2005), 79-85. DOI 10.1353/mpr.2005.0000 | MR 2182152 | Zbl 1113.47005
[4] Bès, J., Chan, K. C., Sanders, R.: Weak$^*$ hypercyclicity and supercyclicity of shifts on $\ell^\infty$. Integral Equations Oper. Theory 55 (2006), 363-376. DOI 10.1007/s00020-005-1394-0 | MR 2244194 | Zbl 1109.47009
[5] Chan, K. C., Sanders, R.: A weakly hypercyclic operator that is not norm hypercyclic. J. Oper. Theory 52 (2004), 39-59. MR 2091459 | Zbl 1114.47006
[6] Costakis, G., Manoussos, A., Parissis, I.: Recurrent linear operators. Complex Anal. Oper. Theory 8 (2014), 1601-1643. DOI 10.1007/s11785-013-0348-9 | MR 3275437 | Zbl 1325.47019
[7] Costakis, G., Parissis, I.: Szemerédi's theorem, frequent hypercyclicity and multiple recurrence. Math. Scand. 110 (2012), 251-272. DOI 10.7146/math.scand.a-15207 | MR 2943720 | Zbl 1246.47003
[8] Crauel, H.: Random Probability Measures on Polish Spaces. Stochastics Monographs 11. CRC Press, New York (2002). DOI 10.4324/9780203219119 | MR 1993844 | Zbl 1031.60041
[9] Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981). DOI 10.1515/9781400855162 | MR 0603625 | Zbl 0459.28023
[10] Gottschalk, W. H., Hedlund, G. A.: Topological Dynamics. American Mathematical Society Colloquium Publications 36. AMS, Providence (1955). DOI 10.1090/coll/036 | MR 0074810 | Zbl 0067.15204
[11] Karim, N., Benchiheb, O., Amouch, M.: Recurrence of multiples of composition operators on weighted Dirichlet spaces. Adv. Oper. Theory 7 (2022), Article ID 23, 15 pages. DOI 10.1007/s43036-022-00186-1 | MR 4409863 | Zbl 495.47023
[12] Poincaré, H.: Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13 (1890), 3-270 French \99999JFM99999 22.0907.01.
[13] Sanders, R.: Weakly supercyclic operators. J. Math. Anal. Appl. 292 (2004), 148-159. DOI 10.1016/j.jmaa.2003.11.049 | MR 2050222 | Zbl 1073.47013
[14] Neerven, J. van: The Asymptotic Behaviour of Semigroups of Linear Operators. Operator Theory: Advances and Applications 88. Birkhäuser, Basel (1996). DOI 10.1007/978-3-0348-9206-3 | MR 1409370 | Zbl 0905.47001
[15] Zaou, A., Amouch, M.: The narrow recurrence of continuous-time Markov chains. Funct. Anal. Approx. Comput. 16 (2024), 29-35. MR 4761148
[16] Zaou, A., Amouch, M.: The narrow recurrence of Markov chains. Rend. Circ. Mat. Palermo (2) 73 (2024), 613-620 \99999DOI99999 10.1007/s12215-023-00937-w . DOI 10.1007/s12215-023-00937-w | MR 4709081 | Zbl 7812637
Partner of
EuDML logo