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Keywords:
hypercyclicity; transitivity; recurrence; the narrow topology; random dynamical system
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.
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