Linear dynamics of random products of weighted shifts

Abstract

The aim of this article is to study the dynamics of random products of weighted shifts on a separable Fr\'echet sequence space. That is, given a measure-preserving dynamical system $(\Omega, \mathcal{F}, \mu, \tau)$, a Fr\'echet sequence space $X$ with a basis $(e_n)_{n \geq 0}$, and a strongly measurable map $T : \Omega \to \mathcal{B}(X)$ taking values in a finite set of weighted shifts on $X$, we study the dynamics of the sequence $(T(\tau^{n-1}\omega) \dotsm T(\tau \omega) T(\omega))_{n \geq 1}$ for almost every $\omega \in \Omega$. After proving criteria to determine whether this sequence is universal, weakly mixing or mixing for almost every $\omega \in \Omega$, we study some examples on the spaces $X = \ell_p$, $X = c_0$ and $X = H(\mathbb{C})$ involving two shifts, first in the commuting case and then in the non-commuting one.