Linear dynamics of random products of operators

We study the linear dynamics of the random sequence $(T_n(\mathrm{.}){)}_{n\ge 1}$ of the operators $T_n(\omega )=T({\tau }^{n-1}\omega )\cdots T(\tau \omega )T(\omega ),n\ge 1$. These products depend on an ergodic measure-preserving transformation $\tau :\mathbb{T}\to \mathbb{T}$ on the probability space $(\mathbb{T},m)$ and on a strongly measurable map $T:\mathbb{T}\to \mathcal{B}(X)$, where $X$ is a separable Fréchet space. We will be focusing on the case where $T(\omega )$ is equal to an operator $T_1$ on $X$ for every $\omega \in A_1$ and equal to an operator $T_2$ on $X$ for every $\omega \in A_2$, where $A_1,A_2$ are two disjoint Borel subsets of $[0,1)$ such that $A_1\cup A_2=[0,1)$ and $m(A_k)>0$ for $k=1,2$. More precisely, we will be focusing on the case where the operators $T_1$ and $T_2$ are adjoints of multiplication operators on the Hardy space $H^2(\mathbb{D})$, as well as the case where $T_1$ and $T_2$ are entire functions of exponential type of the derivation operator on the space of entire functions. Finally, we will study the linear dynamics of a case of a random product $T_n(\omega )$ for which the operators $T({\tau }^i\omega ),i\ge 0$, do not commute. We will give particular importance to the case where the ergodic transformation is an irrational rotation or the doubling map on $\mathbb{T}$.