Confined extensions and non-standard dynamical filtrations
Volume 276 / 2024
Abstract
We explore various ways in which a factor $\sigma $-algebra $\mathscr B$ can sit in a dynamical system $\mathbf X :=(X, \mathscr A, \mu , T)$, i.e. we study some possible structures of the extension $\mathscr A \rightarrow \mathscr B$. We consider the concepts of super-innovations and standardness of extensions, which are inspired by the theory of filtrations. An important aspect of our work is the introduction of the notion of confined extensions, which first interested us because they have no super-innovations. We give several examples and study additional properties of confined extensions, including several lifting results. Then, using $T, T^{-1}$ transformations, we show our main result: the existence of non-standard extensions. Finally, this result finds an application to the study of dynamical filtrations, i.e. filtrations of the form $(\mathscr F_n)_{n \leq 0}$ such that each $\mathscr F_n$ is a factor $\sigma $-algebra. We show that there exist non-standard I-cosy dynamical filtrations.
