## 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*.