Confined extensions and non-standard dynamical filtrations

TL;DR

This paper investigates the structure of extensions within dynamical systems, introducing confined extensions that lack super-innovations, and demonstrates the existence of non-standard dynamical filtrations with complex properties.

Contribution

It introduces the concept of confined extensions in dynamical systems, explores their properties, and proves the existence of non-standard extensions and filtrations.

Findings

Confined extensions have no super-innovation.

Existence of non-standard extensions demonstrated.

Non-standard I-cosy dynamical filtrations exist.

Abstract

In this paper, 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 from the theory of filtrations. An important focus of our work is the introduction of the notion of confined extensions, whose initial interest is that they have no super-innovation. 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.