arXiv subjects

Séverin Benzoni

Publications and source records attributed to Séverin Benzoni.

3 recordsLinked to original sources

arXiv2025

Confined Poisson extensions

This paper follows on from our previous work, where we introduced the notion of \emph{confined extensions}, and our purpose is to widen the context in which such extensions appear. We do so in the setup of Poisson suspensions: we take a $σ$-finite measure-preserving dynamical system $(X, μ, T)$ and a compact extension $(X \times G, μ\otimes m_G, T_ϕ)$, then we consider the corresponding Poisson extension $((X \times G)^*, (μ\otimes m_G)^*, (T_ϕ)_*) \overset{}{\to} (X^*, μ^*, T_*)$. Our results give two different conditions under which that extension is confined. Finally, to show that those conditions are not void, we give an example of a system $(X, μ, T)$ and a cocycle $ϕ$ so that the compact extension $(X \times G, μ\otimes m_G, T_ϕ)$ has an infinite ergodic index.

math.DS↗

arXiv2025

Introduction to weak Pinsker filtrations

We introduce the so-called weak Pinsker dynamical filtrations, whose existence in any ergodic system follows from the universality of the weak Pinsker property, recently proved by Austin. These dynamical filtrations appear as a potential tool to describe and classify positive entropy systems. We explore the links between the asymptotic structure of weak Pinsker filtrations and the properties of the underlying dynamical system. A central question is whether, on a given system, the structure of weak Pinsker filtrations is unique up to isomorphism. We give a partial answer, in the case where the underlying system is Bernoulli. We conclude our work by giving two explicit examples of weak Pinsker filtrations.

math.DS↗

arXiv2023

Confined extensions and non-standard dynamical filtrations

In this paper, we explore various ways in which a factor $σ$-algebra $\mathscr{B}$ can sit in a dynamical system $\mathbf{X} :=(X, \mathscr{A}, μ, T)$, 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 $σ$-algebra. We show that there exist non-standard I-cosy dynamical filtrations.

math.DS↗