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.