Confined Poisson extensions

Abstract

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 $\ddot{I}ƒ$-finite measure-preserving dynamical system $(X,\hat{I}¼,T)$ and a compact extension $(X\times G,\hat{I}¼\otimes m_G,T_{\ddot{I}}•)$, then we consider the corresponding Poisson extension $((X\times G{)}^{\ast },(\hat{I}¼\otimes m_G{)}^{\ast },(T_{\ddot{I}}•{)}_{\ast })\to (X^{\ast },\hat{I}{¼}^{\ast },T_{\ast })$. 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 $(X,\hat{I}¼,T)$ and a cocycle $\ddot{I}•$ so that the compact extension $(X\times G,\hat{I}¼\otimes m_G,T_{\ddot{I}}•)$ has an infinite ergodic index.