An extension of Furstenberg's structure theorem for Noetherian modules   and multiple recurrence theorems III
  Using a recent Furstenberg structure theorem, we obtain Multiple Recurrence Theorems relative to any locally compact second countable Noetherian module $G$ over a syndetic ring $R$, which generalizes Furstenberg's multiple recurrence theorem. In addition we study the multiple Birkhoff center and the pointwise multiple recurrence of a topological $G$-action on a compact metric space. 
Xiongping Dai
math.DS