An extension of Furstenberg's structure theorem for Noetherian modules and multiple recurrence theorems III

TL;DR

This paper extends Furstenberg's structure theorem to Noetherian modules over syndetic rings, establishing new multiple recurrence theorems and analyzing recurrence properties in topological dynamics.

Contribution

It generalizes Furstenberg's multiple recurrence theorem to a broader algebraic setting involving Noetherian modules over syndetic rings.

Findings

Established multiple recurrence theorems for Noetherian modules over syndetic rings.

Analyzed the multiple Birkhoff center and pointwise recurrence in topological G-actions.

Abstract

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.