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

TL;DR

This paper generalizes Furstenberg's structure theorem to a broader class of dynamical systems involving Noetherian modules over syndetic rings, enhancing the understanding of multiple recurrence phenomena.

Contribution

It introduces an extension of Furstenberg's structure theorem applicable to standard Borel G-spaces with Noetherian modules over syndetic rings, broadening the theorem's scope.

Findings

Extended Furstenberg's structure theorem to new algebraic settings

Established multiple recurrence results in the generalized framework

Provided foundational tools for further research in ergodic theory

Abstract

We extend Furstenberg's structure theorem to any standard Borel $G$-space, where $G$ is any locally compact second countable Noetherian module over a syndetic ring.