Ergodic seminorms for commuting transformations and applications

TL;DR

This paper develops new tools involving ergodic seminorms for multiple commuting transformations, aiming to facilitate future research in multiple ergodic theory beyond existing proofs of convergence theorems.

Contribution

It introduces tools for analyzing multiple commuting transformations, extending methods used for single transformations to more complex systems.

Findings

Provides a new framework for multiple commuting transformations

Extends ergodic seminorm techniques to broader settings

Lays groundwork for solving related open problems

Abstract

Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide some tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they will be useful in the solution of other problems.