Norm convergence of multiple ergodic averages on amenable groups

TL;DR

This paper extends Walsh's method to prove norm convergence of multiple ergodic averages for arbitrary amenable groups, including polynomial and triangular averages, generalizing previous results.

Contribution

It introduces a new approach for proving convergence of multiple ergodic averages on amenable groups, covering polynomial and triangular cases.

Findings

Established norm convergence for polynomial actions.

Proved convergence for triangular averages with commuting actions.

Generalized previous results to broader group settings.

Abstract

We apply Walsh's method for proving norm convergence of multiple ergodic averages to arbitrary amenable groups. We obtain convergence in the uniform Ces\`aro sense for their polynomial actions and for ``triangular'' averages associated to commuting homomorphic actions. The latter generalizes a result due to Bergelson, McCutcheon, and Zhang in the case of two actions.