Norm convergence of multiple ergodic averages on amenable groups

Abstract

We apply Walsh's method for proving norm convergence of multiple ergodic averages to arbitrary amenable groups. We obtain convergence in the uniform Cesàro 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.

v8: 16 p., with improved notation and a new introduction

References in corpus (3)

• A quantitative ergodic theory proof of Szemerédi's theorem
• A nilpotent IP polynomial multiple recurrence theorem
• Oscillation and the mean ergodic theorem for uniformly convex Banach spaces

Cited by in corpus (6)

• Cancellation for the simplex Hilbert transform
• Quantitative norm convergence of double ergodic averages associated with two commuting group actions
• Khintchine-type recurrence for 3-point configurations
• Multiple recurrence and large intersections for abelian group actions
• Uniform syndeticity in multiple recurrence
• Model Theory and the mean ergodic theorem for abelian unitary actions