A proof of Walsh's convergence theorem using couplings

TL;DR

This paper reinterprets Walsh's proof of convergence for nonconventional ergodic averages involving polynomial sequences in nilpotent groups, using classical ergodic theory tools like couplings and characteristic factors.

Contribution

It provides a new proof of Walsh's theorem by applying classical ergodic theory methods instead of finitary arguments.

Findings

Walsh's convergence theorem is established using couplings.

The proof connects finitary and classical ergodic theory approaches.

The approach simplifies understanding of polynomial ergodic averages.

Abstract

Walsh has recently proved the norm convergence of all nonconventional ergodic averages involving polynomial sequences in discrete nilpotent acting groups. He deduces this convergence from an equivalent, `finitary' assertion of stability over arbitrarily long time-intervals for these averages, which is proved by essentially finitary means. The present paper shows how the induction at the heart of Walsh's proof can also be implemented using more classical notions of ergodic theory: in particular, couplings and characteristic factors.