Norm-variation of triple ergodic averages for commuting transformations

TL;DR

This paper establishes an $r$-variation estimate for ergodic averages involving three commuting transformations, advancing understanding of their convergence properties in ergodic theory.

Contribution

It proves an $r$-variation estimate for three commuting transformations, a case not previously resolved for all $r>4$ in ergodic averages.

Findings

Proves $r$-variation estimate for three transformations with $r>4$

Highlights open questions for $r o 2$ and more than three transformations

Advances the mathematical understanding of ergodic averages and their convergence

Abstract

We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all $r\ge 2$ as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any $r<\infty$ for more than three commuting transformations.