Multiple recurrence for non-commuting transformations along rationally independent polynomials

Abstract

We prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+p_i(n)$, with rationally independent $p_i$'s with zero constant term. This is in contrast to the single variable case, in which even double recurrence fails unless the transformations generate a virtually nilpotent group. The proof involves reduction to nilfactors and an equidistribution result on nilmanifolds.

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References in corpus (1)

• Ergodic averages of commuting transformations with distinct degree polynomial iterates

Cited by in corpus (2)

• Generic properties of extensions
• Van der Corput's difference theorem for amenable groups and the left regular representation