Ergodic averages of commuting transformations with distinct degree polynomial iterates

TL;DR

This paper proves mean convergence of multiple ergodic averages involving polynomial iterates of distinct degrees for commuting transformations, advancing the understanding of polynomial ergodic theory and its combinatorial implications.

Contribution

It establishes convergence results for polynomial ergodic averages with distinct degrees, extending previous linear cases and identifying the structure of characteristic factors as mixtures of nilsystems.

Findings

Proves mean convergence for polynomial ergodic averages with distinct degrees.

Shows characteristic factors are mixtures of inverse limits of nilsystems.

Provides applications to multiple recurrence and combinatorics.

Abstract

We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)... f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,...,p_\ell$ are integer polynomials with distinct degrees, and $T_1,...,T_\ell$ are commuting, invertible measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman, that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, the corresponding characteristic factors are mixtures of inverse limits of nilsystems. We use this particular structure, together with some equidistribution results on nilmanifolds, to give an application to multiple recurrence and a corresponding one to combinatorics.