Pleasant extensions retaining algebraic structure, II

TL;DR

This paper develops methods to analyze and construct extensions of dynamical systems that allow for concrete descriptions of characteristic factors, leading to proofs of convergence for certain polynomial nonconventional ergodic averages involving multiple transformations.

Contribution

It introduces a new approach to constructing 'pleasant' extensions of $bZ^2$-systems with explicit characteristic factors, enabling convergence results for polynomial ergodic averages.

Findings

Constructed pleasant extensions with explicit characteristic factors.

Proved norm convergence for polynomial averages involving commuting transformations.

Connected algebraic structure with ergodic averages to nilsystem models.

Abstract

In this paper we combine the general tools developed in (arXiv:0905.0518) with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss, Host and Kra and Ziegler to study the averages $\frac{1}{N}\sum_{n=1}^N(f_1\circ T^{n\bf{p}_1})(f_2\circ T^{n\bf{p}_2})(f_3\circ T^{n\bf{p}_3})$ for $f_1,f_2,f_3 \in L^\infty(\mu)$ associated to a triple of directions $\bf{p}_1,\bf{p}_2,\bf{p}_3 \in \mathbb{Z}^2$ that lie in general position along with $0 \in \mathbb{Z}^2$. We will show how to construct a `pleasant' extension of an initially-given $\mathbb{Z}^2$-system for which these averages admit characteristic factors with a very concrete description, involving one-dimensional isotropy factors and two-step pro-nilsystems. We also use this analysis to construct pleasant extensions and then prove norm convergence for the polynomial nonconventional ergodic averages $\frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$ associated to two commuting transformations $T_1$, $T_2$.