Local rigidity of self-joinings and factors of pro-nilsystems

TL;DR

This paper provides a new, structure-theorem-independent proof that factors of ergodic pro-nilsystems are themselves ergodic pro-nilsystems, using a novel local rigidity theorem for nilsystems.

Contribution

It introduces a local rigidity theorem for nilsystems and offers an alternative proof of the factor-closure property without relying on the ergodic structure theorem.

Findings

Proved that factors of ergodic pro-nilsystems are ergodic pro-nilsystems independently of the structure theorem.

Established a local rigidity theorem showing near-diagonal self-joinings are graph joinings of automorphisms.

Provided a new proof of the ergodic inverse theorem connecting to Green, Tao, and Ziegler's combinatorial results.

Abstract

It is an immediate consequence of the ergodic structure theorem of Host and Kra that every factor of an ergodic $k$-step pro-nilsystem is again an ergodic $k$-step pro-nilsystem. It has remained open whether this fact can be proved independently of the structure theorem itself. In this note, we give such a proof, avoiding the machinery behind that theorem entirely. The key new ingredient is a local rigidity theorem for nilsystems: any ergodic self-joining sufficiently close to the diagonal joining is necessarily the graph joining of an automorphism. This rigidity result may be of independent interest. Together with a complementary result of Tao, our proof of the factor-closure of pro-nilsystems yields a new proof of the ergodic inverse theorem of Host and Kra from the combinatorial inverse theorem of Green, Tao, and Ziegler for the Gowers norms on cyclic groups.