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

Pauwel Van Den Eeckhaut, Asgar Jamneshan

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.

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

Abstract

It is an immediate consequence of the ergodic structure theorem of Host and Kra that every factor of an ergodic -step pro-nilsystem is again an ergodic -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.

Paper Structure

This paper contains 6 sections, 9 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. No small subnilmanifolds
  4. Local rigidity of joinings of nilsystems
  5. A special case
  6. Proof of Theorem \ref{['thm-main']}

Key Result

Theorem 1.1

Let $k \geq 1$. An ergodic measure-preserving dynamical system is of order $k$ if and only if it is measure-theoretically isomorphic to a $k$-step pro-nilsystem. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (16)

  • Theorem 1.1: Host--Kra structure theorem
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1: Subnilmanifold
  • Lemma 2.2: No small subnilmanifolds
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 6 more