Minimal idempotents and ergodic Ramsey theory

Summary

INTRODUCTION

What is common between the invertibility of distal maps, partition regularity of diophantine equation x — y = z², and the notion of mild mixing? The answer is: idempotent ultrafilters, and the goal of this survey is to convince the reader of the unifying role and usefulness of idempotent ultrafilters (and, especially, the minimal ones) in ergodic theory, topological dynamics and Ramsey theory.

We start with reviewing some basic facts about ultrafilters. The reader will find the missing details and more information in the self-contained Section 3 of [B2]. (See also [HiS] for a comprehensive presentation of the material related to topological algebra in the Stone-Čech compactification).

An ultrafilter p on N = {1, 2, …} is, by definition, a maximal filter, namely, a nonempty family of subsets of N satisfying the following conditions (the first three of which constitute the definition of a filter):

1. Ø ∉ p;

2. A ∈ p and A ⊂ B imply B ∈ p;

3. A ∈ p and B ∈ p imply A ∩ B ∈ p;

4. (maximality) if r ∈ N and N = A₁ ∪A₂ ∪ … ∪ A_r, then for some i, 1 ≤ i ≤ r, A_i ∈ p.