Applications of idempotent ultrafilters

Question

Recently Justin Moore has posted a solution to the amenability of Thompson's group F. A key(?) step exploits the existence of idempotent ultrafilters on $\mathbb N$ to construct an idempotent measure on the free non-associative semigroup on one-generator.

Previously the main applications I knew of idempotent ultrafilters involved Ramsey theory, most specifically Hindman's theorem.

Question: What are other applications of idempotent ultrafilters?

I have made this a big-list CW question, although a big list would pleasantly surprise me.

Answer 1

There are applications of idempotent ultrafilters (often under the name "idempotent member of the enveloping semigroup") to finding and classifying the structure of topological dynamical systems. Auslander's book "Minimal Flows and Their Extensions" includes some of them.

Answer 2

I think one could make the case that idempotent ultrafilters are so closely related to Ramsey theory that anything which uses them does relate to Ramsey theory in some way (almost by definition). A less obvious example though would be Gowers's result that $c_0$ is oscillation stable --- that every bounded Lipschitz function on the sphere of $c_0$ is $\epsilon$-constant when restricted to an infinite dimensional subspace. That utilizes a system of idempotent ultrafilters on a families of (partial) semigroups known as $\mathrm{FIN}_k$ (where $k$ ranges over the natural numbers).

Answer 3

There are numerous applications of ultrafilters within ergodic theory/combinatorial number theory. Vitaly Bergelson's article gives a very nice review of these. Among the results presented there are the following:

• Strengthened Hindman: In a finite partition, one can find a cell that both and additive IP-set and a multiplicative IP-set.
• Partition regularity of $a + b = cd $: In a finite partition, one can find a cell that contains $a,b,c,d$ with $a + b = cd$.
• Integer approximation of polynomials: If $f:\ \mathbb{R} \to \mathbb{R}$ is a polynomial with $f(0) = 0$, then for fixed $\varepsilon > 0$, the distance from $f(n)$ to a closest integer is less than $\varepsilon$ for $IP^*$-many $n \in \mathbb{Z}$.
• Combinatorial richness of return times: If $f:\ \mathbb{Z} \to \mathbb{Z}$ is a polynomial with $f(0) = 0$, and $(X,T,\mu)$ is an invertible measure preserving system, then for fixed $\varepsilon > 0$, it holds that $\mu(T^{-f(n)}A \cap A) > \mu(A)^2-\varepsilon$ for $IP^*$-many $n \in \mathbb{Z}$. Under Furstenberg correspondence, this turns into the statement that if $E \subset \mathbb{N}$ has positive density $d^*(E) > 0$, then $d^*(E \cap (E − f(n))) > d^∗(E)^2 − \varepsilon$ for $IP^*$-many $n \in \mathbb{Z}$.

In fact, the last two results can be substantially extended, for instance $f$ can be a generalised polynomial, if we take care of some additional assumptions.

A more sophisticated application can be found in another article by Bergelson and McCutcheon, where a version of Szemeredi's theorem is proved for generalised polynomials. They prove a number of amusing auxiliary results, such as characterisation of weak mixing involving ultrafilters, or some convergence results for ultrafilter convergence in weakly mixing systems. There are also nice results for multiple operators re-proved in an article by Schnell.

I read somewhere a slogan saying approximately that ultrafilters allow one to do ergodic theory without ergodic averages. (So, instead of $\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N T^n $, look at $p\!-\!\lim_{n } T^n $, where $p$ is an ultrafilter (normally: idempotent or even minimal)).

There is a MSc thesis by J.G. Zirnstein, which covers some more applications, including a variety of Ramsey theoretical results (which go far beyond Hindman, but include also van der Waerden and Jin theorems), and is written in a very nice and accessible tone.