Ergodic Theorems for Free Groups and Weak Mixing of Markov Measures on their Boundary
Speaker: Anush TserunyanDate of Talk: February 7, 2025
Upstream link: Very Informal Gathering of Logicians
1. A Nice Local-to-Global Principle
There’s a nice theorem characterising ergodic actions by \(\mathbb{Z}\) on a measure space \((X, \mu )\):
Theorem 1. (Birkhoff, 1931)
A measure-preserving action \(\mathbb{Z}\curvearrowright (X, \mu )\) is ergodic if and only if for all \(f\in L^1(X, \mu )\) and almost every \(x\), \[\lim _{n\to\infty} \frac{1}{n}\sum _{j=-n} ^{n} f(n\cdot x) = \int f\ d \mu.\]
I liked the speaker’s description of this: ergodicity, a global property of an action, can be characterised by its local combinatorics.
This theorem has had many generalisations to so-called amenable groups, with the range of the limit replaced by Følner sequences (what are those?).
People care about generalising this theorem (and other ergodic theorems) to free groups, which are non-amenable. Many of these were rekt by Terence Tao in 2015 with a counterexample.
2. A Generalisation to Free Groups
Theorem 2. (Tserunyan and Zomback, 2024)
A measure-preserving action \(\mathbb{F}_r\curvearrowright (X, \mu )\) is ergodic if and only if for each \(f\in L^1(X, \mu )\) and a.e. \(x\in X\), and for each sequence \(\tau _n\) of subtrees of the Cayley graph of \(\mathbb{F} _r\) containing the identity satisfying \(\lim _{n\to\infty} m \left( \tau _n \right) = \infty\), \[\lim _{n\to\infty} \frac{1}{m \left( \tau _n \right)} \sum _{g\in \tau _n} f\left( g\cdot x \right) m(g) = \int f\ d \mu .\]
Here, \(m\) is the “uniform weight”, which I think you can guess what it means.
Later, the speaker also proposed a theorem that characterises “weakly mixing” actions of \(\mathbb{F}_r \curvearrowright\partial \mathbb{F} _r\), and this involved a similar idea of describing the local properties of the action.
3. A Continuous Counterpart?
In general, these ideas reminded me a lot of something Elias told me about dynamical systems. The “paradigmatic question” was, if a particle evolves according to the SDE \[dx = B_0(x) dt + \sum _{j=1}^{k} B_j(x) dB_t ^{(j)},\] does \(x\) eventually “go everywhere”? Here \(B_0,\ldots, B_k\) are vector fields on \(\mathbb{R}^n\) and \(k \ll n\). This problem of a global nature ends up being equivalent to \(\dim \operatorname{Lie} \left( B_0, \ldots, B_k \right) = n\), a condition on the local interactions of these vector fields!
Somehow this feels like a continuous version of this discrete problem, and I wonder if there’s something like Hőrmander’s condition for weak mixing actions of \(\mathbb{F} _r\).