What are some interesting examples of non-classical dynamical systems? (Group action other than $\mathbb{Z}$ or $\mathbb{R}$ )

Question

By classical dynamical system, I mean a measure space together with a measurable action of the integers or the reals. Of course, this action is often interpreted as evolution with respect to discrete or continuous time, respectively.

But there is a large theory out there of dynamics/ergodic theory where the action is taken to be either an arbitrary group, or a group from some large class generalizing $\mathbb{Z}$ and $\mathbb{R}$. For example, Chapter 8 of Ergodic Theory with a View Towards Number Theory by Einsiedler and Ward begins with:

The facet of ergodic theory coming from abstract mathematical models of dynamical systems evolving in time involves a single, iterated, measure- preserving transformation (action of $\mathbb{N}$ or of $\mathbb{Z}$) or a flow (action of the reals). For many reasons—including geometry, number theory, and the origins of ergodic theory in statistical mechanics—it is useful to study actions of groups more general than the integers or the reals.

The rest of the chapter then develops the ergodic theory of amenable group actions. Likewise, there is the classic paper Ergodic theory of amenable groups actions by Ornstein and Weiss, where the authors say in the introduction that in applications they kept encountering groups other than the integers or lattices of the integers, and so it was worth it to develop the theory in the full generality of the amenable setting.

But I haven't actually seen many examples where we care about, say, an ergodic action of a group other than $\mathbb{Z}$ or $\mathbb{R}$! Or $\mathbb{Z}^d$ or $\mathbb{R}^d$, I suppose. (A notable exception would be homogeneous dynamics, where one is concerned with the action of a Lie group on a quotient of itself.) What are some other examples? Either in other areas of pure mathematics, or in applied areas like Einsiedler and Ward allude to?

(This is a cross-post from Math Stack Exchange, where this question has been sitting unanswered for a while.)

Answer 1

As YCor says, this is a community wiki 'big list'-type question. Here are a couple of examples I've heard of:

1. One context where fairly exotic-looking groups arise from dynamical considerations is in the theory of iterated monodromy groups. See for instance this article of Nekrashevych:

https://arxiv.org/pdf/math/0312306.pdf

2. Another source of groups 'of dynamical origin' are various 'full groups' associated to dynamical systems. Even if the system you start with is, say, the integers acting on the Cantor set (either ergodically with respect to an invariant measure, or minimally with respect to the topology), the associated full group (topological full group, L^p full group, and so on) can be rather complicated and actually captures much of the structure of the dynamical system algebraically. See for instance this article of Giordano, Putnam and Skau:

http://www.math.uvic.ca/faculty/putnam/r/fgrp.pdf

In both of the examples above, the groups that arise are also interesting for purely group-theoretic reasons. For example, topological full groups were used by Juschenko and Monod to give the first known examples of infinite finitely generated simple amenable groups, and iterated monodromy groups give rise to many interesting examples of groups of intermediate growth.

Answer 2

One great example of an ergodic action is $SL_2(\mathbb{Z})$ acting on $\mathbb{R} \cup \infty \approx S^1$ by fractional linear transformations: $$\begin{pmatrix}a & b \\ c & d \end{pmatrix} \cdot t = \frac{at+b}{ct+d} $$ This is just one example of the Moore Ergodicity Theorem which is covered in Zimmer's book "Ergodic Theory and Semisimple Lie Groups", among other places.

More examples of Moore's theorem arise from any finite volume, complete hyperbolic $m$-manifold $M$. Let $\mathbb{H}^m$ be hyperbolic $m$-space, let $\mathbb{H}^m \mapsto M$ be the universal covering map, and let $\pi_1 M \circlearrowleft \mathbb{H}^m$ be the deck transformation action. Letting $S^{m-1}_\infty$ be the sphere at infinity of $\mathbb{H}^m$, the deck action $\pi_1 M \circlearrowleft \mathbb{H}^m$ extends continuously to a smooth action $\pi_1 M \circlearrowleft S^{m-1}_\infty$, and Moore's Theorem lets you conclude that this is an ergodic action.

You can also use other rank 1 symmetric spaces in place of hyperbolic space, with little change to the statement.

Answer 3

The simplest example of a non-trivial group action is that of the action of a group on itself. Now suppose that the group is bordified (particular case: compactified) by attaching a certain boundary to it, i.e., the group is embedded into a bigger space. There are numerous constructions of boundaries, which are natural in the sense that the action of the group on itself extends to the boundary (Stone-Cech compactification, end compactification, Martin compactification, hyperbolic compactification, horospheric compactification, etc., etc.). By endowing the boundary with various quasi-invariant measures one can further study ergodic properties of these actions. A notable example is the Patterson (or Patterson-Sullivan) measure on the hyperbolic boundary (in various degrees of generality, the simplest case being the boundary circle of the hyperbolic plane, which was the original setup of Patterson). However, there is also a construction of a boundary action directly in the measure category (without using any topological bordifications). This is the Poisson boundary. Its definition requires an additional piece of data, namely, a probability measure on the group (akin to specifying a Riemannian structure on a smooth manifold).