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 $\mathbb{R}$! (With the notable exception of 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?

Answer 1

A common example can be found in fractal geometry, in particular, the case of Iterated Function Systems. Here the idea is that you have a compact set $X$ and a set of transformations $f_i\colon X \to X$ (with some sort of regularity, usually contractive similarities/affine maps) which you will compose sequentially. The dynamics arising from this kind of examples are so that looking at limits of the kind $(f_{i_1}\circ\dots\circ f_{i_n})(x_0)$ (for a point $x_0 \in X$) the limits of such expressions will converge to points in a fractal object $F\subset X$. The point of view can be switched around where you look at local inverses of those maps so you post-compose instead of pre-compose functions. The cantor set for instance can be generated this way if you take $f_1(x) = x/3$ and $f_2(x) = x/3 + 2/3$. This kind of stuff is treated in details in books like Falconer's Fractal Geometry.

Another common example comes from either deterministic or random compositions of hyperbolic maps of the unit interval, and the corresponding extension of the theory to non-uniformly hyperbolic maps by Saussol, Vaienti, Liverani, (LSV maps), among many others. The case of LSV maps is very interesting: they consist of deformations of the doubling map to include non-hyperbolicity (essentially having a point with derivative equal to 1, such that all iterates of the map also have derivative equal to 1 at that point) in kind of a minimal way. For a given $\alpha\in(0,1)$, define $$ f_\alpha(x) = \begin{cases} x(1+2^\alpha x^\alpha) \text{ if } \quad x\in[0,1/2] \\ 2x-1 \text{ if } \qquad x\in[1/2,1] \end{cases} . $$

Composing elements of this family of maps can be seen as the action of a semigroup (essentially $\mathbb{N}$) on $[0,1]$. In general people are interested in the limit laws of the dynamics associated to this family of maps (is there a LLN, or a CLT for this family? Is this law a.s. if we take the parameter randomly? Does it hold in average? Does it hold for all choices? Etc).

Answer 2

Here are some examples (some amenable, some not; consequently the arguments involved vary in the extent to which basic ergodic theory arguments are used):

• Ghys' paper "Groups Acting on the Circle" for many examples of interesting groups and group actions; the starting point is the classical rotation number invariant of Poincaré.

• Zimmer's book Ergodic Theory and Semisimple Lie Groups is a good book surveying many of his results; the so-called Zimmer program ("large groups can't act on small manifolds nontrivially") is going strong even today; see e.g. https://arxiv.org/abs/1608.04995 .

• Higher rank abelian actions have been mentioned in the comments above; one can consider smooth $\mathbb{Z}^k$ actions as actions of commuting diffeomorphisms and smooth $\mathbb{R}^k$ actions as commuting vector fields. The classical example of this in ergodic theory is the "$\times 2\times 3$" action on the circle: Rudolph-Lyons-Johnson proved that the only Borel probability measure $\mu$ on the circle that is ergodic and invariant under both $x\mapsto 2x$ and $x\mapsto 3x$ is Haar measure, if the metric entropy w/r/t $\mu$ of one of some time-$t$ map of the action is positive, partially answering a conjecture by Furstenberg. This is one of the starting points of measure rigidity (an area which Einsiedler contributed as well). Fundamentals of smooth ergodic theory of $\mathbb{Z}^k$ actions are established here for instance: https://arxiv.org/abs/1610.09997 .