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?

  • $\begingroup$ in crystallography you typically want to study physical states that are invariant under the crystallgraphic group (ex: you have crystal at equilibrium that is modelled by a probabilitty measure invariant under $\mathbb Z^3$) $\endgroup$ – Glougloubarbaki Oct 11 '17 at 7:50
  • $\begingroup$ When you write "With the notable exception of homogeneous dynamics where one is concerned with the action of a Lie group on a quotient of itself" do you actually have an example of interest other than by itself? It actually occurs naturally in some extended systems in statistical physics. For example couple map lattices are invariant under a $\mathbb Z^2$ action, but it doesn't go much further than that. In particular, equilibrium measures can often be constructed and are unique for potentials that decay exponentially. $\endgroup$ – John B Oct 11 '17 at 7:55
  • $\begingroup$ Those studying tiling dynamics regularly consider $\mathbb{Z}^d$ and $\mathbb{R}^d$ actions, and more exotically, actions by subgroups of the full Euclidean group of rigid motions. $\endgroup$ – Dan Rust Oct 11 '17 at 11:45
  • $\begingroup$ @DanRust Do you know of a reference where tilings have been studied with the action of the full group of rigid motions of $\mathbb Z^d$? $\endgroup$ – caffeinemachine Sep 30 '19 at 17:57
  • $\begingroup$ @caffeinemachine maybe something like arxiv.org/abs/1611.05756? More is known about the full group of rigid mothings of $\mathbb{R}^d$. Search for 'rotational tiling spaces' $\endgroup$ – Dan Rust Oct 1 '19 at 16:46

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