Consider a group $G$ acting on a set $\Omega$. For example, let $G=\{g\in A(\mathbb R):(\alpha +1)g=\alpha g+1\}$ for all $\alpha\in\mathbb R$, where $A(\mathbb R)$ are the order-preserving automorphisms of $\mathbb R$. This example is given by Glass in his "Partially Ordered Groups" as an example of a periodic group action (p. 150) and it makes sense to me.
I don't understand the generalization he gives though. First, say a subset $\Delta\subseteq\Omega$ is coterminal if for any $\alpha\in\Omega$ there are $\delta_1,\delta_2\in\Delta$ such that $\delta_1<\alpha<\delta_2$. Then $(G,\Omega)$ is periodic if:
- The centralizer $C_\Omega(G)=\langle z\rangle$ for some $z$, and
- $\{\alpha z^n:n\in\mathbb Z\}$ is coterminal in $\Omega$ for any $\alpha$
Consider the earlier example of actions with period 1. It doesn't seem to me like anything except the identity is in the centralizer here, so I guess condition 1 is trivially satisfied? But then condition 2 fails. So I don't understand what's going on.
I think maybe there is some difference between "Centralizer of $G$" and "Centralizer of $G$ in $\Omega$" but I'm not sure what it is.
The $z$ realizing condition 1 in the example is the automorphism $\varphi$ of $\mathbb{R}$ given by $x\mapsto x+1$, since $G$ is by definition the order-preserving automorphisms of $\mathbb{R}$ that commute with $\varphi$. $\langle \varphi\rangle$ is the group of automorphisms $x\mapsto x+n$ for all $n$. We want to see this is the entire centralizer of $G$ in $\Omega$ (which in this case is just the center of $G$, see below.) So consider a $\psi\in G$ such that, for some $r$, $\psi(r)-r=:\beta\notin \mathbb{Z}$. Without loss of generality $r=0$. Then $\beta=\psi(0)\notin \mathbb{Z}$ and $\psi$ doesn't commute with the automorphism $\chi$ sending $x$ to $x+(x-\lfloor x\rfloor)^2$, where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$. To see this, compute $\psi(\chi(0))=\beta\neq\chi(\psi(0))=\beta+(\beta-\lfloor \beta\rfloor)^2$. Finally, every element of $G$ is either in $\langle\varphi\rangle$ or has the property we required of $\psi$. (The only alternative would be that every element differ from its image by an integer but that that integer vary-and this contradicts preservation of order.) So we've seen in this case $C_\Omega(G)=\langle\varphi\rangle$ and now it's probably clearer that condition 2 is indeed satisfied.
Having worked through the example, the motivation for the definition should be clearer: elements of $G$ are periodic automorphisms of $\mathbb{R}$ with period 1. In the general case the element $z$ generating the centralizer defines a generalized periodicity in that $z$ cuts $\Omega$ up into coterminal orbits and $G$ acts in the same way on any two elements of the same orbit.
Regarding your last sentence: in general for $G$ acting on some set $\Omega$ (not necessarily ordered) and $S\subset G, C_\Omega(S)$ is the set $\{g\in G:gs(x)=sg(x)\text{ for all }s\in S,x\in\Omega\}.$ So $C_\Omega(G)$ is the center of $G$ over $\Omega$, and in the example case two elements of $G$ commute as automorphisms if and only if they commute acting on every element of $\Omega$, which is how we got the reduction to considering the center of $G$. But in general $C_\Omega(G)$ needn't be anything like the center of $G$: consider a non-abelian group acting on a point, for instance.