In general, if one has two functions $f,g\colon A\to A$, then we say that they commute if $f\circ g=g\circ f$. However, you often see statements such as "Let $(T_g^{(1)})_{g\in G}, (T_g^{(2)})_{g\in G},...,(T_g^{(n)})_{g\in G}$ be commuting measure preserving transformations of a group (or semi-group) $G$ on a probability space $(X,\mathcal{B},\mu)$." in ergodic theory. Now these actions are functions $G\times X\to X$ but clearly this means something other than the functions commuting (because of domain issues). I have two possible interpretations of this statement and was wondering which (if any) is correct:
- For all $g\in G$ and any distinct $i,j\in\{1,...,n\}$, we have $T_g^{(i)}\circ T_g^{(j)}=T_g^{(j)}\circ T_g^{(i)}$.
- For all $g,h\in G$ and any distinct $i.j\in\{1,...,n\}$, we have $T_g^{(i)}\circ T_h^{(j)}=T_h^{(j)}\circ T_g^{(i)}$.
More generally, ergodic theory people also talk about IP-systems of transformations. That is, say we're given commuting measure-preserving transformations $T_1,T_2,T_3,...$ of an appropriate space. For each multi-index $\alpha=\{i_1,i_2,...,i_k\}$ with $i_1<i_2<\cdots i_k$, put $T_\alpha=T_{i_1}\circ T_{i_2}\circ\cdots\circ T_{i_k}$. Then we get the IP system $(T_\alpha)_\alpha$. Again, ergodic theory people talk about the "commuting" IP systems $(T^{(1)}_\alpha),(T^{(2)}_\alpha),...,(T^{(n)}_\alpha)$. I assume that the meaning here is totally analogous to the (semi)group case.
I briefly tried seeing if this was defined in any introductory ergodic theory textbooks (such as the notes by Marcelo Viana/Krerley Oliveira and the ones by Omri Sarig) but came up empty-handed. They just mention commuting maps $X\to X$ but not commuting actions. I also checked some standard abstract algebra texts but found nothing.
Thank you!
Let's say that you have a $\mathbb{Z}$-action and two invertible measure-preserving transformations, say $T$ and $S$. In this case $T_n=T^n$, for $n\in \mathbb{Z}$, where the latter is understood as $T\ \circ \cdots \circ \ T$, if $n>0$ or $T^{-1}\ \circ \cdots \circ \ T^{-1}$, if $n<0$, where we compose $|n|$-times. The statement that $T$ and $S$ commute means that $T \circ S=S \circ T$ and the statement that $(T_n)_{n\in \mathbb{Z}}$ and $(S_n)_{n\in \mathbb{Z}}$ commute should be understood as $ T^n \circ S^m = S^m \circ T^n,$ for any $n,m\in \mathbb{Z}$. The logical generalization seems to be number $2$ of your cases.
A reference, for example, can be found in page 1 of this paper: https://link.springer.com/article/10.1007/s11854-016-0019-7