If a group $G$ acts on cantor set $(X,\mu)$ by odometers, my question is what is the explicit automorphism $\alpha_{g}$ for the extended koopman action on $L^{\infty}(X,\mu)$, for $g$ $\in$ $G$? I am not getting by lots of trying, in many book its written that it is adding by $(\cdots,0,0,1)$ on the sequence of 0,1's. Kindly help me figuring it out.

If a group $G$ acts on a Cantor set $(X,\mu)$ by odometers, my question is: what is the explicit automorphism $\alpha_{g}$ for the extended Koopman action on $L^{\infty}(X,\mu)$, for $g$ $\in$ $G$? I am not getting by lots of trying; in many books it's written that it is adding by $(\cdots,0,0,1)$ on the sequence of 0,1's. Kindly help me figuring it out.