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.
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1$\begingroup$ I think the explicit automorphism is $\alpha_g f(x)=f(g^{-1}x)$. For this to make sense, $\mu$ should be non-singular with respect to the $G$-action: that is $\mu(gS)=0$ if and only if $\mu(S)=0$. This ensures that the $L^\infty$ norm is preserved by the action. The most concrete example would be that $G$ is the 2-adic odometer, acting on $G$ itself, equipped with Haar measure. This is the example that you have read about: $G$ is the set of all infinite sequences of 0's and 1's. Addition is "adding with carry to the left". $\endgroup$– Anthony QuasCommented Sep 17, 2019 at 0:59
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1$\begingroup$ If $G$ is equipped with the metric $d(x,y)=2^{-\min \{n\colon x_n\ne y_n\}}$, then $G$ is a Cantor set. The Haar measure is just the (1/2,1/2) coin tossing measure. In ergodic theory, people often study the action of the element $g=(\ldots,0,0,1)$. I believe "extended" here means that you are looking at the action of the whole group $G$, rather than just the iterates of the single element $g$. $\endgroup$– Anthony QuasCommented Sep 17, 2019 at 1:00
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