Here is a Bratelli-Vershik graph: alt text http://www.freeimagehosting.net/newuploads/64jrg.jpg

This graph should (I do not master this topic) define a Vershik adic transformation $T$ on its path space.

• What are the invariant measure(s) on the path space for which $T$ is ergodic ?

• Is $T$ (isomorphic to) a known example of transformation in ergodic theory ?

EDIT 26/08/2012: I finally don't understand why $T$ is isomorphic to the dyadic odometer, as claimed in RW's answer below. Below are some observations about this graph.

• To be sure of the definition of the graph, using Vershik terminology it is given by the Markov compactum $(M_n)$ where $M_n$ is the $n\times (n+1)$-matrix obtained by concatenating the $n \times n$-identity matrix with a column of ones: $$M_n=\begin{array}{cccccc} 1 & 0 & \cdots &\cdots & 0 & 1 \\\ 0 & \ddots & \ddots & & \vdots & \vdots \\\ \vdots & \ddots & \ddots & & \vdots & \vdots \\\ \vdots & & & & 0 & \vdots \\\ 0 & \cdots & & 0 & 1 & 1 \end{array}$$

• In figure below this is an example of the action of $T$, the blue path becomes the red path: alt text http://s16.postimage.org/pj9umy939/bratelli_Copie.jpg

• According to the link between adic and cut-and-stack the transformation $T$ should correspond to a cutting and stacking construction looking like:alt text http://s16.postimage.org/hvmy4naf9/cutandstack.jpg

Is there something wrong in my understanding ? Could you elaborate a little the isomorphism between $T$ and the dyadic odometer ? Perhaps I could convince myself that the cutting and stacking construction asymptotically coincides with the odometer, but I don't see the isomorphism between the paths space of this adic graph and the paths space of the usual adic graph of the odometer.

EDIT (later): I have checked that the cut-and-stack procedure is indeed an approximation of the dyadic odometer. Hence RW is right but still I don't see the isomorphism between the paths space.