Here is a Bratelli-Vershik graph: ![alt text][1] 

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][2]

- 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][3]

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 path spaces of this adic graph and the path space of the usual adic graph of the odometer.

  [1]: http://www.freeimagehosting.net/newuploads/64jrg.jpg
  [2]: http://s16.postimage.org/pj9umy939/bratelli_Copie.jpg
  [3]: http://s16.postimage.org/hvmy4naf9/cutandstack.jpg