Is it a known example of adic transformation ? (1)

Question

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:

       

• According to the link between adic and cut-and-stack the transformation $T$ should correspond to a cutting and stacking construction looking like:

       

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.

Answer 1

There are two edges going out of every vertex of this graph; label the vertical one with 0, and the other one with 1. Thus, paths in your graph can be parametrized by the space of 0,1 sequences. If one removes the vertical path corresponding to the sequence (0,0,0,...), then the tail equivalence relation and the lexicographic order on its classes for the punctured path space coincide with the corresponding objects for the punctured space of sequences. Thus, your transformation in precisely the classical 2-adic shift with all due consequences.

Answer 2

Let me make some trivial remarks about your question. I suppose you are looking for an automorphism of a measure space that is isomorphic to the transformation $T$ defined by the above Bratteli diagram ($T$ is the so called Bratteli-Vershik transformation that acts on the path space $X$ of this diagram). Staying in the context of measurable dynamics, you should clearly define a measure $\mu$ invariant (or quasi-invariant) with respect to $T$. In other words, $T$ must be considered on the support of $\mu$ what is not necessary the whole space $X$. Another thing is that $\mu$ could be non-unique. The concept of isomorphism in measurable dynamics is actually $\rm{mod}\ 0$ isomorphism so that you can ignore sets of measure zero.

I am doubtful that your transformation $T$ is isomorphic to an odometer. In the other question of yours when the right-most edges of the diagram are multiple you will get that an induced transformation $T_A$ is isomorphic to a non-stationary odometer. I suppose this is rather obvious so that let me omit details.