Welcome to MathOverflow

Question

2

Roughly speaking, a theorem by Vershik says that every ergodic invertible measure-preserving transformation is isomorphic to some "adic" transformation on the spaces of paths of a Bratelli-Vershik graph. What is the adic transformation corresponding to a Bernoulli shift ?

flag	asked Feb 16 2012 at 10:51 Stéphane Laurent 172●7

Nikita Sidorov · Accepted Answer · 2012-02-16 13:18:59Z UTC

The short answer is that nobody knows. The reason is that Vershik's proof uses Rokhlin's towers and is thus virtually non-constructive.

As far as I know, the only known examples of explicit adic realizations are substitutional dynamical systems and the irrational rotations of the circle. Even for a simple ergodic rotation of the 2-torus this is an open question, let alone Bernoulli shifts.

Thierry de la Rue · Answer 2 · 2012-02-16 14:51:45Z UTC

1

I think you can realize the Bernoulli shift on $k$ symbols as an adic transformation on the following Bratelli-Vershik diagram: put $k$ nodes on the first level. Suppose levels 1 to $n$ have been defined, and call $L_n$ the set of nodes in the $n$-th level. Then nodes on the $(n+1)$-th level are pairs $(i,j)\in L_n\times L_n$, where $(i,j)$ is connected to $i$ and to $j$ (in this order) to the $n$-th level. Does not this work?

answered Feb 16 2012 at 14:51

Thierry de la Rue
161●2

	Thanks to both of you. @Thierry, hence the graph is "homogeneous", in the sense that the number of paths to some node at level $n$ to a node at level $1$ is the same for all nodes at level $n$ ? – Stéphane Laurent Feb 16 2012 at 15:18
	In fact I am not very easy with the adic representation. Is it easy to check that Thierry's proposal is fine ? – Stéphane Laurent Feb 16 2012 at 15:55
	Indeed, the graph I proposed is homogeneous: Each node of level n has exactly k^n paths to the root of the diagram. But then I must admit that this poses some problem, since this introduces eigenvalues for the associated adic transformation. So this adic transformation is rather an extension of the Bernoulli shift (probably a direct product with an odometer). – Thierry de la Rue Feb 16 2012 at 20:03

What is the adic realization of a Bernoulli shift ?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.