What is the adic realization of a Bernoulli shift ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:34:28Z http://mathoverflow.net/feeds/question/88621 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88621/what-is-the-adic-realization-of-a-bernoulli-shift What is the adic realization of a Bernoulli shift ? Stéphane Laurent 2012-02-16T10:51:37Z 2012-02-16T14:51:45Z <p>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 ?</p> http://mathoverflow.net/questions/88621/what-is-the-adic-realization-of-a-bernoulli-shift/88626#88626 Answer by Nikita Sidorov for What is the adic realization of a Bernoulli shift ? Nikita Sidorov 2012-02-16T13:18:59Z 2012-02-16T13:18:59Z <p>The short answer is that nobody knows. The reason is that Vershik's proof uses Rokhlin's towers and is thus virtually non-constructive. </p> <p>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. </p> http://mathoverflow.net/questions/88621/what-is-the-adic-realization-of-a-bernoulli-shift/88637#88637 Answer by Thierry de la Rue for What is the adic realization of a Bernoulli shift ? Thierry de la Rue 2012-02-16T14:51:45Z 2012-02-16T14:51:45Z <p>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?</p>