Roughly speaking, a theorem by Vershik says that every ergodic invertible measurepreserving transformation is isomorphic to some "adic" transformation on the spaces of paths of a BratelliVershik graph. What is the adic transformation corresponding to a Bernoulli shift ?
The short answer is that nobody knows. The reason is that Vershik's proof uses Rokhlin's towers and is thus virtually nonconstructive. 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 2torus this is an open question, let alone Bernoulli shifts. 


I think you can realize the Bernoulli shift on $k$ symbols as an adic transformation on the following BratelliVershik 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? 

