Feeds:
Posts

## Bratelli diagrams, Vershik maps and Invariant measures

DEFINITION. A Bratteli diagram is a directed graph $(V,E)$ with a distinguished vertex $v_0$ such that (i) any vertex $v$ can be joined from $v_0$ by at least one path; (ii) such paths have all the same length called the level of $v$; (iii) there is a finite, non-zero number of arrows leaving each vertex.

Remark. Property (ii) is equivalent to the fact that there are parititions of the set of vertices $V=\bigcup_{i\geq0} V_i$ and the set of arrows $E=\bigcup_{i\geq0} E_i$ such that each arrow in $E_i$ goes from $E_i$ to $E_{i+1}$.

DEFINITION. An order on a Bratelli diagram is the data, for each vertex $v$, of a total order on the set of all arrows pointing to $v$. A path is maximal, resp. minimal, if each of its arrows is maximal, resp. minimal, among the set of arrows with the same target. $X_{max}, X_{min}$ will denote the set of maximal, minimal paths.

To each Bratelli diagram $B$ is attached the set $X_B$ of infinite path starting at $v_0$. As a subset of $E_0\times E_1\times\dots$, it is compact.

DEFINITION. A Vershik map (or adic transformation) for an ordered Bratelli diagram $B$ is a homeomorphism $\phi:X_B\to X_B$ with the following properties: (i) $latex \phi(X_{max})=X_{min}$; (iii) $\phi(e_0,e_1,\dots) = (m_0,m_1,\dots,m_{k-1},e_k',e_{k+1},e_{k+2},\dots)$ where $k$ is the smallest integer such that $e_k$ is not maximal and $e_k'$ is the arrow following $e_k$ in the diagram order and $m_0,m_1,\dots,m_{k-1}$ is the minimal path joining $v_0$ to the origin of $e_k'$.

Remark. Not all ordered Bratelli diagrams admit Vershik maps (K. Medynets). If the set $X_{max}$ has empty interior, then there is at most one Vershik map.

The recent preprint of S. Bezugly, J. Kwiatkowski, K. Medynets and B. Solomyak describe the invariant measures under a Borel version of the Vershik map