**DEFINITION. ***A ***Bratteli diagram** is a directed graph with a distinguished vertex such that (i) any vertex can be joined from by at least one path; (ii) such paths have all the same length called the* ***level** of ; (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 and the set of arrows such that each arrow in $E_i$ goes from to .

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

To each Bratelli diagram is attached the set of infinite path starting at . As a subset of , it is compact.

**DEFINITION.*** A ***Vershik map** (or adic transformation) for an ordered Bratelli diagram is a homeomorphism with the following properties: (i) $latex \phi(X_{max})=*X_{min}$; (iii) where is the smallest integer such that is not maximal and is the arrow following in the diagram order and is the minimal path joining to the origin of .*

*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

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on November 16, 2009 at 14:47 |David CorfieldThe name is spelled Bratteli (see his site).