In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple

Bratteli diagram In mathematics, a Bratteli diagram is a combinatorial structure: a Graph (discrete mathematics), graph composed of vertices labelled by positive integers ("level") and unoriented edges between vertices having levels differing by one. The notion wa ...

(''V'', ''E'') with a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

on the set of all infinite paths called the Veršhik transformation. It is named after Ola Bratteli and

Anatoly Vershik Anatoly Moiseevich Vershik (russian: Анато́лий Моисе́евич Ве́ршик; born on 28 December 1933 in Leningrad) is a Soviet and Russian mathematician. He is most famous for his joint work with Sergei V. Kerov on representati ...

Definition

Let ''X'' = be the set of all paths in the essentially simple

(''V'', ''E''). Let ''E''_min be the set of all minimal edges in ''E'', similarly let ''E''_max be the set of all maximal edges. Let ''y'' be the unique infinite path in ''E''_max. (Diagrams which possess a unique infinite path are called "essentially simple".) The Veršhik transformation is a homeomorphism φ : ''X'' → ''X'' defined such that φ(''x'') is the unique minimal path if ''x'' = ''y''. Otherwise ''x'' = (''e''₁, ''e''₂,...) , ''e''_''i'' ∈ ''E''_''i'' where at least one ''e''_''i'' ∉ ''E''_max. Let ''k'' be the smallest such integer. Then φ(''x'') = (''f''₁, ''f''₂, ..., ''f''_''k''−1, ''e''_''k'' + 1, ''e''_''k''+1, ... ), where ''e''_''k'' + 1 is the successor of ''e''_''k'' in the total ordering of edges incident on ''r''(''e''_''k'') and (''f''₁, ''f''₂, ..., ''f''_''k''−1) is the unique minimal path to ''e''_''k'' + 1. The Veršhik transformation allows us to construct a pointed topological system (''X'', ''φ'', ''y'') out of any given ordered, essentially simple Bratteli diagram. The reverse construction is also defined.

Equivalence

The notion of

graph minor In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if ...

can be promoted from a

well-quasi-ordering In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements x_0, x_1, x_2, \ldots from X contains an increasing pair x_i \leq x_j with i x_2> \cdots) nor infinite sequenc ...

to an equivalence relation if we assume the relation is

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

. This is the notion of equivalence used for Bratteli diagrams. The major result in this field is that equivalent essentially simple ordered

s correspond to

topologically conjugate In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated func ...

pointed

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

s. This allows us apply results from the former field into the latter and vice versa.

Definition

Equivalence

See also

Notes

Further reading