In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple Bratteli diagram (''V'', ''E'') with a

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

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

Anatoly Vershik Anatoly Moiseevich Vershik (; 28 December 1933 – 14 February 2024) was a Soviet and Russian mathematician. He is most famous for his joint work with Sergei V. Kerov on representations of infinite symmetric groups and applications to the lon ...

Definition

Let ''X'' = be the set of all paths in the essentially simple Bratteli diagram (''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, 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 on a set X is a quasi-ordering of X for which every 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) ...

to an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

if we assume the relation is

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

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

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

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

Definition

Equivalence

See also

Notes

Further reading