THE ABSOLUTE OF THE COMB GRAPH

摘要

In the 1970s, R. Stanley introduced the comb graph 𔼠whose vertices are indexed by the set of compositions of positive integers and branching reflects the ordering of compositions by inclusion. A. Vershik defined the absolute of a ℤ+-graded graph as the set of all ergodic probability central measures on it. We show that the absolute of 𔼠is naturally parametrized by the space Ω = {(α1, α2, …) : αi ≥ 0, ∑iαi ≤ 1}.