Ergodicity of the adic transformation on the Euler graph

Abstract

The Euler graph has vertices labelled (n,k) for n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to (n,k) is the Eulerian number A(n,k), the number of permutations of 1,2,...,n+1 with exactly n-k falls and k rises. We prove that the adic (Bratteli-Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.

8 pages, 8 figures, to appear Math. Proc. Camb. Phil. Soc

Cited by in corpus (3)

• Invariant Measures on Stationary Bratteli Diagrams
• The Euler adic dynamical system and path counts in the Euler graph
• Boundaries from inhomogeneous Bernoulli trials