Ergodicity of the Martin boundary of the Young--Fibonacci graph. I

TL;DR

This paper proves the ergodicity of certain measures on the Young--Fibonacci graph's path space, focusing on the case where the parameter beta equals 1, extending previous results on the Plancherel measure.

Contribution

It establishes the ergodicity of measures parametrized by infinite words and beta=1 on the Young--Fibonacci graph, expanding understanding of the Martin boundary.

Findings

Ergodicity of measures with beta=1 is proven.

Measures are parametrized by infinite words of digits 1 and 2.

Extends previous results on the Plancherel measure to new measures.

Abstract

Among central measures on the path space of the Young--Fibonacci lattice the so-called Plancherel measure has a special role. Its ergodicity was proved by Kerov and Gnedin. The goal of this cycle of two articles is to prove that remaining measures from the Martin boundary of this graph (which were described by Kerov and Goodman) are also ergodic. The measures are parametrized with an infinite word of digits 1 and 2 and the parameter $\beta\in(0,1]$ (the case $\beta=0$ corresponds to the Plancherel measure). In this article we prove the statements which correspond to the case $\beta=1$.