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

TL;DR

This paper completes the proof that all measures on the Young--Fibonacci graph's Martin boundary, beyond the Plancherel measure, are ergodic, using previous results and a parameterized family of measures.

Contribution

It proves the ergodicity of the remaining measures on the Young--Fibonacci graph's Martin boundary, extending prior work on the Plancherel measure.

Findings

All measures on the Martin boundary are ergodic.

The ergodicity is established for a family of measures parametrized by infinite words and a parameter β.

The proof leverages previous results as a black box.

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 finish the proof of their ergodicity using the statements proved in the first paper as a "black box".