  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 $\beta=1$.
