  The ergodic properties of the shift on both full and $m$-truncated $t$-free
$C^*$-algebras are analyzed. In particular, the shift is shown to be uniquely
ergodic with respect to the fixed-point algebra. In addition, for every $m\geq
1$, the invariant states of the shift acting on the $m$-truncated $t$-free
$C^*$-algebra are shown to yield a $m+1$-dimensional Choquet simplex, which
collapses to a segment in the full case. Finally, the spectrum of the position
operators on the $m$-truncated $t$-free Fock space is also determined.
