Let $(X, \zeta)$ be a primitive substitution and let $C_n$ be a non-empty cylinder for the system of length $n$. For the moment, let's assume that $C_n$ is defined by the first $n$ coordinates. Is it true that there is some constant $K>0$ independent of $n$ such that
$\frac{1}{n} \leq K \mu(C_n)$
for each cylinder?
Since the substitution is uniquely ergodic this is equivalent to saying that each frequency is more or less uniform, as the number of cylinders of length $n$ is linear in $n$.
I've tried using estimates from the Perron-Frobenius theorem but haven't gotten very far.