I am thinking a problem: given a subshift of finte type of $\{0,1\}^{\mathbb{N}}$ and $2>q>1$, where $q$ is a real number. Then how can we find the largest and smallest numbers of the projection of this subshift of finite type in base $q$. We know that the projection of the subshift of finite type in base $q$ is a graph-directed self-similar sets. Hence, the problem is that how can we find the extreme points of this set. Here, we only think one-dimensional graph-directed self-similar sets.


The largest and smallest numbers exist as SFT is closed and the projection is compact. 



The upper and lower bound of the projection of a subshift of finite type
It's a rational point for each $q$. Suppose the SFT has forbidden words of length at most $\ell$. Then given your current symbol and $\ell-1$ previous symbols, you can decide what symbol to put to maximize(minimize) your projection going forward. This means that your point is eventually periodic with period at most $2^\ell$. 


