Metric on Shift of Finite Type is Non-Archimedean

Theorem
Let $\struct {X _\mathbf A, \sigma_\mathbf A}$ be a shift of finite type.

Let $\theta \in \openint 0 1$.

Then the metric $d_\theta$ on $X_\mathbf A$ is non-archimedean metric.

That is, $\struct {X _\mathbf A, d _\theta} $ is an ultrametric space.