ergodic theory – Number of permitted words up to permutation in a subshift

Let $A$ be a finite set and let $X subseteq A^{mathbb{N}}$ be a subshift. Let $mathcal{L}_n$ denote the set of words of length $n$ appearing in $X$. For a word $w in mathcal{L}_n$, one can define its empirical distribution to be the probability vector $ P_w := left( frac{N(a | w)}{n} right)_{a in A}$ where $N(a | w)$ denotes the number of occurrences of the letter $a in A$ in the word $w$. I am interested in the quantity $T(n) := # { P_w : w in mathcal{L}_n }$ (equivalently, $T(n) = #(mathcal{L}_n / sim)$, where $sim$ is the equivalence relation on $mathcal{L}_n$ that declares two words to be equivalent if they are permutations of each other).

More specifically, I’m interested in subshifts with the property that $limsup_{n to infty} T(n) = infty$. Is there a name for this property? Are there any more commonly known conditions that imply this property?

Some examples/non-examples for intuition:

  1. $X = {0,1}^mathbb{N}$. All words are permitted, so $T(n) = n+1$.

  2. $X subseteq {0,1}^mathbb{N}$ is the SFT defined by forbidding the word $11$. Words of length $n$ can have up to half of their entries be equal to $1$, so $T(n) approx n/2$.

  3. $X$ is the orbit closure of ${ x in {0,1}^mathbb{N} : x_{2i}x_{2i+1} = 01 text{ or } 10 text{ for all } i }$. Any permitted word $w$ of length $n$ has both $N(0 | w), N(1 | w) in (n/2 – 2, n/2 + 2)$, so $T(n) leq 5$ for all $n$.