# 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 $$n$$.