An explicit condition for boundedly supermultiplicative subshifts

We study some properties of the growth rate of \ (L (A, F) \), that is, the language of words over the alphabet \ (A\) avoiding the set of forbidden factors \ (F\). We first provide a sufficient condition on \ (F\) and \ (A\) for the growth of \ (L (A, F) \) to be boundedly supermultiplicative. That is, there exist constants \ (C›0\) and \ (0\), such that for all \ (n\), the number of words of length \ (n\) in \ (L (A, F) \) is between \ (ⁿ\) and \ (Cⁿ\). In some settings, our condition provides a way to compute \ (C\), which implies that \ (\), the growth rate of the language, is also computable whenever our condition holds. We also apply our technique to the specific setting of power-free words where the argument can be slightly refined to provide better bounds. Finally, we apply a similar idea to \ (F\) -free circular words and in particular we make progress toward a conjecture of Shur about the number of square-free circular words. Mathematics Subject Classifications: 68R15Keywords: Subshift, supermultiplicativity, growth of languages, circular word