Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$.

Its topological entropy is defined as follows: $$ h(\Sigma) = \lim_{n\to\infty}\frac{\log \#\Sigma_n}{2n+1}. $$ Now let $Per_n$ denote all periodic sequences in $\Sigma$ of period $n$. Assume $$ \widetilde h(\Sigma) = \lim_{n\to\infty}\frac{\log \#Per_n}{n}=0. $$

**Question.** I need sufficient conditions for $h(\Sigma)=0$. It is known to hold for the sofic subshifts (Lind & Marcus, Exercise 4.4.3) but in my examples they are not sofic. Nor are they transitive.