The entropies of renewal systems

Renewal systems are symbolic dynamical systems originally introduced by Adler. If W is a finite set of words over a finite alphabet A, then the renewal system generated by W is the subshift X _W ⊂A ^Z formed by bi-infinite concatenations of words from W. Motivated by Adler's question of whether every irreducible shift of finite type is conjugate to a renewal system, we prove that for every shift of finite type there is a renewal system having the same entropy. We also show that every shift of finite type can be approximated from above by renewal systems, and that by placing finite-type constraints on possible concatenations, we obtain all sofic systems.