# Coded Systems and dense subsets

A shift space $(X, \sigma)$ is a coded system if there exist a countable collection of finite words $(\omega^n)_{n \in \mathbb{N}}$, called generators, such that $X$ is the closure of the set of sequences obtained by freely concatenating the generators.

In Lind and Marcus book An introduction to symbolic dynamics and coding, page 451 there is stated a nice topological equivalence: $(X,\sigma)$ is a coded system if and only if $X$ contain an increasing sequence of transitive subshifts of finite type whose union is dense in $X$.

It is mentioned that this equivalence is due to Krieger and it is cited on Blanchard and Hansel paper Sofic constant-to-one extensions of subshifts of finite type, Proc. Amer.Math. Soc. 112 (1991), 259-265. The given reference is a personal communication between W. Krieger and the authors.

I would like to know if there exist a reference where I can find the proof of this statement.

-
I think you could find a proof in the book by Blanchard and Maass, "Topics in Symbolic Dynamics and Applications, but I don't have the book handy so can't check for sure. –  Douglas Lind Feb 15 '12 at 17:53
Thank you very much! I'm going to the library right know. I'll let you know if the proof is there. –  Rafael Alcaraz Feb 15 '12 at 18:28