Question

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.

Answer 1

In case the book referred to in Doug Lind's comment didn't have what you're looking for, a proof of this statement can be found in Section 2 of this paper:

Doris Fiebig and Ulf-Rainer Fiebig, Invariants for subshifts via nested sequences of shifts of finite type, Ergodic Theory and Dynamical Systems 21 (2001), pp 1731-1758.

There are also some other references there, including an article by Krieger with the proof, and another paper by Fiebig and Fiebig with lots more information on coded systems.