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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.

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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
up vote 2 down vote accepted

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.

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Thanks Vaughn, a direct explicit reference is exactly what we needed here! -Doug – Douglas Lind Feb 16 '12 at 6:39
Thanks! that was exactly what I was looking for. – Rafael Alcaraz Feb 16 '12 at 9:39

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