A well known result in Symbolic Dynamics asserts that every two-sided subshift on a finite alphabet necessarily consists of all doubly infinite words not containing any finite word from a given set of "forbidden" words (Proposition 1.3.4, D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge Univ. Press, 1999). This result is also true for one-sided subshifts (essentially the same proof works) but after researching a lot, I cannot find a direct reference to it, namely one that refers explicitly to one-sided subshifts. Can anyone help me find such a reference?
1 Answer
My friend Marcelo Sobottka came up with the reference:
W. Ott, M. Tomforde, P. N. Willis; One-sided shift spaces over infinite alphabets, NYJM Monographs (5) 2014.
which answers my question in:
Theorem 3.16. A subset $X \subseteq \Sigma_\mathcal A$ is a shift space if and only if $X = X_\mathcal F$ for some subset $\mathcal F \subseteq \Sigma^{fin}_\mathcal A$.
This is a bit of an overkill since this result deals with shift spaces over infinite alphabets but, specializing the above result to the finite alphabet case, it makes me 100% satisfied!