Shift of finite type; sofic; multidimensional
In this paper, we consider a Z d extension of the well-known fact that subshifts with only finitely many follower sets are sofic. As in , we adopt a natural Z d analogue of a follower set called an extender set. The extender set of a finite word w in a Z d subshift X is the set of all configurations of symbols on the rest of Z d which form a point of X when concatenated with w. As our main result, we show that for any d ≥ 1 and any Z d subshift X, if there exists n so that the number of extender sets of words on a d-dimensional hypercube of side length n is less than or equal to n, then X is sofic. We also give an example of a non-sofic system for which this number of extender sets is n + 1 for every n. We prove this theorem in two parts. First we show that if the number of extender sets of words on a d-dimensional hypercube of side length n is less than or equal to n for some n, then there is a uniform bound on the number of extender sets for words on any sufficiently large rectangular prisms; to our knowledge, this result is new even for d = 1. We then show that such a uniform bound implies soficity. Our main result is reminiscent of the classical Morse-Hedlund theorem, which says that if X is a Z subshift and there exists an n such that the number of words of length n is less than or equal to n, then X consists entirely of periodic points. However, most proofs of that result use the fact that the number of words of length n in a Z subshift is nondecreasing in n,and we present an example (due to Martin Delacourt) which shows that this monotonicity does not hold for numbers of extender sets (or follower sets) of words of length n.
Ormes, Nic and Pavlov, Ronnie, "Extender Sets and Multidimensional Subshifts" (2013). Mathematics Preprint Series. Paper 18.