We prove that every non-minimal transitive subshift X satisfying a mild aperiodicity condition satisfies lim sup cn(X)−1.5n = ∞, and give a class of examples which shows that the threshold of 1.5n cannot be increased. As a corollary, we show that any transitive X satisfying lim sup cn(X) − n = ∞ and lim sup cn(X)−1.5n < ∞ must be minimal. We also prove some restrictions on the structure of transitive non-minimal X satisfying lim inf cn(X) − 2n = −∞, which imply unique ergodicity (for a periodic measure) as a corollary, which extends a result of Boshernitzan [2] from the minimal case to the more general transitive case.
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 Kass and Madden [A sufficient condition for non-soficness of higher-dimensional subshifts. Proc. Amer. Math. Soc. 141 (2013), 3803-3816], 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 prism; 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.
For any subshift, define F X (n) to be the collection of distinct follower sets of words of length n in X. Based on a similar result proved in [4], we conjecture that if there exists an n for which |F X (n)| ≤ n, then X is sofic. In this paper, we prove several results related to this conjecture, including verifying it for n ≤ 3, proving that the conjecture is true for a large class of coded subshifts, and showing that if there exists n for which |F X (n)| ≤ log 2 (n + 1), then X is sofic.
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