Extender sets and multidimensional subshifts

Abstract

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.

Free ergodic ℤ²-systems and complexity

“…A simple example of such a rate follows from the classical Morse-Hedlund Theorem [10]: if there exists n ∈ N such that P X (R n ) ≤ n, then X contains only doubly periodic Z 2 -colorings (see e.g. the proof of Theorem 1.2 in [13]). In fact this bound is sharp: there exist Z 2 -colorings that are not doubly periodic and yet satisfy P X (R n ) = n + 1 for all n ∈ N. Many other subquadratic growth rates can also be realized by strictly ergodic Z 2 -subshifts that do not contain doubly periodic points (see, for example, [12]).…”

Characterizing follower and extender set sequences

“…In fact, the converse is also true: if the follower set or extender set sequence of a shift X is bounded, then X is necessarily sofic. (See [6]) Results about shifts presented by graphs which are not irreducible may often be found by considering the reducible graph's irreducible components; for this reason, results in Sections 4 and 5 of this paper will focus on the irreducible case. In fact, every one-dimensional sofic shift has a presentation G which is rightresolving, follower-separated, and contains a right-synchronizing word.…”

“…It is well-known that for a Z shift X, finiteness of {F X (w) | w in the language of X} is equivalent to X being sofic, that is, the image of a shift of finite type under a continuous shift-commuting map. (see [4]) (In fact it is true that finiteness of {E X (w) | w in the language of X} is equivalent to X being sofic as well, see [6]).…”

“…Martin Delacourt discovered the first such example in 2013. (see page 8 of [6]) In fact, a wide class of eventually periodic sequences may be realized: Theorem 1.2. Let n ∈ N, and A = {A 1 , A 2 , A 3 , ..., A k } be a nontrivial partition of {0, 1, ..., n − 1}.…”

Extender sets and measures of maximal entropy for subshifts

“…(As usual, the measure of a finite word is understood to mean the measure of its cylinder set; see Section for details.) A formal statement of our hypothesis uses extender sets ; the condition ‘ is replaceable by ’ is equivalent to the containment , where denotes the extender set of a word .…”