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Extender sets and multidimensional subshifts
Published online by Cambridge University Press: 02 October 2014
Abstract
In this paper, we consider a $\mathbb{Z}^{d}$ 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
$\mathbb{Z}^{d}$ analogue of a follower set called an extender set. The extender set of a finite word
$w$ in a
$\mathbb{Z}^{d}$ subshift
$X$ is the set of all configurations of symbols on the rest of
$\mathbb{Z}^{d}$ which form a point of
$X$ when concatenated with
$w$. As our main result, we show that for any
$d\geq 1$ and any
$\mathbb{Z}^{d}$ 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
$\mathbb{Z}$ 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
$\mathbb{Z}$ subshift is non-decreasing 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
$n$.
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- Research Article
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- © Cambridge University Press, 2014
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