On subshifts with slow forbidden word growth

Abstract

In this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$ , where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result, which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of the measure of maximum entropy and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of $x \mapsto \alpha + \beta x$ (the so-called $\alpha $ - $\beta $ shifts of Hofbauer [Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. verw. Geb.52(3) (1980), 289–300]) and the bounded density subshifts of Stanley [Bounded density shifts. Ergod. Th. & Dynam. Sys.33(6) (2013), 1891–1928].

Finding lower bounds on the growth and entropy of subshifts over countable groups

“…Let us also mention that the result of Miller has identical conditions but the conclusion only implies non-emptiness of the subshift [8]. The ideas behind our proof and the proof from [8,10] share some similarities, which explains the similarities of the conditions. The main difference lies in the fact that they consider the number of possible extensions of a word instead of looking at the suffixes of the words (intuitively, they look ahead at what could go wrong when one tries to extend the word further, while we look at the past to see what could have gone wrong when building the current word).…”

“…The main difference lies in the fact that they consider the number of possible extensions of a word instead of looking at the suffixes of the words (intuitively, they look ahead at what could go wrong when one tries to extend the word further, while we look at the past to see what could have gone wrong when building the current word). In [10], Pavlov had a similar but weaker result that can be restated as follows (by making the replacement c = β −1 in his result).…”

“…We then focus our attention on the special case G = Z and we provide a condition that is probably strictly better than the one given in [10]. In practice, the improvement on the resulting bounds seems to be considerable.…”

“…If instead of any group we restrict our attention to G = Z, we can use the structure of Z to obtain more precise results. In this setting, Pavlov gave a condition on F that implies lower bounds on the entropy of X F [10]. His proof was based on an idea introduced by Miller that only gave the nonemptiness of the subshift [8].…”

“…In the setting of combinatorics on words a similar technique was already known under the name power series method [2,3,9,12]. The conditions in [8,10] resemble the conditions obtained by the power series methods, but it appears that the link was never established. In fact, if G = Z, Lemma 11 can also be deduced as a particular case of the conditions from [9].…”