On subshifts with slow forbidden word growth

Ronnie Pavlov

doi:10.1017/etds.2020.138

Abstract

AbstractIn 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].

Highlights

In this work, we study subshifts, which are symbolically defined topological dynamical systems
The induced subshift is called a shift of finite type and the behavior of SFTs is in many senses very well understood
General arguments for uniqueness of MME and Gibbs bounds we describe a general theorem which implies uniqueness of MME, which is an extension of the main result of [22] using a new measure-theoretic specification property

Summary

Introduction

We study (one-dimensional) subshifts, which are symbolically defined topological dynamical systems. If X is a subshift satisfying the following hypotheses: X has unique MME μ, there exists C so that |Ln(X)| < Cenh(X) for all n, there exist G′ ⊂ L(X) and R ∈ N so that for all u, v, w ∈ G′, there exist y, z ∈ AR for which uyvzw ∈ L(X), and there exist ǫ > 0 and a syndetic S so that μ(G′n) > ǫ for every n ∈ S, G′ has the following Gibbs property: there exists D so that for all w ∈ G′, μ([w]) ≥ De−|w|h(X) Since this requires a stronger specification property than that of Theorem 3.2, verifying this for general subshifts with ‘small’ F requires a slightly different definition; we define G′ to be the set of words which neither begin nor end with more than one-fourth of a word in F. Denote by S′ the subset of S for which this inequality holds

Since the lower density of is greater

The conclusion