On subshifts with slow forbidden word growth

TL;DR

This paper investigates subshifts with slowly growing forbidden words, demonstrating they have unique measures of maximal entropy and Gibbs properties, using a general framework linking supermultiplicativity and specification.

Contribution

It introduces a new approach connecting supermultiplicativity and measure-theoretic properties to establish uniqueness and Gibbs bounds for subshifts with slow forbidden word growth.

Findings

Subshifts with slow forbidden word growth are well-behaved and have unique MME.

They satisfy Gibbs bounds on large measure sets.

The results apply to $eta$-shifts and bounded density subshifts.

Abstract

In this work, we treat subshifts, defined in terms of an alphabet $A$ and (usually infinite) forbidden list $F$, where the number of $n$-letter words in $F$ has "slow growth rate" in $n$. We show that such subshifts are well-behaved in several ways; for instance, they are boundedly supermultiplicative as defined by Baker and Ghenciu 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 MME 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 f(x) = $\alpha + \beta x$ (the so-called $\alpha$-$\beta$ shifts) and the bounded density subshifts of Stanley.