Perturbing Subshifts of Finite Type

TL;DR

This paper develops a general criterion for bounding entropy differences in subshifts of finite type when avoiding certain finite word sets, showing that the entropy perturbation diminishes under specific conditions.

Contribution

It introduces a new criterion to bound entropy perturbations in subshifts of finite type and demonstrates the entropy difference tends to zero with appropriate word sets.

Findings

Entropy difference can be bounded using the new criterion.

Entropy perturbation tends to zero with suitable sequences of word sets.

The results apply under various assumptions on the word sets.

Abstract

Given an SFT $\Sigma$ and a finite set $S$ of finite words, let $\Sigma\langle S\rangle$ denote the subshift of $\Sigma$ that avoids $S$. We establish a general criterion under which we can bound the entropy perturbation $h(\Sigma)-h(\Sigma\langle S\rangle)$ from above. As an application, we prove that this entropy difference tends to zero with a sequence of such sets $S_1, S_2, \dots$ under various assumptions on the $S_i$.