A note on the perturbations of subshifts

TL;DR

This paper studies how different classes of subshifts change when forbidden words are introduced, showing that some properties like soficness are preserved and analyzing entropy decay under perturbations.

Contribution

It provides new results on the preservation of soficness and synchronization under perturbations, and explicitly calculates entropy drops for specific subshifts.

Findings

Perturbations of sofic shifts remain sofic.

Sufficient conditions for synchronized systems to stay synchronized after perturbation.

Explicit entropy decay rates for $ ext{S}$-gap shifts and finite type subshifts.

Abstract

In this paper, we consider different classes of subshifts and study their perturbations obtained by forbidding sequences that contain a given word as a subword. We show that the perturbations of sofic shifts are sofic. Though not true for general coded systems, we provide sufficient conditions under which perturbations of synchronized systems remain synchronized. We investigate which systems remain conjugate under perturbation by a word from a coded system. We also explicitly calculate the drop in entropy under perturbation when the subshift is an $\mathcal{S}$-gap shift, and obtain its exponential decay as the length of the word increases. In the case of subshifts of finite type, we explore the drop in entropy under multi-word perturbations.