Cantor-Bendixson ranks of countable SFTs

TL;DR

This paper characterizes the possible Cantor-Bendixson ranks of countable subshifts of finite type, showing they are exactly finite ordinals and certain computable ordinals, correcting a previous proof.

Contribution

It provides a complete characterization of Cantor-Bendixson ranks for countable SFTs and fixes an error in the author's earlier proof.

Findings

Ranks are exactly finite ordinals and ordinals of the form λ+3 for computable λ

Corrects a previous proof regarding these ranks

Clarifies the structure of countable SFTs in terms of Cantor-Bendixson analysis

Abstract

We show that the possible Cantor-Bendixson ranks of countable SFTs are exactly the finite ordinals and ordinals of the form $\lambda + 3$, where $\lambda$ is a computable ordinal. This result was claimed by the author in his PhD dissertation, but the proof contains an error, which is fixed in this note.