On Derivatives and Subpattern Orders of Countable Subshifts

TL;DR

This paper investigates the structural and computational properties of countable two-dimensional subshifts, introducing novel examples with complex derivatives, subpattern posets, and Cantor-Bendixson ranks.

Contribution

It constructs specific countable subshifts with unique properties, including maximally complex derivatives and intricate subpattern posets, advancing understanding of their structural complexity.

Findings

An SFT with maximally complex iterated derivatives

A sofic shift with an infinite descending chain in its subpattern poset

A family of SFTs with finite subpattern posets containing arbitrary finite posets

Abstract

We study the computational and structural aspects of countable two-dimensional SFTs and other subshifts. Our main focus is on the topological derivatives and subpattern posets of these objects, and our main results are constructions of two-dimensional countable subshifts with interesting properties. We present an SFT whose iterated derivatives are maximally complex from the computational point of view, a sofic shift whose subpattern poset contains an infinite descending chain, a family of SFTs whose finite subpattern posets contain arbitrary finite posets, and a natural example of an SFT with infinite Cantor-Bendixon rank.