Comparing the Effective Content of Subshifts

TL;DR

This paper explores the relationship between subshifts in symbolic dynamics, introducing concepts like finite determination and Ziegler reducibility, and investigates existential closure properties.

Contribution

It establishes a new characterization of finite determination between subshifts using Ziegler reducibility, inspired by computability theory.

Findings

Finite determination between subshifts is characterized by Ziegler reducibility.

Introduces the notion of one subshift being finitely determined over another.

Investigates existential closure properties for subshifts.

Abstract

Motivated by Ziegler's computability-theoretic characterisation of finite absolute presentability between groups, we prove an analogous theorem in symbolic dynamics. We introduce the notion of one subshift being finitely determined over another and show that this relation is characterised by Ziegler reducibility between their co-languages. We further investigate a notion of existential closure for subshifts.