Turing degree spectra of minimal subshifts

TL;DR

This paper investigates the Turing degree spectra of minimal subshifts, proving that some spectra can contain uncountably many disjoint cones, expanding understanding of their computational complexity.

Contribution

It constructs a minimal subshift with a spectrum comprising uncountably many disjoint Turing cones, showing spectra can be more complex than previously known.

Findings

Spectra can contain uncountably many disjoint cones.

Not all minimal subshifts have spectra forming a single cone.

The construction demonstrates greater diversity in Turing degree spectra.

Abstract

Subshifts are shift invariant closed subsets of $\Sigma^{\mathbb{Z}^d}$ , minimal subshifts are subshifts in which all points contain the same patterns. It has been proved by Jeandel and Vanier that the Turing degree spectra of non-periodic minimal subshifts always contain the cone of Turing degrees above any of its degree. It was however not known whether each minimal subshift's spectrum was formed of exactly one cone or not. We construct inductively a minimal subshift whose spectrum consists of an uncountable number of cones with disjoint base.