Turing Degree Spectra of Minimal Subshifts

Subshifts are shift invariant closed subsets of Σ 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 nonperiodic 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. A Z-subshift is a closed shift invariant subset of Σ d . Subshifts may be seen as sets of colorings of Z, with a finite number of colors, avoiding some set of forbidden patterns. Minimal subshifts are subshifts containing no proper subshift, or equivalently subshifts in which all configurations have the same patterns. They are fundamental in the sense that all subshifts contain at least a minimal subshift. Degrees of unsolvability of subshifts have now been studied for a few years, Cenzer, Dashti, and King [CDK08] and Cenzer, Dashti, Toska, and Wyman [CDTW10; CDTW12] studied computability of one dimensional subshifts and proved some results about their Turing degree spectra: the Turing degree spectrum of a subshift is the set of Turing degrees of its points, see Kent and Lewis [KL10]. Simpson [Sim11], building on the work of Hanf [Han74] and Myers [Mye74], noticed that the Medvedev and Muchnik degrees of subshifts of finite type (SFTs) are the same as the Medvedev degrees of Π1 classes: Π 0 1 classes are the subsets of {0, 1} N for which there exists a Turing machine halting only on oracles not in the subset. Subsequently Jeandel and Vanier [JV13] focused on Turing degree spectra of different classes of multidimensional subshifts: SFTs, sofic and effective subshifts. They proved in particular that the Turing degree spectra of SFTs are almost the same as the spectra of Π1 classes: adding a computable point to the spectrum of any Π1 class, one can construct an SFT with this spectrum. In order to prove that one cannot get a stronger statement, they proved that the This work was partially supported by grants EQINOCS ANR 11 BS02 004 03, TARMAC ANR 12 BS02 007 01 and ISF grant 1409/11.