Turing degree spectra of minimal subshifts

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.