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Turing degrees of multidimensional SFTs. (English) Zbl 1417.03241
Summary: In this paper, we are interested in computability aspects of subshifts and in particular Turing degrees of two-dimensional subshifts of finite type (SFTs) (i.e., tilings). To be more precise, we prove that, given any \(\Pi_1^0\) class \(P\) of \(\{0,1\}^\mathbb N\), there is an SFT \(X\) such that \(P\times\mathbb Z^2\) is recursively homeomorphic to \(X\setminus U\), where \(U\) is a computable set of points. As a consequence, if \(P\) contains a computable member, \(P\) and \(X\) have the exact same set of Turing degrees. On the other hand, we prove that, if \(X\) contains only non-computable members, some of its members always have different but comparable degrees. This gives a fairly complete study of Turing degrees of SFTs.

MSC:
03D28 Other Turing degree structures
37B10 Symbolic dynamics
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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