Pascal Vanier

Turing Degree Spectra of Minimal Subshifts

Subshifts are shift invariant closed subsets of

Hardness of Conjugacy, Embedding and Factorization of multidimensional Subshifts of Finite Type

\Sigma^{\mathbb{Z}^d}

Pi01 sets and tilings

, 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 subshifts 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.

Hardness of conjugacy, embedding and factorization of multidimensional subshifts

Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts of finite type (SFTs) in dimension d > 1. In particular, we prove that the factorization problem is Sigma^0_3-complete and that the conjugacy and embedding problems are Sigma^0_1-complete in the arithmetical hierarchy.

Slopes of 3-dimensional Subshifts of Finite Type

In this paper, we prove that given any \(\it \Pi^0_1\) subset P of {0,1}ℕ there is a tileset τ with a countable set of configurations C such that P is recursively homeomorphic to C ∖ U where U is a computable set of configurations. As a consequence, if P is countable, this tileset has the exact same set of Turing degrees.

SLOPES OF TILINGS

Knowing whether two SFTs of dimension higher than two are conjugate or if one embeds in the other is Σ 1 0 -complete.Deciding whether one of two multidimensional SFTs, effective or sofic shifts factor onto the other is Σ 3 0 -complete.For effective subshifts, all the results are also true in dimension one. Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts in dimensions d 1 for subshifts of finite type and sofic shifts and in dimensions d ? 1 for effective shifts. In particular, we prove that the conjugacy, factorization and embedding problems are Σ 3 0 -complete for sofic and effective subshifts and that they are Σ 1 0 -complete for SFTs, except for factorization which is also Σ 3 0 -complete.

Hardness of conjugacy and factorization of multidimensional subshifts of finite type

In this paper we study the directions of periodicity of three-dimensional subshifts of finite type (SFTs) and in particular their slopes. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope being the angle of the periodicity vectors. In this paper, we prove that any

Aperiodic Points in Z 2 -subshifts.

\Sigma^0_2

Aperiodic points in

set may be realized as a a set of slopes of an SFT.