The expressiveness of quasiperiodic and minimal shifts of finite type

Abstract

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for the effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension d can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension d + 1. Supported by the ANR grant Racaf ANR-15-CE40-0016-01. Preliminary versions of some of the presented results we published in conference papers on MFCS-2015 [23] (Theorems 3-4) and MFCS-2017 [26] (Theorems 6-7).

Fixed Point Constructions in Tilings and Cellular Automata

“…The set of valid tilings over T contains X α , and T can quite easily be modified to guarantee that these subshifts are equal. In [7], Durand and Romashchenko presented a variant of the construction in which α is primitive, meaning that for some k ≥ 0, every t ∈ T occurs in α k (s) for every s ∈ T . In this case X α is a minimal subshift.…”

“…Some dynamical properties of the subshift X in theorem 2 can be lifted to the SFT cover Z of Y . One of the main results of Durand and Romashchenko in [7] states that if X is minimal or quasiperiodic, then the same property can be imposed on Z. They modify the construction so that every macrotile contains a zone of "diversification slots" where all valid 2 × 2 patterns of tiles are manually forced to appear.…”

“…This is overcome by choosing a canonical configuration from each and implementing the minimal shift generated by their combination. In the same article, the authors characterize the Turing degree spectra of quasiperiodic SFTs, which are exactly the upper closures of effectively closed subsets of {0, 1} N (see [7,Theorem 4] for details). In particular, there exist nonempty quasiperiodic SFTs with only noncomputable configurations.…”

“…The entropies of two-and higher-dimensional SFTs were characterized by Hochman in [22] as the nonnegative right-computable numbers, that is, limits of computable decreasing sequences of rational numbers. Independently in the preprint [15] using a variant of Robinson tiles and in [7] using fixed point tile sets, it was shown that they can be realized with transitive SFTs. The topological entropies of three-and higher-dimensional CA were characterized in [20].…”