Hierarchy and Expansiveness in 2D Subshifts of Finite Type

Abstract

Subshifts are sets of configurations over an infinite grid defined by a set of forbidden patterns. In this thesis, we study two-dimensional subshifts of finite type (2D SFTs), where the underlying grid is Z 2 and the set of forbidden patterns is finite. We are mainly interested in the interplay between the computational power of 2D SFTs and their geometry, examined through the concept of expansive subdynamics. 2D SFTs with expansive directions form an interesting and natural class of subshifts that lie between dimensions 1 and 2. An SFT that has only one non-expansive direction is called extremely expansive. We prove that in many aspects, extremely expansive 2D SFTs display the totality of behaviours of general 2D SFTs.For example, we construct an aperiodic extremely expansive 2D SFT and we prove that the emptiness problem is undecidable even when restricted to the class of extremely expansive 2D SFTs. We also prove that every Medvedev class contains an extremely expansive 2D SFT and we provide a characterization of the sets of directions that can be the set of non-expansive directions of a 2D SFT. Finally, we prove that for every computable sequence of 2D SFTs with an expansive direction, there exists a universal object that simulates all of the elements of the sequence. We use the so called hierarchical, self-simulating or fixed-point method for constructing 2D SFTs which has been previously used by Gács, Durand, Romashchenko and Shen.

The expressiveness of quasiperiodic and minimal shifts of finite type

“…• We extend to the classes of minimal/quasiperiodic shifts of finite type some known results on subdynamics: 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, which answers positively a question by E. Jeandel, [22] All constructions in this paper involve the technique of self-simulating tilings developed in [19] (see also variants of this technique in [24,28]).…”

Fixed Point Constructions in Tilings and Cellular Automata

“…Building on Hochman's proof, Guillon and Zinoviadis have characterized the sets of nonexpansive directions of SFTs in an unpublished manuscript [18], which also contains several realization results of SFTs with a unique nonexpansive direction. The results are reported in Zinoviadis's PhD dissertation [34]. In the next result, a subset A ⊆ Ṙ is effectively closed if there exists a computable sequence (I n ) n∈N of open intervals (which may have the form (a, ∞] ∪ (−∞, b)) with rational endpoints such that Ṙ \ A = n∈N I n .…”

Distortion in One-Head Machines and Cellular Automata

“…Several examples of such extremely expansive two-dimensional subshifts are known. A general self-simulating construction is given in [8], and effectivized (so that f is obtained as a partial local rule from the full shift) in [9,10].…”