Hierarchy and Expansiveness in Two-Dimensional Subshifts of Finite Type

Subshifts are sets of configurations over an infinite grid defined by a set of forbidden patterns. In this thesis, we study two-dimensional subshifts offinite type (2D SFTs), where the underlying grid is Z2 and the set of for-bidden 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 Ga´cs, Durand, Romashchenko and Shen.Siirretty Doriast

Hierarchy and Expansiveness in Two-Dimensional Subshifts of Finite Type

Using a deterministic version of the self-similar (or hierarchical, or fixed-point ) method for constructing 2-dimensional subshifts of finite type (SFTs), we construct aperiodic 2D SFTs with a unique direction of non-expansiveness and prove that the emptiness problem of SFTs is undecidable even in this restricted case. As an additional application of our method, we characterize the sets of directions that can be the set of non-expansive directions of 2D SFTs.Comment: 72 pages, main body of the author's PhD Thesis, most of the results obtained in collaboration with Pierre Guillo

Truncated horseshoes and formal languages in chaotic scattering

In this paper we study parameter families of truncated horseshoes as models of multiscattering systems which show a transition to chaos without losing hyperbolicity, so that the topological features of the transition are completely describable by a parameterized family of symbolic dynamics. At a fixed parameter value the corresponding horseshoe represents the set of orbits trapped in the scattering region. The bifurcations are a pure boundary effect and no other bifurcations such as saddle center bifurcations occur in this transition scenario. Truncated horseshoes actually arise in concrete potential scattering under suitable conditions. It is shown that a simple scattering model introduced earlier can realize this scenario in a certain parameter range (the "truncated sawshoe") . For this purpose, we solve the inverse scattering problem of finding the central potential associated to the sawshoe model. Furthermore, we review classification schemes for the transition to chaos of truncated horseshoes originating from symbolic dynamics and formal language theory and apply them to the truncated double horseshoe and the truncated sawshoe.Comment: 39 pages postscript; use uudecode and uncompress ! 4 figures available as hardcopies on reques

27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021)

The fixed point construction is a method for designing tile sets and cellular automata with highly nontrivial dynamical and computational properties. It produces an infinite hierarchy of systems where each layer simulates the next one. The simulations are implemented entirely by computations of Turing machines embedded in the tilings or spacetime diagrams. We present an overview of the construction and list its applications in the literature.</p

Language and Automata Theory and Applications

International audienc

Subshifts with sparse traces

We study two-dimensional subshifts whose horizontal trace (also called projective subdynamics) contains only configurations of finite support. Our main result is a classification result for such subshifts satisfying a minimality property. As corollaries, we obtain new proofs for various known results on traces of SFTs, nilpotency, and decidability results for cellular automata and topological full groups. We also construct various (sofic) examples illustrating the concepts

Finitely Constrained Groups

This work investigates three aspects of the theory of finitely constrained groups, motivated by questions first asked by Rostislav Grigorchuk when he introduced the subject in 2005. The first topic is Hausdorff dimension of finitely constrained groups of p-adic tree automorphisms. The set of possible values of Hausdorff dimension for such a group is known, and we are able to show that every value in this set actually occurs. The second topic, related to the first, is topological finite generation of finitely constrained groups of p-adic tree automorphisms. Relatively little is known about which values of Hausdorff dimension occur for topologically finitely generated, finitely constrained groups of p-adic tree automorphisms. We are able to show that certain values can not occur as the Hausdorff dimension a topologically finitely generated, finitely constrained group of p-adic automorphisms defined by patterns of size d. We discuss finitely constrained groups of binary tree automorphisms with pattern size d ≥ 5 and Hausdorff dimension 1 – 2/2^d-1 ; the issue of topological finite generation for these groups is more challenging. We provide explicit constructions of new examples of finitely constrained groups and calculate their Hausdorff dimension. Finally, we study the portraits of self-similar groups using well-known ideas from the theory of tree automata, with particular focus on examples which separate certain classes of tree languages. These self-similar groups generalize the usual notion of self-similar groups, and we show that some well-known results extend to this more general case. From the tree language perspective, self-similar groups whose portraits form sofic tree shifts are of particular interest. We conclude by posing many questions for future study