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Mathieu Sablik

Publications and source records attributed to Mathieu Sablik.

At least 37 records · Page 2Linked to original sources
arXiv2018

Quantified block gluing, aperiodicity and entropy of multidimensional SFT

It is possible to define mixing properties for subshifts according to the intensity which allows to concatenate two rectangular blocks. We study the interplay between this intensity and computational properties. In particular we prove that there exists linearly block gluing subshift of finite type which are aperiodic and that all right-recursively enumerable positive number can be realized as entropy of linearly block gluing Z 2-subshift of finite type. Like linearly block gluing imply transitivity, this last point answer a question asked in [HM10] about the characterization of the entropy of transitive subshift of finite type.

math.DS↗
arXiv2017

Weak colored local rules for planar tilings

A linear subspace $E$ of $\mathbb{R}^n$ has colored local rules if there exists a finite set of decorated tiles whose tilings are digitizations of $E$. The local rules are weak if the digitizations can slightly wander around $E$. We prove that a linear subspace has weak colored local rules if and only if it is computable. This goes beyond the previous results, all based on algebraic subspaces. We prove an analogous characterization for sets of linear subspaces, including the set of all the linear subspaces of $\mathbb{R}^n$.

math.DS↗
arXiv2016

A generalization of the simulation theorem for semidirect products

We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed $\mathbb{Z}^d$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^2$. Let $H$ be a finitely generated group and $G = \mathbb{Z}^2 \rtimes H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,f)$ where $Y$ is a Cantor set, there exists a $G$-subshift of finite type $(X,σ)$ such that the $H$-subaction of $(X,σ)$ is an extension of $(Y,f)$. In the case where $f$ is an expansive action of a recursively presented group $H$, a subshift conjugated to $(Y,f)$ can be obtained as the $H$-projective subdynamics of a $G$-sofic subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.

math.DS↗
arXiv2016

A notion of effectiveness for subshifts on finitely generated groups

We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of a group. This new class of subshifts forms a conjugacy class that contains all sofic subshifts. Motivated by the question of whether there exists a group where the class of sofic subshifts coincides with that of effective subshifts, we show that the inclusion is strict for several groups, including recursively presented groups with undecidable word problem, amenable groups and groups with more than two ends. We also provide an extended model of Turing machine which uses the group itself as a tape and characterizes our extended notion of effectiveness. As applications of these machines we prove that the origin constrained domino problem is undecidable for any group of the form $G \times \mathbb{Z}$ subject to a technical condition on $G$ and we present a simulation theorem which is valid in any finitely generated group.

math.GR↗
arXiv2016

Characterisation of sets of limit measures after iteration of a cellular automaton on an initial measure

The asymptotic behavior of a cellular automaton iterated on a random configuration is well described by its limit probability measure(s). In this paper, we characterize measures and sets of measures that can be reached as limit points after iterating a cellular automaton on a simple initial measure, in the same spirit as SRB measures. In addition to classical topological constraints, we exhibit necessary computational obstructions. With an additional hypothesis of connectivity, we show these computability conditions are sufficient by constructing a cellular automaton realising these sets, using auxiliary states in order to perform computations. Adapting this construction, we obtain a similar characterization for the Cesàro mean convergence, a Rice theorem on the sets of limit points, and we are able to perform computation on the set of measures, i.e. the cellular automaton converges towards a set of limit points that depends on the initial measure. Last, under non-surjective hypotheses, it is possible to remove auxiliary states from the construction.

math.DS↗
arXiv2016

Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type

In this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction...). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d -- that is a subshift whose set of forbidden patterns can be generated by a Turing machine -- can be obtained by applying dynamical operations on a subshift of finite type of dimension d + 1 -- a subshift that can be defined by a finite set of forbidden patterns. This result improves Hochman's [Hoc09].

math.DS↗
arXiv2015

$μ$-Limit Sets of Cellular Automata from a Computational Complexity Perspective

This paper concerns $μ$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $μ$-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, $μ$-limit sets can have a $Σ\_3^0$-hard language, second, they can contain only $α$-complex configurations, third, any non-trivial property concerning them is at least $Π\_3^0$-hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution.

cs.DM↗
arXiv2013

Speed of convergence for the realization of an effective subshift by a multidimensional SFT or Sofic

Realization of $d$-dimensional effective subshifts as projective sub-actions of $d+d'$-dimensional sofic subshifts for $d'\geq 1$ is now well know~\cite{Hochman-2009,Durand-Romashchenko-Shen-2010,Aubrun-Sablik-2010}. In this paper we are interested in the speed of convergence of this realization. That is to say given an effective subshift $Σ$ realized as projective sub-action of a sofic $\T$, we study the function which on input an integer $k$ returns the smallest width of the strip which verify the local rules of $\T$ necessary to obtain exclusively the language of size $k$ of $Σ$ in the central row of the strip. We study this topological conjugacy invariant for effective subshifts in order to exhibit algorithmic properties of these subshifts.

math.DS↗
arXiv2012

Local Rules for Computable Planar Tilings

Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question is to characterize, among a class of non-periodic tilings, the aperiodic ones. In this paper, we answer this question for the well-studied class of non-periodic tilings obtained by digitizing irrational vector spaces. Namely, we prove that such tilings are aperiodic if and only if the digitized vector spaces are computable.

cs.FL↗
arXiv2012

Entry times in automata with simple defect dynamics

In this paper, we consider a simple cellular automaton with two particles of different speeds that annihilate on contact. Following a previous work by K\r urka et al., we study the asymptotic distribution, starting from a random configuration, of the waiting time before a particle crosses the central column after time n. Drawing a parallel between the behaviour of this automata on a random initial configuration and a certain random walk, we approximate this walk using a Brownian motion, and we obtain explicit results for a wide class of initial measures and other automata with similar dynamics.

cs.FL↗
arXiv2010

Construction of $μ$-Limit Sets

The $μ$-limit set of a cellular automaton is a subshift whose forbidden patterns are exactly those, whose probabilities tend to zero as time tends to in- finity. In this article, for a given subshift in a large class of subshifts, we propose the construction of a cellular automaton which realizes this subshift as $μ$-limit set where $μ$ is the uniform Bernoulli measure.

cs.DM↗
arXiv2010

Directional Dynamics along Arbitrary Curves in Cellular Automata

This paper studies directional dynamics in cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behaviour of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviours inherited from topological dynamics (equicontinuity, sensitivity, expansivity) are thus considered along arbitrary curves in space-time. The main contributions of the paper concern equicontinuous dynamics which can be connected to the notion of consequences of a word. We show that there is a cellular automaton with an equicontinuous dynamics along a parabola, but which is sensitive along any linear direction. We also show that real numbers that occur as the slope of a limit linear direction with equicontinuous dynamics in some cellular automaton are exactly the computably enumerable numbers.

cs.DM↗
arXiv2009

Topological Dynamics of Cellular Automata: Dimension Matters

Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.

cs.DM↗
arXiv2009

An Order on Sets of Tilings Corresponding to an Order on Languages

Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view, leading to the notion of subshift. In this article we establish a correspondence between an order on subshifts based on dynamical transformations on them and an order on languages of forbidden patterns based on computability properties.

cs.DM↗
arXiv2008

Topological Dynamics of 2D Cellular Automata

Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on 2D CA and aims at showing that the situation is different and more complex. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants and the existence of CA having only non-recursive equicontinuous points. They all show a difference between the 1D and the 2D case. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case.

cs.DM↗