57 research outputs found
Hierarchy and Expansiveness in Two-Dimensional Subshifts of Finite Type
Subshifts are sets of conïŹgurations over an inïŹnite grid deïŹned by a set of forbidden patterns. In this thesis, we study two-dimensional subshifts ofïŹnite type (2D SFTs), where the underlying grid is Z2 and the set of for-bidden patterns is ïŹnite. 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 ïŹxed-point method for constructing 2D SFTs which has been previously used by GaÂŽcs, 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
An Algebraic Approach to Nivat's Conjecture
This thesis introduces a new, algebraic method to study multidimensional configurations, also sometimes called words, which have low pattern complexity. This is the setting of several open problems, most notably Nivatâs conjecture, which is a generalization of Morse-Hedlund theorem to two dimensions, and the periodic tiling problem by Lagarias and Wang.
We represent configurations as formal power series over d variables where d is the dimension. This allows us to study the ideal of polynomial annihilators of the series. In the two-dimensional case we give a detailed description of the ideal, which can be applied to obtain partial results on the aforementioned combinatorial problems.
In particular, we show that configurations of low complexity can be decomposed into sums of periodic configurations. In the two-dimensional case, one such decomposition can be described in terms of the annihilator ideal. We apply this knowledge to obtain the main result of this thesis â an asymptotic version of Nivatâs conjecture. We also prove Nivatâs conjecture for configurations which are sums of two periodic ones, and as a corollary reprove the main result of Cyr and Kra from [CK15].Algebrallinen lĂ€hestymistapa Nivatân konjektuuriin
TĂ€ssĂ€ vĂ€itöskirjassa esitetÀÀn uusi, algebrallinen lĂ€hestymistapa moniulotteisiin,matalan kompleksisuuden konfiguraatioihin. NĂ€istĂ€ konfiguraatioista, joita moniulotteisiksi sanoiksikin kutsutaan, on esitetty useita avoimia ongelmia. TĂ€rkeimpinĂ€ nĂ€istĂ€ ovat Nivatân konjektuuri, joka on Morsen-Hedlundin lauseen kaksiulotteinen yleistys, sekĂ€ Lagariaksen ja Wangin jaksollinen tiilitysongelma.
VÀitöskirjan lÀhestymistavassa d-ulotteiset konfiguraatiot esitetÀÀn d:n muuttujan formaaleina potenssisarjoina. TÀmÀ mahdollistaa konfiguraation polynomiannihilaattoreiden ihanteen tutkimisen. VÀitöskirjassa selvitetÀÀn kaksiulotteisessa tapauksessa ihanteen rakenne tarkasti. TÀtÀ hyödyntÀmÀllÀ saadaan uusia, osittaisia tuloksia koskien edellÀ mainittuja kombinatorisia ongelmia.
Tarkemmin sanottuna vĂ€itöskirjassa todistetaan, ettĂ€ matalan kompleksisuuden konfiguraatiot voidaan hajottaa jaksollisten konfiguraatioiden summaksi. Kaksiulotteisessa tapauksessa erĂ€s tĂ€llainen hajotelma saadaan annihilaattori-ihanteesta. TĂ€mĂ€n avulla todistetaan asymptoottinen versio Nivatân konjektuurista. LisĂ€ksi osoitetaan Nivatân konjektuuri oikeaksi konfiguraatioille, jotka ovat kahden jaksollisen konfiguraation summia, ja tĂ€mĂ€n seurauksena saadaan uusi todistus Cyrin ja Kran artikkelin [CK15] pÀÀtulokselle
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
On some one-sided dynamics of cellular automata
A dynamical system consists of a space of all possible world states and a transformation of said space. Cellular automata are dynamical systems where the space is a set of one- or two-way infinite symbol sequences and the transformation is defined by a homogenous local rule. In the setting of cellular automata, the geometry of the underlying space allows one to define one-sided variants of some dynamical properties; this thesis considers some such one-sided dynamics of cellular automata.
One main topic are the dynamical concepts of expansivity and that of pseudo-orbit tracing property. Expansivity is a strong form of sensitivity to the initial conditions while pseudo-orbit tracing property is a type of approximability. For cellular automata we define one-sided variants of both of these concepts. We give some examples of cellular automata with these properties and prove, for example, that right-expansive cellular automata are chain-mixing. We also show that left-sided pseudo-orbit tracing property together with right-sided expansivity imply that a cellular automaton has the pseudo-orbit tracing property.
Another main topic is conjugacy. Two dynamical systems are conjugate if, in a dynamical sense, they are the same system. We show that for one-sided cellular automata conjugacy is undecidable. In fact the result is stronger and shows that the relations of being a factor or a susbsystem are undecidable, too
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