scholarly journals Glider automata on all transitive sofic shifts

2021 ◽  
pp. 1-29
Author(s):  
JOHAN KOPRA
Keyword(s):  
Automorphism Group ◽  
Cellular Automaton ◽  
Finite Collection ◽  
Finite Point ◽  
Element Subset ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Reversible Cellular Automaton

Abstract For any infinite transitive sofic shift X we construct a reversible cellular automaton (that is, an automorphism of the shift X) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible cellular automaton which is sensitive with respect to all directions. As another application we prove a finitary version of Ryan’s theorem: the automorphism group $\operatorname {\mathrm {Aut}}(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of S-gap shifts these results do not extend beyond the sofic case.

scholarly journals Glider automorphisms and a finitary Ryan’s theorem for transitive subshifts of finite type

2019 ◽  
Vol 19 (4) ◽  
pp. 773-786
Author(s):  
Johan Kopra
Keyword(s):  
Automorphism Group ◽  
Finite Type ◽  
Finite Collection ◽  
Finite Point ◽  
Element Subset ◽  
Continuous Map ◽  
Full Shift ◽  
Subshifts Of Finite Type

Abstract For any mixing SFT X we construct a reversible shift-commuting continuous map (automorphism) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. As an application we prove a finitary Ryan’s theorem: the automorphism group $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) contains a two-element subset S whose centralizer consists only of shift maps. We also give an example which shows that a stronger finitary variant of Ryan’s theorem does not hold even for the binary full shift.

scholarly journals Normal amenable subgroups of the automorphism group of sofic shifts

2020 ◽  
pp. 1-14
Author(s):  
KITTY YANG
Keyword(s):  
Automorphism Group ◽  
Finite Type ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Amenable Subgroup

Let $(X,\unicode[STIX]{x1D70E})$ be a transitive sofic shift and let $\operatorname{Aut}(X)$ denote its automorphism group. We generalize a result of Frisch, Schlank, and Tamuz to show that any normal amenable subgroup of $\operatorname{Aut}(X)$ must be contained in the subgroup generated by the shift. We also show that the result does not extend to higher dimensions by giving an example of a two-dimensional mixing shift of finite type due to Hochman whose automorphism group is amenable and not generated by the shift maps.

scholarly journals Limit sets of stable cellular automata

2013 ◽  
Vol 35 (3) ◽  
pp. 673-690 ◽  
Author(s):  
ALEXIS BALLIER
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Point Of View ◽  
Limit Set ◽  
Limit Sets ◽  
Sofic Shift ◽  
Almost Everywhere ◽  
Sofic Shifts ◽  
Full Shift ◽  
Factor Map

AbstractWe study limit sets of stable cellular automata from a symbolic dynamics point of view, where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps, and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto the minimal right-resolving cover of the image. To translate this into terms of cellular automata, a sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it is the factor of a full shift and there exists a right-closing almost-everywhere factor map from the sofic shift onto its minimal right-resolving cover. A sofic shift can be the limit set of a stable cellular automaton reaching its limit set with a right-continuing almost-everywhere factor map if and only if it is the factor of a full shift and there exists a factor map from the sofic shift onto its minimal right-resolving cover. Finally, as a consequence of the previous results, we provide a characterization of the almost of finite type shifts (AFT) in terms of a property of steady maps that have them as range.

scholarly journals A note on subgroups of automorphism groups of full shifts

2016 ◽  
Vol 38 (4) ◽  
pp. 1588-1600 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Endomorphism Monoid ◽  
Direct Products ◽  
Free Products ◽  
Group Case ◽  
Sofic Shift ◽  
Full Shift

We discuss the set of subgroups of the automorphism group of a full shift and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.

scholarly journals Two-species hardcore reversible cellular automaton: matrix ansatz for dynamics and nonequilibrium stationary state

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Marko Medenjak ◽  
Vladislav Popkov ◽  
Tomaz Prosen ◽  
Eric Ragoucy ◽  
Matthieu Vanicat
Keyword(s):  
Cellular Automaton ◽  
Time Evolution ◽  
Stationary State ◽  
Finite Lattice ◽  
Matrix Product ◽  
Density Profiles ◽  
Reversible Cellular Automaton ◽  
Product Expression ◽  
Particle Current ◽  
Exact Matrix

In this paper we study the statistical properties of a reversible cellular automaton in two out-of-equilibrium settings. In the first part we consider two instances of the initial value problem, corresponding to the inhomogeneous quench and the local quench. Our main result is an exact matrix product expression of the time evolution of the probability distribution, which we use to determine the time evolution of the density profiles analytically. In the second part we study the model on a finite lattice coupled with stochastic boundaries. Once again we derive an exact matrix product expression of the stationary distribution, as well as the particle current and density profiles in the stationary state. The exact expressions reveal the existence of different phases with either ballistic or diffusive transport depending on the boundary parameters.

Complete set of Genera of Compact Riemann Surfaces with reference to the point Group of Sulphur (S8) Molecule

2020 ◽  
Vol 32 (7) ◽  
pp. 88-92
Author(s):  
RAFIQUL ISLAM ◽  
◽  
CHANDRA CHUTIA ◽  
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Point Group ◽  
Finite Point ◽  
Group Of Automorphisms ◽  
Complete Set ◽  
Compact Riemann Surfaces ◽  
Group Of Symmetries

In this paper we consider the group of symmetries of the Sulphur molecule (S8 ) which is a finite point group of order 16 denote by D16 generated by two elements having the presentation { u\upsilon/u2= \upsilon8 = (u\upsilon)2 = 1} and find the complete set of genera (g ≥ 2) of Compact Riemann surfaces on which D16 acts as a group of automorphisms as follows: D16 the group of symmetries of the sulphur (S8) molecule of order 16 acts as an automorphism group of a compact Riemann surfaces of genus g ≥ 2 if and only if there are integers \lambda and \mu such that \lambda \leq 1 and \mu \geq 1 and g=\lambda +8\mu (\geq2) , \mu\geq |\lambda|

scholarly journals Representing Reversible Cellular Automata with Reversible Block Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Jérôme Durand-Lose
Keyword(s):  
System Theory ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Linear Time ◽  
Fixed Size ◽  
Mathematical System ◽  
Reversible Cellular Automaton ◽  
International Audience ◽  
Infinite Lattices ◽  
Reversible Cellular Automata

International audience Cellular automata are mappings over infinite lattices such that each cell is updated according tothe states around it and a unique local function.Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks.We prove that any d-dimensional reversible cellular automaton can be exp ressed as thecomposition of d+1 block permutations.We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus <i>(Physica D 45)</i> improved by Kari in 1996 <i>(Mathematical System Theory 29)</i>.

scholarly journals -ALGEBRAS ASSOCIATED WITH LAMBDA-SYNCHRONIZING SUBSHIFTS AND FLOW EQUIVALENCE

2013 ◽  
Vol 95 (2) ◽  
pp. 241-265 ◽  
Author(s):  
KENGO MATSUMOTO
Keyword(s):  
Algebraic Structure ◽  
Isomorphism Class ◽  
Cartan Subalgebra ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Flow Equivalence ◽  
Graph System

AbstractThe class of $\lambda $-synchronizing subshifts generalizes the class of irreducible sofic shifts. A $\lambda $-synchronizing subshift can be presented by a certain $\lambda $-graph system, called the $\lambda $-synchronizing $\lambda $-graph system. The $\lambda $-synchronizing $\lambda $-graph system of a $\lambda $-synchronizing subshift can be regarded as an analogue of the Fischer cover of an irreducible sofic shift. We will study algebraic structure of the ${C}^{\ast } $-algebra associated with a $\lambda $-synchronizing $\lambda $-graph system and prove that the stable isomorphism class of the ${C}^{\ast } $-algebra with its Cartan subalgebra is invariant under flow equivalence of $\lambda $-synchronizing subshifts.

Core dimension group constraints for factors of sofic shifts

1993 ◽  
Vol 13 (1) ◽  
pp. 213-224 ◽  
Author(s):  
Paul Trow ◽  
Susan Williams
Keyword(s):  
Characteristic Polynomial ◽  
The Core ◽  
Sofic Shift ◽  
Dimension Group ◽  
Core Matrix ◽  
Sofic Shifts ◽  
Factor Maps

AbstractWe give constraints on the existence of factor maps between sofic shifts. These constraints yield examples of sofic shifts of entropy lognwhich do not factor onto the fulln-shift. We also show that any prime which divides the degree of an endomorphism of a sofic shift must divide the non-leading coefficients of the characteristic polynomial of the core matrix of the shift.

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