On the Interplay of Direct Topological Factorizations and Cellular Automata Dynamics on Beta-Shifts

2021 ◽  
pp. 1-21
Author(s):  
Johan Kopra
Keyword(s):  
Cellular Automata ◽  
Finite Type ◽  
Pisot Number ◽  
Simple Criterion ◽  
Subshift Of Finite Type

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts [Formula: see text] and its relation to direct topological factorizations. We show that any reversible CA [Formula: see text] has an almost equicontinuous direction whenever [Formula: see text] is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations [Formula: see text] of two nontrivial subshifts [Formula: see text] and [Formula: see text]. We also give a simple criterion to determine whether [Formula: see text] is conjugate to [Formula: see text] for a given integer [Formula: see text] and a given real [Formula: see text] when [Formula: see text] is a subshift of finite type. When [Formula: see text] is strictly sofic, we show that such a conjugacy is not possible at least when [Formula: see text] is a quadratic Pisot number of degree [Formula: see text]. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.

Languages, equicontinuity and attractors in cellular automata

1997 ◽  
Vol 17 (2) ◽  
pp. 417-433 ◽  
Author(s):  
PETR KŮRKA
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
State Space ◽  
Finite Type ◽  
Measure Zero ◽  
Initial Conditions ◽  
The State ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
The Third

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

scholarly journals Embedding Bratteli–Vershik systems in cellular automata

2009 ◽  
Vol 30 (5) ◽  
pp. 1561-1572 ◽  
Author(s):  
MARCUS PIVATO ◽  
REEM YASSAWI
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Finite Type ◽  
Two Dimensional ◽  
Time Diagram ◽  
Constant Length ◽  
Subshift Of Finite Type ◽  
One Dimensional ◽  
Toeplitz Systems

AbstractMany dynamical systems can be naturally represented as Bratteli–Vershik (or adic) systems, which provide an appealing combinatorial description of their dynamics. If an adic system X is linearly recurrent, then we show how to represent X using a two-dimensional subshift of finite type Y; each ‘row’ in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the ‘successor’ map of X. Any Y-admissible configuration can then be recoded as the space-time diagram of a one-dimensional cellular automaton Φ; in this way X is embedded in Φ (i.e. X is conjugate to a subsystem of Φ). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

scholarly journals Periodic Intermediate β-Expansions of Pisot Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 903
Author(s):  
Blaine Quackenbush ◽  
Tony Samuel ◽  
Matt West
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Markov Property ◽  
Periodic Points ◽  
Pisot Number ◽  
Subshift Of Finite Type ◽  
Pisot Numbers ◽  
Type Property ◽  
Perron Number ◽  
Dense Set

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are β -shifts, namely transformations of the form T β , α : x ↦ β x + α mod 1 acting on [ − α / ( β − 1 ) , ( 1 − α ) / ( β − 1 ) ] , where ( β , α ) ∈ Δ is fixed and where Δ ≔ { ( β , α ) ∈ R 2 : β ∈ ( 1 , 2 ) and 0 ≤ α ≤ 2 − β } . Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045–2055, 2019), that the set of ( β , α ) such that T β , α has the subshift of finite type property is dense in the parameter space Δ . Here, they proposed the following question. Given a fixed β ∈ ( 1 , 2 ) which is the n-th root of a Perron number, does there exists a dense set of α in the fiber { β } × ( 0 , 2 − β ) , so that T β , α has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269–278, 1980) from the case when α = 0 to the case when α ∈ ( 0 , 2 − β ) . That is, we examine the structure of the set of eventually periodic points of T β , α when β is a Pisot number and when β is the n-th root of a Pisot number.

scholarly journals On classification of some surfaces of revolution of finite type

1993 ◽  
Vol 17 (1) ◽  
pp. 287-298 ◽  
Author(s):  
Bang-yen Chen ◽  
Susumu Ishikawa
Keyword(s):  
Finite Type ◽  
Surfaces Of Revolution

scholarly journals EVOLVING LOCALIZATIONS IN REACTION-DIFFUSION CELLULAR AUTOMATA

2008 ◽  
Vol 19 (04) ◽  
pp. 557-567 ◽  
Author(s):  
ANDREW ADAMATZKY ◽  
LARRY BULL ◽  
PIERRE COLLET ◽  
EMMANUEL SAPIN
Keyword(s):  
Cellular Automata ◽  
Chemical Species ◽  
Reaction Diffusion ◽  
Discrete Systems ◽  
Chemical Systems ◽  
Transition Functions ◽  
Phenomenological Studies ◽  
Spatially Extended ◽  
Limited Diffusion

We consider hexagonal cellular automata with immediate cell neighbourhood and three cell-states. Every cell calculates its next state depending on the integral representation of states in its neighbourhood, i.e., how many neighbours are in each one state. We employ evolutionary algorithms to breed local transition functions that support mobile localizations (gliders), and characterize sets of the functions selected in terms of quasi-chemical systems. Analysis of the set of functions evolved allows to speculate that mobile localizations are likely to emerge in the quasi-chemical systems with limited diffusion of one reagent, a small number of molecules are required for amplification of travelling localizations, and reactions leading to stationary localizations involve relatively equal amount of quasi-chemical species. Techniques developed can be applied in cascading signals in nature-inspired spatially extended computing devices, and phenomenological studies and classification of non-linear discrete systems.

Automatic Texture Based Classification of the Dynamics of One-Dimensional Binary Cellular Automata

2019 ◽  
Vol 8 (4) ◽  
pp. 41-61
Author(s):  
Marcelo Arbori Nogueira ◽  
Pedro Paulo Balbi de Oliveira
Keyword(s):  
Cellular Automata ◽  
Dynamic Behavior ◽  
Temporal Evolution ◽  
Computational Cost ◽  
Great Variability ◽  
Texture Descriptor ◽  
One Dimensional ◽  
Binary Images ◽  
Elementary Cellular Automata

Cellular automata present great variability in their temporal evolutions due to the number of rules and initial configurations. The possibility of automatically classifying its dynamic behavior would be of great value when studying properties of its dynamics. By counting on elementary cellular automata, and considering its temporal evolution as binary images, the authors created a texture descriptor of the images - based on the neighborhood configurations of the cells in temporal evolutions - so that it could be associated to each dynamic behavior class, following the scheme of Wolfram's classic classification. It was then possible to predict the class of rules of a temporal evolution of an elementary rule in a more effective way than others in the literature in terms of precision and computational cost. By applying the classifier to the larger neighborhood space containing 4 cells, accuracy decreased to just over 70%. However, the classifier is still able to provide some information about the dynamics of an unknown larger space with reduced computational cost.

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