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 characterization of ω-limit sets in shift spaces

2009 ◽  
Vol 30 (1) ◽  
pp. 21-31 ◽  
Author(s):  
ANDREW BARWELL ◽  
CHRIS GOOD ◽  
ROBIN KNIGHT ◽  
BRIAN E. RAINES
Keyword(s):  
Finite Type ◽  
Limit Set ◽  
Invariant Subset ◽  
Limit Sets ◽  
Tent Map ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Shift Space ◽  
Full Shift

AbstractA set Λ is internally chain transitive if for any x,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point z∈X with ω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is the ω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space Z𝒢 (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the ω-limit set of any point in Z𝒢.

On the sofic limit sets of cellular automata

1995 ◽  
Vol 15 (4) ◽  
pp. 663-684 ◽  
Author(s):  
Alejandro Maass
Keyword(s):  
Fixed Point ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Finite Type ◽  
Finite Time ◽  
Limit Set ◽  
Limit Sets ◽  
One Dimensional ◽  
Sofic System

AbstractIt is not known in general whether any mixing sofic system is the limit set of some one-dimensional cellular automaton. We address two aspects of this question. We prove first that any mixing almost of finite type (AFT) sofic system with a receptive fixed point is the limit set of a cellular automaton, under which it is attained in finite time. The AFT condition is not necessary: we also give examples of non-AFT sofic systems having the same properties. Finally, we show that near Markov sofic systems (a subclass of AFT sofic systems) cannot be obtained as limit sets of cellular automata at infinity.

Models and Paradigms of Cellular Automata With a Variable Set of Active Cells

2021 ◽  
pp. 75-98
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Variable Number ◽  
Point Of View ◽  
Modes Of Operation ◽  
Von Neumann ◽  
Asynchronous Cellular Automata ◽  
Cad Models

The chapter describes the functioning model of an asynchronous cellular automaton with a variable number of active cells. The rules for the formation of active cells with new active states are considered. Codes of active states for the von Neumann neighborhood are presented, and a technique for coding active states for other forms of neighborhoods is described. Several modes of operation of asynchronous cellular automata from the point of view of the influence of active cells are considered. The mode of coincidence of active cells and the mode of influence of neighboring active cells are considered, and the mode of influence of active cells of the surroundings is briefly considered. Algorithms of cell operation for all modes of the cellular automata are presented. Functional structures of cells and their CAD models are constructed.

scholarly journals EVENTUALLY NUMBER-CONSERVING CELLULAR AUTOMATA

2007 ◽  
Vol 18 (01) ◽  
pp. 35-42 ◽  
Author(s):  
NINO BOCCARA
Keyword(s):  
Cellular Automaton ◽  
Active Sites ◽  
Sufficient Conditions ◽  
Limit Set ◽  
Constant Number ◽  
Interacting Particles ◽  
Limit Sets ◽  
One Dimensional ◽  
Preliminary Study ◽  
Conservation Rule

Although it is undecidable whether a one-dimensional cellular automaton obeys a given conservation law over its limit set, it is however possible to obtain sufficient conditions to be satisfied by a one-dimensional cellular automaton to be eventually number-conserving. We present a preliminary study of two-input one-dimensional cellular automaton rules called eventually number-conserving cellular automaton rules whose limit sets, reached after a number of time steps of the order of the cellular automaton size, consist of states having a constant number of active sites. In particular, we show how to find rules having given limit sets satisfying a conservation rule. Viewed as models of systems of interacting particles, these rules obey a kind of Darwinian principle by either annihilating unnecessary particles or creating necessary ones.

Lattice invariants for sofic shifts

1991 ◽  
Vol 11 (4) ◽  
pp. 787-801 ◽  
Author(s):  
Susan Williams
Keyword(s):  
Finite Type ◽  
Necessary Conditions ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Dimension Group ◽  
Sofic Shifts ◽  
Factor Map ◽  
Factor Maps

AbstractTo a factor map φ from an irreducible shift of finite type ΣAto a sofic shiftS, we associate a subgroup of the dimension group (GA, Â) which is an invariant of eventual conjugacy for φ. This invariant yields new necessary conditions for the existence of factor maps between equal entropy sofic shifts.

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 Porosity in Conformal Dynamical Systems

2021 ◽  
pp. 1-69
Author(s):  
VASILEIOS CHOUSIONIS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Continued Fractions ◽  
Julia Set ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Markov Systems ◽  
Other Hand ◽  
Almost Everywhere ◽  
Complex Continued Fractions ◽  
Dense Set

Abstract In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that their limit sets are porous in large (in the sense of category and dimension) subsets. We also provide natural geometric and dynamic conditions under which the limit set of a GDMS is upper porous or mean porous. On the other hand, we prove that if the limit set of a GDMS is not porous, then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction. We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterisation of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are uniformly upper porous and mean porous almost everywhere. We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

scholarly journals Mean equicontinuity and mean sensitivity on cellular automata

2020 ◽  
pp. 1-18
Author(s):  
LUGUIS DE LOS SANTOS BAÑOS ◽  
FELIPE GARCíA-RAMOS
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
The Other ◽  
Other Hand ◽  
Full Shift ◽  
Mean Sensitivity

Abstract We show that a cellular automaton (or shift-endomorphism) on a transitive subshift is either almost equicontinuous or sensitive. On the other hand, we construct a cellular automaton on a full shift (hence a transitive subshift) that is neither almost mean equicontinuous nor mean sensitive.

EXTENSIONS IN REVERSIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA ARE EQUIVALENT WITH THE FULL SHIFT

2003 ◽  
Vol 14 (08) ◽  
pp. 1143-1160 ◽  
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
MANUEL GONZÁLEZ HERNÁANDEZ ◽  
GENARO JUÁREZ MARTÍNEZ ◽  
SERGIO V. CHAPA VERGARA
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Special Kind ◽  
Discrete Dynamical Systems ◽  
Graph Representation ◽  
Global Behavior ◽  
One Dimensional ◽  
Full Shift ◽  
Graph Presentation ◽  
The One

Cellular automata are discrete dynamical systems based on simple local interactions among its components, but sometimes they are able to yield quite a complex global behavior. A special kind of cellular automaton is the one where the global behavior is invertible, this type of cellular automaton is called reversible. In this paper we expose the graph representation provided by de Bruijn diagrams of reversible one-dimensional cellular automata and we define the distinct types of paths between self-loops in such diagrams. With this, we establish the way in which a reversible one-dimensional cellular automaton generates sequences composed by subsequences produced by the undefined repetition of a single state. Using this graph presentation, we define Welch diagrams which will be useful for proving that all the extensions of the ancestors in reversible one-dimensional cellular automata are equivalent to the full shift. In this way an important result of this paper is that we understand and classify the behavior of a reversible automaton analyzing the extensions of the ancestors of a given sequence by means of symbolic dynamics tools. A final example illustrates the results exposed in the paper.

Simulating Self-Regeneration and Self-Replication Processes Using Movable Cellular Automata with a Mutual Equilibrium Neighborhood

2020 ◽  
Vol 29 (4) ◽  
pp. 741-757
Author(s):  
Kateryna Hazdiuk ◽  
◽  
Volodymyr Zhikharevich ◽  
Serhiy Ostapov ◽  
◽  
...  
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Three Dimensional ◽  
Computer Models ◽  
The Self ◽  
Model Construction ◽  
Natural Phenomena ◽  
Different Types ◽  
Movable Cellular Automata ◽  
Self Replication

This paper deals with the issue of model construction of the self-regeneration and self-replication processes using movable cellular automata (MCAs). The rules of cellular automaton (CA) interactions are found according to the concept of equilibrium neighborhood. The method is implemented by establishing these rules between different types of cellular automata (CAs). Several models for two- and three-dimensional cases are described, which depict both stable and unstable structures. As a result, computer models imitating such natural phenomena as self-replication and self-regeneration are obtained and graphically presented.

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