Diameter mean equicontinuity and cellular automata (original) (raw)
Local non-periodic order and diam-mean equicontinuity on cellular automata
2021
Diam-mean equicontinuity is a dynamical property that has been of use in the study of non-periodic order. Using some type of "local" skew product between a shift and an odometer looking cellular automaton (CA) we will show there exists an almost diam-mean equicontinuous CA that is not almost equicontinuous, and hence not almost locally periodic.
Mean equicontinuity and mean sensitivity on cellular automata
2020
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.
Strictly periodic points and periodic factors of cellular automata
arXiv (Cornell University), 2023
We show that the set of strictly temporally periodic points of cellular automata with almost equicontinuous points is dense in the topological support of the measure. This extends a result of Lena, Margara and Dennunzio about the density of the set of strictly temporally periodic of cellular automata with equicontinuous points.
Strictly Temporally Periodic Points in Cellular Automata
Electronic Proceedings in Theoretical Computer Science, 2012
We study the set of strictly temporally periodic points in surjective cellular automata, i.e., the set of those configurations which are temporally periodic for a given automaton but are not spatially periodic. This set turns out to be residual for equicontinuous surjective cellular automata, dense for almost equicontinuous surjective cellular automata, while it is empty for the positively expansive ones. In the class of additive cellular automata, the set of strictly temporally periodic points can be either dense or empty. The latter happens if and only if the cellular automaton is topologically transitive.
Lecture Notes in Computer Science, 2009
In this paper we begin the study the dynamical behavior of non-uniform cellular automata and compare it to the behavior of "classical" cellular automata. In particular we focus on surjectivity and equicontinuity.
On the quantitative behavior of the linear cellular automata
2013
In this paper, we study the quantitative behavior of one-dimensional linear cellular automata T f [−r,r] , defined by local rule f (x −r , . . . , xr) = r i=−r λ i x i (mod m), acting on the space of all doubly infinite sequences with values in a finite ring Zm, m ≥ 2. Once generalize the formulas given by Ban et al. [J. Cellular Automata 6 (2011) 385-397] for measure-theoretic entropy and topological pressure of one-dimensional cellular automata, we calculate the measure entropy and the topological pressure of the linear cellular automata with respect to the Bernoulli measure on the set Z Z m . Also, it is shown that the uniform Bernoulli measure is the unique equilibrium measure for linear cellular automata. We compare values of topological entropy and topological directional entropy by using the formula obtained by Akın [J. Computation and Appl. Math. 225 (2) (2009) 459-466]. The topological directional entropy is interpreted by means of figures. As an application, we demonstrate that the Hausdorff of the limit set of a linear cellular automaton is the unique root of Bowen's equation. Some open problems remain to be of interest.
Periodicity and Transitivity for Cellular Automata in Besicovitch Topologies
Lecture Notes in Computer Science, 2003
We study cellular automata (CA) behavior in Besicovitch topology. We solve an open problem about the existence of transitive CA. The proof of this result has some interest in its own since it is obtained by using Kolmogorov complexity. At our knowledge it if the first result on discrete dynamical systems obtained using Kolmogorov complexity. We also prove that every CA (in Besicovitch topology) either has a unique fixed point or a countable set of periodic points. This result underlines that CA have a great degree of stability and may be considered a further step towards the understanding of CA periodic behavior.
Stochastic Processes and their Applications
We provide an example of a discrete-time Markov process on the three-dimensional infinite integer lattice with -invariant Bernoulli-increments which has as local state space the cyclic group . We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere . The update mechanism we construct is exponentially well localized on the lattice.
Some results about the chaotic behavior of cellular automata
Theoretical Computer Science, 2005
We study the behavior of cellular automata (CA for short) in the Cantor, Besicovitch and Weyl topologies. We solve an open problem about the existence of transitive CA in the Besicovitch topology. The proof of this result has some interest of its own since it is obtained by using Kolmogorov complexity. To our knowledge it is the first result about discrete dynamical systems obtained using Kolmogorov complexity. We also prove that in the Besicovitch topology every CA has either a unique periodic point (thus a fixed point) or an uncountable set of periodic points. This result underlines the fact that CA have a great degree of stability; it may be considered a further step towards the understanding of CA periodic behavior.
Cellular automata with almost periodic initial conditions
Nonlinearity, 1995
Cellular automata are dynamical system on the compact metric space of subshifls. They leave many classes of sukhifts invariant. Here we show that c e l l u l~ automata leave 'circle subhhifls' invariant. These are the stricUy ergodic subshifls of (0. obtained by a circle sequence zn = I,(n .e), where J is a finite union of half-open intervals. For such initial conditions, the evolution of the whole infinite confi'&tion mi be computed by evolving the finitely many parameters defining the set J. Moreover, many macroswpic quantities can be computed exactly for the infinite system. We illusmate that in one dimension by rule 18 and in W O dimensions by the Game of Life. The ides, also apply to cellular automata acting on (0,. . ., Nl)zd. This we illWrate by the HPP model, a lattice gas aulomalon with N = 16.
Growth patterns of ordered cellular automata
Journal of Computer and System Sciences, 1981
Let F be the transition rule of an ordered cellular automaton. The author studies the geometry of a certain subset of Euclidean space associated to the sequence w, Fw, F*o,..., F"w,.... This subset yields information on the geometry and cardinality of Fpw for large p.
A non-ergodic probabilistic cellular automaton with a unique invariant measure
Stochastic Processes and their Applications, 2011
We exhibit a Probabilistic Cellular Automaton (PCA) on {0, 1} Z with a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA.
Conservation of some dynamical properties for operations on cellular automata
Theoretical Computer Science, 2009
We consider the family of all the Cellular Automata (CA) sharing the same local rule but having different memory. This family contains also all the CA with memory m ≤ 0 (one-sided CA) which can act both on A Z and on A N . We study several set theoretical and topological properties for these classes. In particular, we investigate if the properties of a given CA are preserved when considering the CA obtained by changing the memory of the original one (shifting operation). Furthermore, we focus our attention to the one-sided CA acting on A Z starting from the one-sided CA acting on A N and having the same local rule (lifting operation). As a particular consequence of these investigations, we prove that the long-standing conjecture [Surjectivity ⇒ Dense Periodic Orbits (DPO)] can be restated in several different (but equivalent) ways. Furthermore, we give some results on properties conserved under iteration of the CA global map.