scholarly journals Conservation Laws and Invariant Measures in Surjective Cellular Automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Jarkko Kari ◽  
Siamak Taati
Keyword(s):  
Cellular Automata ◽  
Conservation Laws ◽  
Arbitrary Dimension ◽  
Close Link ◽  
One Dimensional ◽  
Full Support ◽  
Markov Measure ◽  
Bernoulli Measure ◽  
International Audience ◽  
Markov Measures

International audience We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.

scholarly journals The Measure-Theoretic Entropy and Topological Entropy of Actions overℤm

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Chih-Hung Chang ◽  
Yu-Wen Chen
Keyword(s):  
Invariant Measure ◽  
Cellular Automata ◽  
Topological Entropy ◽  
Finite Ring ◽  
One Dimensional ◽  
Maximal Measure ◽  
Bernoulli Measure ◽  
Markov Measures ◽  
Infinite Sequences

This paper studies the quantitative behavior of a class of one-dimensional cellular automata, named weakly permutive cellular automata, acting on the space of all doubly infinite sequences with values in a finite ringℤm,m≥2. We calculate the measure-theoretic entropy and the topological entropy of weakly permutive cellular automata with respect to any invariant measure on the spaceℤmℤ. As an application, it is shown that the uniform Bernoulli measure is the unique maximal measure for linear cellular automata among the Markov measures.

AN UPPER BOUND OF THE DIRECTIONAL ENTROPY WITH RESPECT TO THE MARKOV MEASURES

2012 ◽  
Vol 22 (11) ◽  
pp. 1250263 ◽  
Author(s):  
HASAN AKIN
Keyword(s):  
Upper Bound ◽  
Stochastic Matrix ◽  
Natural Extension ◽  
Short Paper ◽  
Probability Vector ◽  
Compact Metric Space ◽  
One Dimensional ◽  
Markov Measure ◽  
Shift Map ◽  
Markov Measures

In this short paper, without considering the natural extension we study the directional entropy of a Z2-action Φ generated by an invertible one-dimensional linear cellular automaton [Formula: see text] and [Formula: see text], over the ring Zpk(with p a prime number and k ≥ 2), where gcd (p, λr) = 1 and p ∣ λifor all i ≠ r, and the shift map acting on the compact metric space [Formula: see text]. Without loss of generality, we consider k = 2. We prove that the directional entropy hv(Φ)(v = (s, q) ∈ R) of a Z2-action with respect to a Markov measure μπPover space [Formula: see text] defined by a stochastic matrix P = (aij) and a probability vector π = {π0, π1, …, πp2-1} is bounded above by [Formula: see text].

scholarly journals Orbits of the Bernoulli measure in single-transition asynchronous cellular automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Henryk Fukś ◽  
Andrew Skelton
Keyword(s):  
Cellular Automata ◽  
Initial Density ◽  
Response Curves ◽  
Asynchronous Cellular Automata ◽  
Final Density ◽  
Short Cylinder ◽  
Bernoulli Measure ◽  
Single Transition ◽  
International Audience ◽  
Transition Rules

International audience We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.

scholarly journals Product decomposition for surjective 2-block NCCA

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Felipe García-Ramos
Keyword(s):  
Cellular Automata ◽  
Product Decomposition ◽  
One Dimensional ◽  
International Audience

International audience In this paper we define products of one-dimensional Number Conserving Cellular Automata (NCCA) and show that surjective NCCA with 2 blocks (i.e radius 1/2) can always be represented as products of shifts and identites. In particular, this shows that surjective 2-block NCCA are injective.

One-Dimensional Cellular Automata, Conservation Laws and Partial Differential Equations

2007 ◽  
Vol 62 (10-11) ◽  
pp. 569-572
Author(s):  
Willi-Hans Steeb ◽  
Yorick Hardy
Keyword(s):  
Partial Differential Equations ◽  
Cellular Automata ◽  
Differential Equations ◽  
Conservation Laws ◽  
Partial Differential ◽  
One Dimensional ◽  
Reversible Cellular Automata

The existence of conservation laws (invariants) are discussed for various one-dimensional cellular automata. The cellular automata are derived from partial differential equations. Both irreversible and reversible cellular automata are investigated.

scholarly journals Conservation laws and integrability of a one-dimensional model of diffusing dimers

1997 ◽  
Vol 86 (5-6) ◽  
pp. 1237-1263 ◽  
Author(s):  
Gautam I. Menon ◽  
Mustansir Barma ◽  
Deepak Dhar
Keyword(s):  
Conservation Laws ◽  
Dimensional Model ◽  
One Dimensional

The dynamics of defect ensembles in one-dimensional cellular automata

1994 ◽  
Vol 76 (5-6) ◽  
pp. 1377-1398 ◽  
Author(s):  
Kari Eloranta
Keyword(s):  
Cellular Automata ◽  
One Dimensional
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