Evolution of probability measures by cellular automata on algebraic topological Markov chains

Maass, Alejandro; Martı́nez, Servet

doi:10.4067/s0716-97602003000100010

Cited by 5 publications

(6 citation statements)

References 6 publications

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“…(2.1), then the finite formal power series (fps for brevity) F associated with f is [3], [10] for details). The technique of fps is well known for the study of these problems.…”

Section: Preliminariesmentioning

confidence: 99%

“…. , λ r ∈ Z m (see [3], [10] for details). The technique of fps is well known for the study of these problems.…”

Section: Preliminariesmentioning

confidence: 99%

“…Hedlund's paper started investigation of current problems in symbolic dynamics. The study of such dynamics called CA from the point of view of the ergodic theory has received remarkable attention in the last few years ( [1], [6], [10]), because CA have been widely investigated in a number of disciplines (e.g., mathematics, physics, computer sciences, etc.). In [11], Shereshevsky has defined nth iteration of a permutative CA and shown that if the local rule f is right (left) permutative, then its nth iteration also is right (left) permutative.…”

Section: Introductionmentioning

confidence: 99%

“…In [10], Mass and Martinez have studied the dynamics of Markov measures by a particular linear cellular automata (LCA). They have reviewed some results on the evolution of probability measures under CA acting on a fullshift.…”

Section: Introductionmentioning

confidence: 99%

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The Entropy of Linear Cellular Automata with Respect to Any Bernoulli Measure

Akın¹

2009

ComplexSystems

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In this paper we study the measure-theoretical entropy of the one-dimensional linear cellular automata (CA hereafter) T f [−l,r] , generated by local rule f (x −l , . . . , xr) = r i=−l λ i x i (mod m), where l and r are positive integers, acting on the space of all doubly infinite sequences with values in a finite ring Zm, m ≥ 2, with respect to a Markov measure. We prove that if the local rule f is bipermutative, then the measure-theoretical entropy of linear CA T f [−l,r] with respect to a Markov measure µ πP is hµ πP (T f [−l,r] ) = −(l + r) m−1 i,j=0 p i p ij log p ij .

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“…(2.1), then the finite formal power series (fps for brevity) F associated with f is [3], [10] for details). The technique of fps is well known for the study of these problems.…”

Section: Preliminariesmentioning

confidence: 99%

“…. , λ r ∈ Z m (see [3], [10] for details). The technique of fps is well known for the study of these problems.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Entropy of Linear Cellular Automata with Respect to Any Bernoulli Measure

Akın¹

2009

ComplexSystems

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“…The general question of finding the iterates of the Bernoulli measure under a given cellular automaton (CA) has been subject of many recent studies, including, among others, [2,3,4,5,6,7,8] and [9]. A more specific question of this type is sometimes called the density response problem: If the probability of occurence of a certain state in the initial configuration drawn from a Bernoulli distribution is given, what is the probability of occurence of this state after n iterations of the CA rule?…”

mentioning

confidence: 99%

An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

Fukś¹,

Midgley-Volpato²

2016

Acta Phys. Pol. B Proc. Suppl.

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In a recent paper [1] we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also calculated densities of 0, 1 and 2 after n iterations of this rule, using finite state machines to conjecture patterns present in preimage sets. Here we rederive the same formulae in a rigorous way, without resorting to any semiempirical methods. This is done by analyzing the behaviour of continuous clusters of symbols and by considering their interactions.(1)

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Probabilistic initial value problem for cellular automaton rule 172

Fukś¹

2010

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience We present a method of solving of the probabilistic initial value problem for cellular automata (CA) using CA rule 172 as an example. For a disordered initial condition on an infinite lattice, we derive exact expressions for the density of ones at arbitrary time step. In order to do this, we analyze topological structure of preimage trees of finite strings of length 3. Level sets of these trees can be enumerated directly using classical combinatorial methods, yielding expressions for the number of $n$-step preimages of all strings of length 3, and, subsequently, probabilities of occurrence of these strings in a configuration obtained from the initial one after $n$ iterations of rule 172. The density of ones can be expressed in terms of Fibonacci numbers, while expressions for probabilities of other strings involve Lucas numbers. Applicability of this method to other CA rules is briefly discussed.

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Evolution of probability measures by cellular automata on algebraic topological Markov chains

Cited by 5 publications

References 6 publications

The Entropy of Linear Cellular Automata with Respect to Any Bernoulli Measure

The Entropy of Linear Cellular Automata with Respect to Any Bernoulli Measure

An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

Probabilistic initial value problem for cellular automaton rule 172

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