Limit measures for affine cellular automata

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385702000548How to cite this article: MARCUS PIVATO and REEM YASSAWI (2002). Limit measures for afne cellular automata.Abstract. Let M be a monoid (e.g. N, Z, or M D ), and A an abelian group. A M is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A M −→ A M that commutes with all shift maps. Let µ be a (possibly non-stationary) probability measure on A M ; we develop sufficient conditions on µ and F so that the sequence {F N µ} ∞ N=1 weak* converges to the Haar measure on A M in density (and thus, in Cesàro average as well). As an application, we show that, if A = Z /p (p prime), F is any 'non-trivial' LCA on A (Z D ) , and µ belongs to a broad class of measures (including most Bernoulli measures (for D ≥ 1) and 'fully supported' N-step Markov measures (when D = 1)), then F N µ weak* converges to the Haar measure in density.Card S(m j ) + log p (J ) + 2. stands for 'gap', and is the size of the gaps we will require.

Asymptotic randomization of sofic shifts by linear cellular automata

“…We say that Φ is dispersive if, for any S > 0, and any character χ ∈ A M , there is a subset J ⊂ N of density 1 such that lim Proof. See the proof of Theorem 12 in [PY02]. ✷…”

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

“…Lind's point of view is that of harmonic analysis: the cellular automaton is viewed as an endomorphism of the product group and the uniform Bernoulli measure is its Haar measure. This approach was followed in Pivato andYassawir (2001 and2002), where the authors proved the same kind of results for the class of probability measures they called harmonically mixing, and the family of additive cellular automata verifying a diffusive condition. The previous results motivated us to study those sensitive cellular automata which can be decomposed as the product of an additive cellular automaton with one having an equi-continuous direction (Host et al, 2002).…”

“…The behavior of such a sequence has been studied in particular contexts in recent years: (Lind, 1984;Mass and Martínez, 1998;Ferrari et al, 2000;Pivato andYassawir, 2001, 2002). One of the primary objectives of this study is to link the limit behavior of this sequence with probability measures of maximal entropy, in other words, the way in which the CA goes to a maximal excited state in equilibrium.…”

Probability and algorithmics: a focus on some recent developments

“…In this case, a randomisation phenomenon appears: as soon as the initial measure is in a large class which contains Markov measures, the Cesàro mean sequence converges to the uniform Bernoulli measure [18,36,37,41]. In other words, V 1 pF, µq " tλ A Z u.…”