ISSN:
1572-9613
Keywords:
Permutative cellular automata
;
mod 2 automaton
;
Cesàro means
;
Bernoulli measures
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract We study Cesàro means (time averages) of the evolution measures of the class of permutative cellular automata over {0, 1}ℕ defined by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabew% 9aQnaaBaaaleaacaWGcbaabeaakiaadIhacaGGPaWaaSbaaSqaaiaa% d6gaaeqaaOGaeyypa0JaamiEamaaBaaaleaacaWGUbGaaGjcVlaayI% W7cqGHRaWkcaaMi8UaaGjcVlaadkfaaeqaaOGaey4kaSIaeuiOda1a% a0baaSqaaiaadQgacaaMi8UaaGjcVlabg2da9iaayIW7caaMi8UaaG% imaaqaaiaadkfacaaMi8UaaGjcVlabgkHiTiaayIW7caaMi8UaaGym% aaaakiaayIW7caaMi8UaaGjcVlaayIW7caaMi8UaaGjcVlaacIcaca% aIXaGaey4kaSIaamOyamaaBaaaleaacaWGQbaabeaakiabgUcaRiaa% dIhadaWgaaWcbaGaamOBaiaayIW7caaMi8Uaey4kaSIaaGjcVlaayI% W7caWGQbaabeaakiaacMcaaaa!7530! $$(\varphi _B x)_n = x_{n{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} R} + \Pi _{j{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 0}^{R{\kern 1pt} {\kern 1pt} - {\kern 1pt} {\kern 1pt} 1} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1 + b_j + x_{n{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} j} )$$ where B=b 0 ⋯ b R-1is an aperiodic block in {0, 1} R and operations are taken mod 2. If the initial measure is Bernoulli, we prove that the limit of the Cesàro mean of the first column distribution exists. When R = 1 and B = 1, φ B is the mod 2 sum automaton. For this automaton we show that the limit is the (1/2, 1/2(-Bernoulli measure, and if the initial measure is Markov, we show that the limit of Cesàro mean of the one-site distribution is equidistributed.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1023276306998
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