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Ergodic Theory of Cellular Automata [chapter]

Marcus Pivato
2012 Computational Complexity  
in the ergodic theory of cellular automata, because of the following result.  ...  Can we generalize any of the theory of cellular automata to cellular networks? Is it possible to develop a nontrivial ergodic theory for such systems?  ...  Proof: (a) follows from the fact that, for any a ∈ A Z , the sequence [ (Shereshevsky, 1992b , Theorem 1), and follows from Kingman's subadditive ergodic theorem.  ... 
doi:10.1007/978-1-4614-1800-9_62 fatcat:6chdeuo5n5b3dj434scnmeplwq

Ergodic Theory of Cellular Automata [chapter]

Marcus Pivato
2009 Encyclopedia of Complexity and Systems Science  
in the ergodic theory of cellular automata, because of the following result.  ...  Can we generalize any of the theory of cellular automata to cellular networks? Is it possible to develop a nontrivial ergodic theory for such systems?  ...  Proof: (a) follows from the fact that, for any a ∈ A Z , the sequence [ (Shereshevsky, 1992b , Theorem 1), and follows from Kingman's subadditive ergodic theorem.  ... 
doi:10.1007/978-0-387-30440-3_178 fatcat:phj2lgyf3veeboz7vbyspuonje

The ergodic theory of cellular automata

Marcus Pivato
2012 International Journal of General Systems  
in the ergodic theory of cellular automata, because of the following result.  ...  Can we generalize any of the theory of cellular automata to cellular networks? Is it possible to develop a nontrivial ergodic theory for such systems?  ...  Proof: (a) follows from the fact that, for any a ∈ A Z , the sequence [ (Shereshevsky, 1992b , Theorem 1), and follows from Kingman's subadditive ergodic theorem.  ... 
doi:10.1080/03081079.2012.695898 fatcat:difdkmdqpfbi3hkpqslugjh6pq

Ergodic aspects of cellular automata

Mike Hurley
1990 Ergodic Theory and Dynamical Systems  
This paper contains a study of attractors in cellular automata, particularly the minimal attractors as defined by J Milnor Milnor's definition of attractor uses a measure on the state space, the measures  ...  attractor by the purely topological definition used by C Conley The examples also show that certain invariant sets of cellular automata are less well-behaved than one might hope for instance the periodic  ...  Acknowledgment The author thanks the referee, who pointed out an error in the original version of this paper, and who suggested example 4 C  ... 
doi:10.1017/s0143385700005848 fatcat:mjjhww4f7behhdkbj7glzee5te

Limit sets of stable cellular automata

ALEXIS BALLIER
2013 Ergodic Theory and Dynamical Systems  
AbstractWe study limit sets of stable cellular automata from a symbolic dynamics point of view, where they are a special case of sofic shifts admitting a steady epimorphism.  ...  To translate this into terms of cellular automata, a sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it  ...  , thus providing characterization of the limit sets of certain stable cellular automata.  ... 
doi:10.1017/etds.2013.72 fatcat:fdutugyrajhqtm4pshsvgvlrqe

Large semigroups of cellular automata

YAIR HARTMAN
2011 Ergodic Theory and Dynamical Systems  
In this article we consider semigroups of transformations of cellular automata which act on a fixed shift space.  ...  the cellular automata transformations which are linear (when the symbols set is the ring of integers mod n).  ...  Hillel Furstenberg, for introducing me to the area and for sharing with me the joy of doing mathematics. I would like also to thank Dr. Uri Shapira and Dr. Jehuda Hartman for their help.  ... 
doi:10.1017/s0143385711000551 fatcat:6cpgrelkqjhftm3t7kndcnveze

Asymptotic randomization of sofic shifts by linear cellular automata

MARCUS PIVATO, REEM YASSAWI
2006 Ergodic Theory and Dynamical Systems  
A continuous function : Suppose that µ is a probability measure on A M whose support is a subshift of finite type or sofic shift.  ...  To see that density (J 2 ) = 1, note that J(S 0 ) \ J 2 ⊂ {N ∈ N; N (r) = 0 for all r ∈ (log p (S 0 ) . . . 1 2 log p (N)]}, which is a set of density zero.  ...  This research was partially supported by NSERC Canada, and was also supported by the kind hospitality of the Universidad de Chile during July 2003.  ... 
doi:10.1017/s0143385706000228 fatcat:css5ktlxlvbm5hhx3nbhzi6e7e

Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

M. KESSEBÖHMER, J. D. M. RADEMACHER, D. ULBRICH
2020 Ergodic Theory and Dynamical Systems  
In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine  ...  system of coupled shift dynamics.  ...  In what follows, the step sizes s m p (·), as elements of the communicating classes, should not be confused with the states of the cellular automata since both are contained in A but obey different transition  ... 
doi:10.1017/etds.2020.18 fatcat:xliv33wn4feyvje6qke22ligye

Some applications of information theory to cellular automata

Michael S. Waterman
1984 Physica D : Non-linear phenomena  
Assuming a probability measure on the initial line that is stationary and ergodic for the shift, a consistent ergodic theory is derived for any finite or infinite collection of lines of the automata.  ...  In this paper general deterministic one-dimensional cellular automata are identified with mappings of the unit interval into itself.  ...  Hopefully this will begin non-empirical studies of ergodic theory of one-dimensional cellular automata. Many numerical phenomena need explanation.  ... 
doi:10.1016/0167-2789(84)90247-1 fatcat:o3x5ac2djzctddrbo7ufctrw3y

Page 6998 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
Ferrariet al., Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1657-1670; MR 2001k:37017; D. A. Lind, Phys. D 10 (1984), no. 1-2, 36-44; MR 86g:68128; M. Pivato and R. Yassawi, Ergodic Theory Dynam.  ...  Sci. 259 (2001), no. 1-2, 99-128; MR DYNAMICAL SYSTEMS AND ERGODIC THEORY 6998 2002e:68098].  ... 

Page 7821 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
Roland Vollmar (Karlsruhe) 68Q Theory of computing 97m:68148 68Q80 Sato, Tadakazu Ergodicity of linear cellular automata over Z,,,,. Inform. Process. Lett. 61 (1997), no. 3, 169-172.  ...  Introduction: “The study of parallel maps is centrally located in cellular automata theory and has been discussed from various points of view. S. J. Willson [Math.  ... 

Synchronization and Linearity

1994 Discrete Applied Mathematics  
DNA sequences and cellular automata. L-systems and the growth of plants. The game of life. A cellular automation model of vertebrate skin patterns. Self-reproducing automata. Artificial life.  ...  Local properties of one-dimensional automata. Global properties of one-dimensional automata. Algebraic properties of additive cellular automata. Languages: formal and natural.  ... 
doi:10.1016/0166-218x(94)90156-2 fatcat:hfmmezgygvah5fqwozklyxfcya

Reality rules:1 picturing the world in mathematics. The fundamentals

1994 Discrete Applied Mathematics  
DNA sequences and cellular automata. L-systems and the growth of plants. The game of life. A cellular automation model of vertebrate skin patterns. Self-reproducing automata. Artificial life.  ...  Local properties of one-dimensional automata. Global properties of one-dimensional automata. Algebraic properties of additive cellular automata. Languages: formal and natural.  ... 
doi:10.1016/0166-218x(94)90157-0 fatcat:uq7qzduqt5flxpld5eo2b5qyxe

Page 4579 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
Summary: “We study the ergodic behavior of linear cellular au- tomata over Z,,.  ...  linear cellular automata over Z,,,.  ... 

Page 1525 of Mathematical Reviews Vol. 45, Issue 5 [page]

1973 Mathematical Reviews  
Using the notion of cellular matrices the author defines a weakly ergodic probabilistic automaton. Any probabilistic automaton with quasi-definite table is weakly ergodic, but not conversely.  ...  V. 8464 The problem of completeness in the structural theory of probabilistic automata. (Russian. English summary insert) Kibernetika (Kiev) 1971, no. 1, 17-20.  ... 
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