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Ergodicity of some classes of cellular automata subject to noise. (English) Zbl 07055679
Summary: Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise.We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J05 Discrete-time Markov processes on general state spaces
37B15 Dynamical aspects of cellular automata
37A50 Dynamical systems and their relations with probability theory and stochastic processes
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