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Journal Articles Ergodic Theory and Dynamical Systems Year : 2016

Characterisation of sets of limit measures of a cellular automaton iterated on a random configuration

Mathieu Sablik

Abstract

The asymptotic behaviour of a cellular automaton iterated on a random configuration is well described by its limit probability measure(s). In this paper, we characterise measures and sets of measures that can be reached as limit points after iterating a cellular automaton on a simple initial measure. In addition to classical topological constraints, we exhibit necessary computational obstructions. With an additional hypothesis of connectivity, we show these computability conditions are sufficient by constructing a cellular automaton realising these sets, using auxiliary states in order to perform computations. Adapting this construction, we obtain a similar characterisation for the Ces\`aro mean convergence, a Rice theorem on the sets of limit points, and we are able to perform computation on the set of measures, i.e. the cellular automaton converges towards a set of limit points that depends on the initial measure. Last, under non-surjective hypotheses, it is possible to remove auxiliary states from the construction.
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Dates and versions

hal-01299001 , version 1 (07-04-2016)

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Benjamin Hellouin de Menibus, Mathieu Sablik. Characterisation of sets of limit measures of a cellular automaton iterated on a random configuration. Ergodic Theory and Dynamical Systems, 2016, ⟨10.1017/etds.2016.46⟩. ⟨hal-01299001⟩
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