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