The generic limit set of cellular automata

Abstract

In topological dynamics, the generic limit set is the smallest closed subset which has a comeager realm of attraction. We study some of its topological properties, and the links with equicontinuity and sensitivity. We emphasize the case of cellular automata, for which the generic limit set is included in all subshift attractors, and discuss directional dynamics, as well as the link with measure-theoretical similar notions.

Complexity of Generic Limit Sets of Cellular Automata

“…The pair (A Z , f ) is a dynamical system. Generic limit sets were defined by Milnor in [9] for general dynamical systems, and were first considered in the context of cellular automata in [5].…”

“…By Lemma 1, ρ is surjective, hence a permutation, and f maps each X i surjectively to X ρ(i) . Corollary 4.12 in [5] implies that ρ must be a cyclic permutation. Proposition 4.…”

“…They also provide a combinatorial characterization of the generic limit set of a CA, which allows us to study its descriptional and structural complexity and carry out complex constructions. This point of view was not present in [5], where relatively simple examples of generic limit sets were provided to highlight the main classification results.…”

“…In [5], Djenaoui and Guillon studied the generic limit sets of dynamical systems in general and directed cellular automata in particular. They related dynamical properties of a given CA to the structure of its generic limit set in different directions and its relation to the set of equicontinuity points and the limit set.…”