Browsing by Author "Herrera Núñez, Alonso Eduardo"

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    Study of attractors complexity in cellular automata and the topological structure of Z^d-subshifts
    (2025) Herrera Núñez, Alonso Eduardo; Rojas, Cristóbal; Sablik, Mathieu; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    This work explores two notions of typicality in dynamical systems within the frameworkof symbolic dynamics. The first concerns typical systems from a topological viewpoint: thosethat are “large” in a topological sense, often referred to as generic systems. We focus onthe topological space S d of all subshifts AZd, where A is a finite subset of Z. For d = 1, R.Pavlov and S. Schmieding, [57], have shown that isolated subshifts are generic. To navigatethis question for d ≥ 2, we introduce the notion of maximal subsystem—a subsystem thatis inclusion-wise maximal—and use it to characterise isolated systems in Sd as follows: asubshift is isolated if and only if it is of finite type and it has a finite class of maximalsubsystems that contains every proper subsystem. This class is not generic as in the d = 1case, but any generic class must contain it, hence the interest in it. Later, we provideinsights into how the number of maximal subsystems a subshift has relates to its dynamicaland structural properties. Finally, we use some of the machinery developed to show that theCantor-Bendixon rank of Sd is infinite when d ≥ 2, which drastically differs from the casefor d = 1, where the Cantor-Bendixon rank is 1. The second notion changes focus to typically observable behaviours in the form of attractors. In the context of cellular automata, we investigate how complicated the generic and likely limit sets are, originally introduced by J. Milnor. To measure this, we rely on the well known arithmetical hierarchy and find that in general, the language of the likely limit set is a Σ3 set—for the generic limit set, the same upper bound was already known from [72]. Under the restriction of an automaton with equicontinuity points, we show that both attractors coincide and the complexity decreases to Σ1, with tight bounds. In the case the attractors are inclusion-wise minimal, we find an upper bound of Π2, matching known results for general systems, [62]. Finally, we prove the following realisation theorem: for any pair of chain-mixing Π2 subshifts Y ⊆ X, there exists a cellular automaton whose generic and likely limit sets are precisely X and Y , respectively. Altogether, this work offers new insights into the interplay between structure and observability in symbolic dynamical systems, highlighting how typical behaviours may vary dramatically across dimensions and under different dynamical constraints.