Soutenances

Study of attractors complexity in cellular automata and the topological structure of Z^d-subshifts

by Alonso Herrera Nunez

Europe/Paris
Salas de usos múltiples 1 y 2, (Batîment Felipe Villanueva)

Salas de usos múltiples 1 y 2,

Batîment Felipe Villanueva

Faculté de Matematiques. Pontificia Universidad Católica de Chile
Description

This work explores two notions of typicality in dynamical systems within the framework of symbolic dynamics. The first concerns typical systems from a topological viewpoint: those that are “large” in a topological sense, often referred to as generic systems. We focus on the topological space S^d of all subshifts A^(Z^d), where A is a finite subset of Z. For d=1, R. Pavlov and S. Schmieding have shown that isolated subshifts are generic. To navigate this queson for d>2, we introduce the notion of maximal subsystem—a subsystem that is inclusion-wise maximal—and use it to characterise isolated systems in S^d as follows: a subshift is isolated if and only if it is of finite type and it has a finite class of maximal subsystems that contains every proper subsystem. This class is not generic as in the d=1 case, but any generic class must contain it, hence the interest in it. Later, we provide insights into how the number of maximal subsystems a subshift has relates to its dynamical and structural properties. Finally, we use some of the machinery developed to show that the Cantor-Bendixon rank of S^d is infinite when d>1, which drascally differs from the case for 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 invesgate how complicated the generic and likely limit sets are, originally introduced by J. Milnor. To measure this, we rely on the well-known arithmecal hierarchy and find that, in general, the language of the likely limit set is a Sigma_3 set—for the generic limit set, the same upper bound was already known from an arcle by I. Törmä. Under the restricon of an automaton with equicontinuity points, we show that both attractors coincide and the complexity decreases to Sigma_1, with tight bounds. In the case where the attractors are inclusion-wise minimal, we find an upper bound of Pi_2, matching known results for general systems by C. Rojas and M. Sablik. Finally, we prove the following realisaon theorem: for any pair of chain-mixing Pi_2 subshifts Y subset of 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.