Journée Systèmes Dynamiques Marseille-Avignon 2023 : dynamique symbolique

Date(s) : 21/06/2023   iCal
9 h 30 min - 17 h 30 min

Marie-Pierre Béal a dû annuler sa venue.

  • 9h45 : Ilkka Törmä (Université de Turku, Finlande) : Generic limit sets of cellular automata: structure and applications

    A cellular automaton is a dynamical system consisting of an infinite line of cells, each of which stores a state that comes from a finite set. The line evolves in discrete time steps. On each step, each cell assumes a new state based on the previous states of itself an some finite set of neighboring cells, using the same transition rule everywhere.
    One way to capture the long-term behavior of a dynamical system is its limit set. The Ω-limit set contains those points of the phase space with an infinite chain of predecessors. The µ-limit set aims to capture the probable behavior starting from a randomly chosen initial conditon, and the generic limit set contains the limit points of a topologically large set of initial conditions.
    We prove various constraints on the structure of the generic limit set of a cellular automaton, and in some cases, present corresponding realization results. We also show how the generic limit set can be used to constrain the probable evolution of a cellular automaton model starting from a random initial configuration, such as a deterministic system of interacting particles.


  • 11h15 : Pierre Arnoux (Université d’Aix-Marseille) : Some very much curious dynamical systems coming from number theory


  • 12h15 : buffet


  • 14h00 : Dominik Kwietniak (Université Jagellonne, Pologne) : An anti-classification theorem for the topological conjugacy problem of Cantor minimal systems

    The isomorphism problem in dynamics dates back to a question of von Neumann from 1932: Is it possible to classify (in some reasonable sense) the ergodic measure-preserving diffeomorphisms of a compact manifold up to isomorphism? We would like to study a similar problem: let C be the Cantor set and let Min(C) stand for the space of all minimal homeomorphisms of the Cantor set. Recall that a Cantor set homeomorphism f is in Min(C) if every orbit of f is dense in C. We say that f and g in Min(C) are topologically conjugate if there exists a Cantor set homeomorphism h such that f∘h=h∘g. We prove an anti-classification result showing that even for very liberal interpretations of what a « reasonable » classification scheme might be, a classification of minimal Cantor set homeomorphism up to topological conjugacy is impossible. We see it as a consequence of the following: we prove that the topological conjugacy relation of Cantor minimal systems TopConj treated as a subset of Min(C)×Min(C) is complete analytic. In particular, TopConj is a non-Borel subset of Min(C)×Min(C). Roughly speaking, it means that it is impossible to tell if two minimal Cantor set homeomorphisms are topologically conjugate using only a countable amount of information and computation.
    Our result is proved by applying a Foreman-Rudolph-Weiss-type construction used for an anti-classification theorem for ergodic automorphisms of the Lebesgue space. We find a continuous map F from the space of all trees over non-negative integers with arbitrarily long branches into the class of minimal homeomorphisms of the Cantor set. Furthermore, F is a reduction, which means that a tree T is ill-founded if and only if F(T) is topologically conjugate to its inverse. Since the set of ill-founded trees is a well-known example of a complete analytic set, we see that it is essentially impossible to classify which minimal Cantor set homeomorphisms are topologically conjugate to their inverses.
    This is joint work with Konrad Deka, Felipe García-Ramos, Kosma Kasprzak, Philipp Kunde (all from the Jagiellonian University in Kraków).


  • 15h30 : soutenance de thèse de Firas Ben Ramdhane (Université d’Aix-Marseille et Université de Sfax, Tunisie) : Symbolic dynamical systems in topological spaces defined via edit distances

    In this thesis, we study symbolic dynamical systems on spaces defined from edit distances, in particular the spaces of Besicovitch and Weyl. These are metric spaces defined using pseudo-metrics and quotients by the relation of pseudo-metric zero. For this purpose, we start by studying these two pseudo-metrics which depend on the Hamming distance.
    We give a generalization of these two pseudo-metrics (centered and sliding) by replacing the Hamming distance by any distance defined on the set of finite words. Then, we present some properties of these two pseudo-metrics: measurability, continuity, shift invariance and behavior on periodic configurations. On the other hand, these two pseudo-metrics are defined as an upper limit. For this reason, we study the existence of the limit for each pseudo-metric.
    We show that the centered pseudo-metric is not always a limit. Moreover, we show that in some class of subshifts equipped with the Cantor topology, the set where the centered pseudo-metric reaches the maximum and the lower limit is zero is a dense Gδ. Furthermore, we show that the set where this pseudo-metric is a limit is of full measure for any weakly-mixing measure and that this limit does not depend on the choice of configurations.
    In contrast, we show that the sliding pseudo-metric is always a limit. Moreover, in some class of subshifts equipped with the Cantor topology, the set where this pseudo-metric reaches the maximum is a dense Gδ. In addition, the set where this pseudo-metric is maximum (within the support of a weakly-mixing measure) is of full measure.
    Finally, we give a first study of dill maps (which generalize cellular automata and substitutions) over the Besicovitch, Weyl and the Feldman-Katok spaces (the latter is obtained by changing the Hamming distance by that of Levenshtein).
    We prove that all dill maps are well-defined over the Feldman-Katok space, in contrast to the Besicovitch and the Weyl spaces where only uniform and constant dill maps are well defined. Furthermore, we show that the Feldman-Katok space is a suitable playground to study the dynamics of dill maps. Indeed, we prove that the shift is equal to the identity, there are no expansive cellular automata, every substitution admits at least one equicontinuous point.


FRUMAM, St Charles (2ème étage)


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