Approche graphique de la dynamique symbolique

Offre de thèse

Date limite de candidature

10-05-2025

Date de début de contrat

01-10-2025

Directeur de thèse

JEANDEL Emmanuel

Encadrement

Réunion régulières avec le doctorant ou la doctorante

Type de contrat

Concours pour un contrat doctoral

Candidater à cette offre

école doctorale

IAEM - INFORMATIQUE - AUTOMATIQUE - ELECTRONIQUE - ELECTROTECHNIQUE - MATHEMATIQUES

équipe

MOCQUA

contexte

Symbolic dynamics in dimension 1 is a modelling by simply infinite or bi-infinite words of a dynamical system. We encode the dynamical system by the set of all of its trajectories after discretization of the space. If it has good properties (typically expansivity), there is no loss of information between the system and its model. The easiest models of dynamical systems are those for which the model is represented by a finite graph, that is for which the set of trajectories can be transformed in a set of (bi-)infinite paths on a finite graph. We then talk about sofic subshift}. Moreover, if those models are described only by (time) local constraints, they are called subshifts of finite type. One of the most important question in symbolic dynamics is to determine when two systems are conjugated (isomorphic), which is expressed in terms of (bi-)infinite word by the existence of a reversible cellular automata transforming the set of trajectories of the first system to the set of trajectories of the second system. This problem is solved for non-reversible systems (simply infinite word) but is open since the '60 for reversible systems (bi-infinite word). The aim of this thesis is to study the conjugacy problem and more generally the symbolic dynamics in the framework of diagrammatical languages. Diagrammatical languages are a tool coming from category theory and that recently proved its interests in other domains from computer science, mathematics, and physics; in particular in quantum computation [CK], in topological field theory [TV], and in linear algebra [Z]. More recently, it has been proven that this languages can as well provide a new point of view on symbolic dynamics [J,CM,D]. The scope of this thesis is to follow these works in many directions. First by constructing diagrammatical languages to represent graphically some symbolic dynamics invariants, especially the full group. Then the generalization of existing results to any groups by using tensors networks.

spécialité

Informatique

laboratoire

LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications

Mots clés

dynamique symbolique, catégories

Détail de l'offre

Le but de cette thèse est d'étudier la possibilité d'utiliser des méthodes issues des langages graphiques et des catégories monoïdales afin de mieux comprendre le problème de la conjugaison en dynamique symbolique. Plus précisément, le candidat ou la candidate examinera comment certains invariants (comme le groupe plein topologique) peuvent être représentés à l'aide de langages diagrammatiques. Le candidat ou la candidate devra créer un langage graphique capable de représenter les sous-systèmes symboliques sofiques et les sous-systèmes symboliques de type fini sur des groupes finiment engendrés. Enfin, les liens étroits entre les bialgèbres et l'équivalence de flot sur les sous-systèmes symboliques seront étudiés.

Keywords

symbolic dynamics, category theory

Subject details

The purpose of this thesis is to study the possibility of using methods from graphical languages and monoidal categories to better understand the conjugacy problem in unidimensional symbolic dynamics. More precisely, we'll investigate how some invariants (as the topological full group) can be represented in terms of diagrammatical languages. In addition, we'll create a graphical language able to represent sofic subshifts and subshifts of finite type on finitely generated groups. Finally, we'll study the strong links between bialgebra and flow equivalence on subshifts.

Profil du candidat

Diplôme de M2 en Informatique ou en Mathématiques.

Candidate profile

Masters Degree in Computer Science and/or Mathematics

Référence biblio

[CK] Bob Coecke and Aleks Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press, 2017.

[D] Titouan Carette, Marc de Visme, Vivien Ducros, Victor Lutfalla, Etienne Moutot, Complete Compositional Syntax for Finite Transducers on Finite and Bi-Infinite Words. arXiv:2502.06450

[LM] Douglas A. Lind and Brian Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, New York, NY, USA, 1995.

[BK] Bruce Kitchens. Symbolic Dynamics. Springer, 1988.

[HM] Hiroki Matui. Topological full groups of one-sided shifts of finite type. Journal für die reine und angewandte Mathematik, 2013.

[J] Emmanuel Jeandel, Strong Shift Equivalence as a Category Notion, arXiv:2107.10734.

[TV] Vladimir Turaev and Alexis Virelizier. Monoidal Categories and Topological Field Theory. Birkhaüser, 2017.

[Z] Fabio Zanasi. Interacting hopf algebras: the theory of linear systems. Ph.D. dissertation, Ecole Normale Supérieure de Lyon, 2015.

[CM] Titouan Carette and Etienne Moutot. Aperiodicity in Quantum Wang Tilings. arXiv:2302.04503v3