Analyse de systèmes dynamiques par discrétisation. Exemples d'applications en théorie des nombres et en biologie moléculaire

Anne Siegel 1
1 SYMBIOSE - Biological systems and models, bioinformatics and sequences
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : Applications focus on several mathematical fields and molecular biology. Our purpose is to illustrate how one can couple algorithmic approaches with discretization methods of dynamical system in order to exploit (even partial) knowledge on the system.

The first type of system we adress consist in toral autormorphisms and translations. To obtain informations on their applications needs to abstract their dynamical properties. Inspired by the one-dimensional case (beta-numeration and sturmian sequences); the main question is to exhibit a toral fundamental domain and a partition of this domain in which the dynamics of the systems is coded by simple symbolic systems. We used a decidable approach to describe the fractal boundary of the domain in the Pisot case and its tiling properties and we derived criteria to decide whether it generates a good coding. From their topological properties we deduced applications in number theory (characterization of rational numbers with a finite or purely periodic expansion in a non-integer basis) and discrete geometry (condition for the iterative generation of discrete planes).

The second class of question we adress in the framework of large-scale dynamical systems in molecular biology is how to produce information on a system that is only partially known and observed. We introduced a formalism to interpret observations in molecular biology, with a dynamical background. The main purpose was to guide model correction and, in the future, experiment design. Qualitative aspects are replaced by properties of shift equilibria,
so that analyzing data is reduced to writting and then solving constraints on discrete sets.
Document type :
Habilitation à diriger des recherches
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Anne Siegel. Analyse de systèmes dynamiques par discrétisation. Exemples d'applications en théorie des nombres et en biologie moléculaire. Mathématiques [math]. Université Rennes 1, 2008. ⟨tel-00358996⟩

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