1,720,966 research outputs found
Dynamics of physical systems , normal forms and Markov chains
No full textCette thèse porte sur le comportement asymptotique des systèmes dynamiques et contient cinq chapitres indépendants.Nous considérons dans la première partie de la thèse trois systèmes dynamiques concrets. Les deux premiers chapitres présentent deux modèles de systèmes physiques : dans le premier, nous étudions la structure géométrique des langues d'Arnold de l'équation modélisant le contact de Josephson; dans le deuxième, nous nous intéressons au problème de Lagrange de recherche de la vitesse angulaire asymptotique d'un bras articulé sur une surface. Dans le troisième chapitre nous étudions la géométrie plane du billard elliptique avec des méthodes de la géométrie complexe.Les quatrième et cinquième chapitres sont dédiés aux méthodes générales d'étude asymptotique des systèmes dynamiques. Dans le quatrième chapitre nous prouvons la convergence des moyennes sphériques pour des actions du groupe libre sur un espace mesuré. Dans le cinquième chapitre nous fournissons une forme normale pour un produit croisé qui peut s'avérer utile dans l'étude des attracteurs étranges de systèmes dynamiques.This thesis deals with the questions of asymptotic behavior of dynamical systems and consists of six independent chapters. In the first part of this thesis we consider three particular dynamical systems. The first two chapters deal with the models of two physical systems: in the first chapter, we study the geometric structure and limit behavior of Arnold tongues of the equation modeling a Josephson contact; in the second chapter, we are interested in the Lagrange problem of establishing the asymptotic angular velocity of the swiveling arm on the surface. The third chapter deals with planar geometry of an elliptic billiard.The forth and fifth chapters are devoted to general methods of studying the asymptotic behavior of dynamical systems. In the forth chapter we prove the convergence of markovian spherical averages for free group actions on a probablility space. In the fifth chapter we provide a normal form for skew-product diffeomorphisms that can be useful in the study of strange attractors of dynamical systems
Tiling billiards and Dynnikov's helicoid
No full text18 pages, 5 figuresInternational audienceHere are two problems. First, understand the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describe the topology of connected components of plane sections of a centrally symmetric subsurface of genus . In this note we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov's problem in higher genus seems promising, as we show in the end of this note
Arbres et fleurs sur une table de billard
Full text linkIn this work we study the dynamics of triangle tiling billiards. We unite geometric and combinatorial approaches in order to prove several conjectures. In particular, we prove the Tree Conjecture and the 4n+2 Conjecture, both stated by Baird-Smith, Davis, Fromm and Iyer. Moreover, we study the set of exceptional trajectories which is closely related to the set of minimal Arnoux-Rauzy maps and prove that all of such trajectories pass by all tiles. Finally, we prove that the arithmetic orbits of the Arnoux-Yoccoz map converge, up to rescaling, to the Rauzy fractal, as conjectured by Hooper and Weiss.Nous étudions ici la dynamique des billards dans les pavages triangulaires périodiques. Nous réunissons des approches géométrique et combinatoire pour prouver quelques conjectures. En particulier, nous prouvons la Conjecture d'arbre et la Conjecture formulées par Baird-Smith, Davis, Fromm et Iyer. Puis, nous étudions l'ensemble des trajectoires exceptionelles qui est lié à l'ensemble des applications d'Arnoux-Rauzy minimales, et nous prouvons que ces trajectoires passent par toute tuile. Finalement, nous prouvons que les orbites arithmetiques de l'application d'Arnoux-Yoccoz convergent à la fractale de Rauzy, modulo changement d'échelle, comme c'était conjecturé par Hooper et Weiss
Automorphisms of compact K\"ahler manifolds with slow dynamics
No full text53 pages, 1 figureInternational audienceWe study the automorphisms of compact K\"ahler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in dimensions and . We prove that every automorphism with sublinear derivative growth is an isometry ; a counter-example is given in the context, answering negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial entropy. Finally, we classify minimal automorphisms in dimension and prove they exist only on tori. We conjecture that this is true for any dimension
Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket
No full text47 pages, 18 figuresInternational audienceConsider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to −1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N + 2. We also give a precise description of the exceptional family of trajectories (of zero measure) : these trajectories escape non-linearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips
Free actions of large groups on complex threefolds
No full text14 pagesInternational audienceWe classify compact K\"ahler threefolds with a free group of automorphisms acting freely on
Géométrie plane des q-rationnels et les opérations de Springborn
No full textInternational audienceWe study the geometry of q-rational numbers, introduced by Morier-Genoud and Ovsienko, for positive real q. In particular, we construct and analyse the deformed Farey triangulation and the deformed modular surface. We interpret every q-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on q-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition. Geometrically, the Springborn operations correspond to taking the homothety centers of a pair of two circles. Contents 1. Introduction 1 2. From q-integers to q-rationals to q-reals 5 3. Hyperbolic geometry and deformed Farey tesselation 9 4. Deformed Farey determinants and operations 18 5. Classical Springborn operations 24 6. Springborn operations for q-rationals 32 7.</div
Géométrie plane des q-rationnels et les opérations de Springborn
No full textInternational audienceWe study the geometry of q-rational numbers, introduced by Morier-Genoud and Ovsienko, for positive real q. In particular, we construct and analyse the deformed Farey triangulation and the deformed modular surface. We interpret every q-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on q-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition. Geometrically, the Springborn operations correspond to taking the homothety centers of a pair of two circles. Contents 1. Introduction 1 2. From q-integers to q-rationals to q-reals 5 3. Hyperbolic geometry and deformed Farey tesselation 9 4. Deformed Farey determinants and operations 18 5. Classical Springborn operations 24 6. Springborn operations for q-rationals 32 7.</div
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