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Dynamical Systems and Ordinary Differential Equations  | enregistrements trouvés : 123

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Classifying regular systems of first order linear ordinary equations is a classical subject going back to Poincare and Dulac. There is a gauge group whose action can be described and an integrable normal form produced. A similar problem for higher order differential equations was never addressed, perhaps because the corresponding equivalence relationship is not induced by any group action. Still one can develop a reasonable classification theory, largely parallel to the classical theory. This is a joint work with Shira Tanny from the Weizmann Institiute, see http://arxiv.org/abs/1412.7830. Classifying regular systems of first order linear ordinary equations is a classical subject going back to Poincare and Dulac. There is a gauge group whose action can be described and an integrable normal form produced. A similar problem for higher order differential equations was never addressed, perhaps because the corresponding equivalence relationship is not induced by any group action. Still one can develop a reasonable classification ...

34C20 ; 34M35

We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in each moduli space. The proofs use recent results in Teichmüller dynamics, especially joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. Joint work with McMullen and Mukamel as well as Eskin, McMullen and Mukamel shows that exotic examples of "higher dimensional Teichmüller discs" do exist. We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in ...

30F60 ; 32G15

Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on which one can act by means of a control, like for example a car, a robot, a space shuttle, a chemical reaction or in more general a process that one aims at steering to some desired target state.
Emmanuel Trélat will overview the range of applications of that theory through several examples, sometimes funny, but also historical. He will show you that the study of simple cases of our everyday life, far from insignificant, allow to approach problems like the orbit transfer or interplanetary mission design.
control theory - optimal control - stabilization - optimization - aerospace - Lagrange points - dynamical systems - mission design
Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on ...

49J15 ; 93B40 ; 93B27 ; 93B50 ; 65H20 ; 90C31 ; 37N05 ; 37N35

I will present results on the dynamics of horocyclic flows on the unit tangent bundle of hyperbolic surfaces, density and equidistribution properties in particular. I will focus on infinite volume hyperbolic surfaces. My aim is to show how these properties are related to dynamical properties of geodesic flows, as product structure, ergodicity, mixing, ...

37D40

Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the corresponding nonlinear group of morphims of affine three space. Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the ...

11G05 ; 37A45

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

37Cxx ; 37Jxx ; 53D25 ; 53D40 ; 53D42

I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, and iceberg model. I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, ...

37B50 ; 37B10 ; 37B40

I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, and iceberg model. I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, ...

37B50 ; 37B10 ; 37B40

I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, and iceberg model. I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, ...

37B50 ; 37B10 ; 37B40

We study cascades of bifurcations in a simple family of maps on the circle, and connect this behavior to the geometry of an absolute period leaf in genus $2$. The presentation includes pictures of an exotic foliation of the upper half plane, computed with the aid of the Möller-Zagier formula.

30F10 ; 30F30

Post-edited  Darcy problem and crowd motion modeling
Maury, Bertrand (Auteur de la Conférence) | CIRM (Editeur )

We describe here formal analogies between the Darcy equations, that describe the flow of a viscous fluid in a porous medium, and some problems arising from the handing of congestion in crowd motion models.
At the microscopic level, individuals are identified to rigid discs, and the dual handling of the non overlapping constraint leads to discrete Darcy-like equations with a unilateral constraint that involves the velocities and interaction pressures, and that are set on the contact network. At the macroscopic level, a similar problem is obtained, that is set on the congested zone.
We emphasize the differences between the two settings: at the macroscopic level, a straight use of the maximum principle shows that congestion actually favors evacuation, which is in contradiction with experimental evidence. On the contrary, in the microscopic setting, the very particular structure of the discrete differential operators makes it possible to reproduce observed "Stop and Go waves", and the so called "Faster is Slower" effect.
We describe here formal analogies between the Darcy equations, that describe the flow of a viscous fluid in a porous medium, and some problems arising from the handing of congestion in crowd motion models.
At the microscopic level, individuals are identified to rigid discs, and the dual handling of the non overlapping constraint leads to discrete Darcy-like equations with a unilateral constraint that involves the velocities and interaction ...

34A60 ; 34D20 ; 35F31 ; 35R70 ; 70E50 ; 70E55

The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

In this international farewell address about 35 years of research on Dynamic energy Budget theory, I review the ontogeny of the theory, and discuss background, motivation, start, milestones and outlook from a personal perspective. The effort is framed as a case study in Theoretical Biology.

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant $\sum \left ( G \right )$ containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of $\sum \left ( G \right ) $ and $\sum \left ( G \right )$ is connected then for every splitting of $G$ as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible.
This talk is based on joint work with Spencer Dowdall and Christopher Leininger.
An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully ...

20F65 ; 57Mxx ; 37BXX ; 37Dxx

I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions. I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close ...

28A80 ; 37A10 ; 03D99 ; 54H20

I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely many factor representations of type $II_1$. I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely ...

20E08 ; 20F65 ; 37B05

The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.
The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform ...

35B40 ; 35B15 ; 35Q55 ; 37K15 ; 47B35

Post-edited  A universal hypercyclic representation
Glasner, Eli (Auteur de la Conférence) | CIRM (Editeur )

For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about the development of this idea and its applications as expounded in a subsequent work of Sophie Grivaux. For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about ...

37A15 ; 37A05 ; 37A25 ; 37A30 ; 47A16 ; 47A67 ; 47D03

We will consider (sub)shifts with complexity such that the difference from $n$ to $n+1$ is constant for all large $n$. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most $d/2$ ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss further improvements when more assumptions are allowed. This is ongoing work with Michael Damron. We will consider (sub)shifts with complexity such that the difference from $n$ to $n+1$ is constant for all large $n$. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most $d/2$ ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss ...

37B10 ; 37A25 ; 68R15

Post-edited  Taming the coloured multizetas
Ecalle, Jean (Auteur de la Conférence) | CIRM (Editeur )

1. We shall briefly describe the ARI-GARI structure; recall its double origin in Analysis and mould theory; explain what makes it so well-suited to the study of multizetas; and review the most salient results it led to, beginning with the exchanger $adari(pal^\bullet)$ of double symmetries $(\underline{al}/\underline{il}) \leftrightarrow (\underline{al}/\underline{al})$, and culminating in the explicit decomposition of multizetas into a remarkable system of irreducibles, positioned exactly half-way between the two classical multizeta encodings, symmetral resp. symmetrel.

2. Although the coloured, esp. two-coloured, multizetas are in many ways more regular and better-behaved than the plain sort, their sheer numbers soon make them computationally intractable as the total weight $\sum s_i$ increases. But help is at hand: we shall show a conceptual way round this difficulty; make explicit its algebraic implementation; and sketch some of the consequences.

A few corrections and comments about this talk are available in the PDF file at the bottom of the page.
1. We shall briefly describe the ARI-GARI structure; recall its double origin in Analysis and mould theory; explain what makes it so well-suited to the study of multizetas; and review the most salient results it led to, beginning with the exchanger $adari(pal^\bullet)$ of double symmetries $(\underline{al}/\underline{il}) \leftrightarrow (\underline{al}/\underline{al})$, and culminating in the explicit decomposition of multizetas into a ...

11M32

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