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  • 5 minutes Lebesgue
    May 23, 2017

    Les vidéos des exposés seront mises en ligne quelques jours après l'exposé. Vidéothèque

    Prochain exposé :

    23-05-2017:  Jean-Philippe Nicolas
    Trous noirs et superradiance

    Certains trous noirs présentent au voisinage de leur horizon une « ergo-région » dans laquelle l'énergie de certaines particules de spin entier peut devenir négative. Roger Penrose a imaginé un processus tirant partie de cette stucture pour en extraire de l'énergie. Nous décrirons le processus de Penrose et une application industrielle imaginée par Subrahmanyan Chandrasekhar. Puis nous verrons la notion de superradiance qui est un phénomène analogue pour les champs de spin entier, champs électromagnétiques par exemple. Nous concluerons en décrivant une application militaire qui pourrait intéresser les forces de l'Empire : les Black Hole Bombs.



    Exposés à venir:

    06-06-2017:  David Lubicz

    07-11-2017:  Guy Casale

  • Séminaire Quimpériodique
    Jun 1, 2017 to Jun 2, 2017

    Ce séminaire de géométrie, complètement à l'Ouest, réunit à Quimper, trois fois l'an, pour deux journées, le jeudi et le vendredi, des géomètres venus des régions Bretagne et Pays de Loire.


    Inas Amacha (LMBA): TBA;
    Vincent Colin (LMJL): TBA;
    Bertrand Deroin (ENS Paris): TBA;
    Victor Kleptsyn (IRMAR): TBA;
    Paul Laurain (univ. Paris VII): TBA.


    Guillaume Deschamps, LMBA, Brest
    Laurent Meersseman, LAREMA, Angers
    Gaël Meigniez, LMBA, Vannes
    Yann Rollin, LMJL, Nantes
    Frédéric Touzet, IRMAR, Rennes


    Historique du séminaire


    Télecharger l'affiche ici

  • Conference - Infinite measure Dynamics
    Jun 6, 2017 to Jun 9, 2017

    Brest, from June 6th to June 9th

    Organization board: Yves Coudène, François Maucourant, Françoise Pène, Barbara Schapira, Samuel Tapie, Annick Nicolle

    Scientific board: Jon Aaronson, Jean-Pierre Conze, Gilles Courtois, Domokos Szasz

    This conference will focus on dynamical systems which naturally preserve a measure with infinite mass. These systems appear in a geometric or probabilistic context, or may come from natural sciences. When the invariant measure has infinite mass, recurrence is no longer automatic, usual mixing properties disappear and new asymptotical properties (such as rationnal ergodicity) may occur. Such systems may develop various and subtil behaviours, which could not exists in finite measure dynamics.

    This conference will gather international experts on this topic, and will allow young researcher to have an easy access to the large recent developpment on such questions.

    See also here here


    Jon Aaronson (Tel Aviv Univ.),

    Sara Brofferio (Univ. Paris Sud),

    Jon Chaika (Univ. Utah),

    Françoise Dal'bo (Univ. Rennes 1),

    Dmitry Dolgopyat (Univ. Maryland),

    Rhiannon Dougall (Univ. Warwick),

    Olivier Glorieux (Luxembourg),

    Sebastien Gouëzel (Univ. Nantes),

    Alba Malaga Sabogal (Univ. Paris 8),

    Emmanuel Roy (Univ. Paris 13),

    Manuel Stadlbauer (Univ. Federal Rio de Janeiro),

    Dalia Terhesiu (Univ. Vienna),

    Damien Thomine (Univ. Paris Sud),

    Michael Bromberg (Univ. Bristol),

    Roland Zweimüller (Univ. Vienna).

    Titles and abstracts

    Jon Aaronson (Tel Aviv Univ.) Rational ergodicity properties and distributional limits of infinite ergodic transformations.

    In infinite ergodic theory, various weak and distributional limits replace the absolutely normalized pointwise ergodic theorem. We’ll review the subject and then see that every random variable on the positive reals occurs as the distributional limit of some infinite ergodic transformation. As a corollary, we obtain a complete classification of the possible ”A-rational ergodicity properties” for an infinite ergodic transformation.

    The main construction follows by ”inversion” from a cutting and stacking construction showing that every random variable on the positive reals occurs as the distributional limit of the partial sums some positive, ergodic stationary process normalized by a 1-regularly varying normalizing sequence (indeed, here the process can be chosen over any EPPT).

    Joint work with Benjamin Weiss. See arXiv:1604.03218

    Sara Brofferio (Univ. Paris Sud), On unbounded invariant measures of stochastic dynamical systems

    We consider stochastic dynamical systems Xn = Yn(Xn−1), where Yn are i.i.d. random continuous transformations of R. We assume that Yn(x) behave asymptotically like Anx, for some random positive number An. The main example is the stochastic affine recursion Xn = AnXn−1+Bn, but this class includes other interesting processes such as reflecting random walks or branching process. Our aim is to describe invariant Radon measures of the process {Xn} in the critical case, when ElogA = 0. Under optimal assumptions, we prove that those measures behave at infinity like dx/x. In the proof we strongly use some properties of random walks on the affine group. The talk will be based on a joint paper with Dariusz Buraczewski.

    Michael Bromberg Temporal distributional limit theorem for cocycles over rotations

    For a measure preserving system (X,B,μ,T ) and a real valued function f on X, temporal random variables along an orbit of a fixed point x in X are obtained by considering the Birkhoff sums Sn(f,x), n = 1,...,N and choosing n randomly uniformly from 1, ..., N. These r.v’s, measure the fraction of time that Birkhoff sums spend in various sets. If, under proper normalization, as N tends to infinity, these variables converge to a non-atomic distribution, we say that f satisfies a temporal limit theorem along the orbit of x (when the limit is Gaussian, we refer to this as temporal CLT). The aim of the talk is to introduce the relevant concepts and sketch a proof of a temporal CLT for piecewise constant cocycles with a single breakpoint, over an irrational rotation with a badly approximable rotation number. This result generalises earlier results by J.Beck and by D.Dolgopyat and O.Sarig. This is joint work with C.Ulcigrai.

    Jon Chaika (Univ. Utah), Ergodicity of typical skew products over some interval exchange transformations

    Let T be a linear recurrent interval exchange transformation. This is ameasure zero, but full Hausdorff dimension set of interval exchange transformations that are analogous to badly approximable rotations. We show that an R valued skew product over such an IET by an integral 0 function that is a linear combination of characteristic functions of intervals is typically ergodic. Relevant terms will be defined. This is joint work with Donald Robertson.

    Françoise Dal’bo (Univ. Rennes 1), An example of a nonuniform lattice with infinite Bowen-Margulis measure

    Joint work with M.Peign´e, J-C Picaud,A.Sambusetti. I will explain how to construct a noncompact negatively curved Riemannian surface with finite volume admitting an infinite Bowen-Margulis measure.

    Dmitry Dolgopyat (Univ. Maryland) On Local Limit Theorems for hyperbolic flows

    I describe an approach to proving local limit theorems and related for flows based on (multidimensional) local limit theorem for associated Poincare map. Both finite and infinite measure case will be discussed. Based on a joint work with Peter Nandori.

    Rhiannon Dougall (Univ. Warwick) Growth of closed geodesics for infinite covers

    We are interested in the dynamics of the geodesic flow for infinite volume manifolds M which arise as a regular cover of a fixed compact (or convex cocompact) negatively curved manifold M0. Writing hM for the exponential growth rate of closed geodesics in M, we have that hM ≤ h0, where h0 is the topological entropy of the geodesic flow for M0. We answer the question of when there is a uniform gap hM < h0 in M in terms of the permutation representations given by the covering M of M0. The proof uses the symbolic dynamics for the flow, and so we formulate the analogous statements for countable state shifts obtained as group extensions of a finite state shift.

    Olivier Glorieux (IMPA) Hausdorff dimension and critical exponent of Quasi-Fuchsian Anti-de Sitter manifolds

    The aim of my talk will be to explain how classical invariants and theorems for groups acting on the hyperbolic space, can be extended to the Anti-de Sitter (AdS) setting. We will recall the notion of critical exponent and Hausdorff dimension for discrete action on the hyperbolic space and explain how we can define similar notions for a certain type of groups acting on AdS manifolds. We will finally explain how to get a rigid bound for these invariants in dimension 3 which is a result equivalent a famous result obtained by R. Bowen in ’79 . This is a joint work with D. Monclair.

    Sébastien Gouëzel Quantitative Pesin theory for subshifts of finite type

    In non-uniformly hyperbolic dynamics, Pesin sets are measurable sets where the dynamics is very well understood. However, their definition makes these sets hard to control in a quantitative way, even when the underlying dynamics is hyperbolic. We will explain why such a control is useful, and what kind of bounds we can obtain. Joint work with L. Stoyanov.

    Emmanuel Roy (Univ. Paris 13) Ergodic splittings of Poisson processes

    If N denotes a Poisson process, a splitting of N is formed by two point processes N1 and N2 such that N = N1 +N2. If N1 and N2 are independent Poisson processes then the splitting is said to be Poisson and such a splitting is always available (We allow the possibility to enlarge the ambient probability space). In general, a splitting is not Poisson but the situation changes if we require that the distributions of the point processes are invariant by a common underlying map that acts at the level of each point of the processes. We will prove that if this map has infinite ergodic index, then a splitting is necessarily Poisson if the environment is ergodic.

    This is a work in progress, with Elise Janvresse and Thierry de la Rue.

    Dalia Terhesiu, Exploiting semistable laws for random variables

    We recall that semistable laws is a class of infinitely divisible laws, which complements the more well known stable laws. I will recall some main, previously established, results on necessary and sufficient conditions for the existence of semistable laws for random variables. I will report on work in progress with Peter Kevei which aims toward a complete understanding of a limit law for null recurrent renewal chains, assuming that the involved return function is in the domain of a semistable law (as such, no strict regular variation is required). Some analogies with the Darling Kac law will be discussed. If time remains, I will present some results of work in progress with Douglas Coates on semistable laws for interval intermittent maps.

  • Conference - Families of algebraic dynamical systems
    Jun 12, 2017 to Jun 16, 2017

    Rennes, from June 12th to June 16th

    Organization board: Serge Cantat, Christophe Dupont

    Scientific board: Matthew Baker, Eric Bedford, Serge Cantat, Christophe Dupont, Mattias Jonsson

    Mini-courses :

    • Bertrand Deroin (Ecole Normale Supérieure, Paris) Holomorphic families of representations in SL(2,C)

    We will survey some aspect of the theory of holomorphic families representations in SL(2,C):
    1. Sullivan's stability theory
    2. Bifurcation currents
    3. Harmonic measures of complex projective structures

    • Charles Favre (Ecole Polytechnique, Palaiseau) Degeneration of rational maps of the Riemann sphere

    We shall describe how one can control the dynamics of a meromorphic family of rational maps of the Riemann sphere parameterized by the punctured unit disk as one approaches the puncture. Our analysis is based in a crucial way on the interplay between complex and non-archimedean dynamics. We shall also review how this control can be combined with technics from arithmetic geometry to the description of the special curves in the parameter space that contain infinitely many post-critically finite maps.

    • Laura de Marco (Northwestern University, Chicago) Rational maps, elliptic curves, and heights

    We will study the geometry and arithmetic of families of rational maps and families of elliptic curves. The focus will be on "canonical height functions", introduced by Tate and Neron around 1960 in the setting of abelian varieties and further developed by Call and Silverman (1993) for algebraic dynamical systems. My aim is to present recent results -- both in the setting of elliptic curves and of rational maps -- and to present open questions inspired by the connections between holomorphic dynamics and arithmetic geometry.

    Talks :

    • François Berteloot (Toulouse)
    • Simon Brandhorst (Hannover): On the dynamical spectrum of projective K3 surfaces.
    • Romain Dujardin (Université Paris 6)
    • Alexander Gamburd (City University of New-York)
    • Thomas Gauthier (Université de Picardie Jules Verne, Amiens): The support of the bifurcation measure has positive volume
    • Martin Hils (Paris)
    • Sarah Koch (Ann Harbor): Irreducibility of curves in parameter space: cubic polynomials vs. quadratic rational maps.
    • Holly Krieger (Cambridge University)
    • Juan Rivera-Letelier (Rochester)
    • Thomas Scanlon (Berkeley University)
    • Tom Tucker (Rochester): Towards a finite index conjecture for iterated Galois groups
    • Junyi Xie (Université de Rennes 1): Invariant pencils for polynomial selfmaps of the affine plane

    Abstracts :

    Simon Brandhorst : On the dynamical spectrum of projective K3 surfaces.

    The dynamical degree of a surface automorphism is a Salem number, that is, an algebraic integer lambda>1 which is conjugate to 1/\lamda and all whose other conjugates lie on the unit circle. We prove that for each Salem number lambda of degree at most 20, there is a power lambda^n, n in N, which is the dynamical degree of an automorphism of some projective K3 surface.

    Thomas Gauthier : The support of the bifurcation measure has positive volume.

    The moduli space M_d of degree d>=2 rational maps can naturally be endowed with a measure mu_bif detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure mu_bif has positive Lebesgue measure. To do so, we establish a general criterion for the conjugacy class of a rational map to belong to the support of mu_bif and we exhibit a "large" set of Collet-Eckmann rational maps which satisfy that criterion. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps. This is a joint work with Matthieu Astorg, Nicolae Mihalache and Gabriel Vigny.

    Sarah Koch : Irreducibility of curves in parameter space: cubic polynomials vs. quadratic rational maps

    Living inside the space of monic centered cubic polynomials, are the curves S_n, which consist of all polynomials f which possess a superattracting cycle of period n. Recently, Arfeux and Kiwi announced a proof that S_n is irreducible for all n>=1. In this talk, we consider the analogous curves which live in the moduli space of quadratic rational maps. It is currently unknown if these curves are irreducible. We discuss some unexpected challenges that arise in the quadratic rational map setting which are absent in the cubic polynomial setting. This talk is based on joint work with E. Hironaka.

    Tom Tucker : Towards a finite index conjecture for iterated Galois groups

    Let f be a polynomial over a global field. Let G denote the inverse limits of the Galois groups of f^n, where f^n denotes n-th iterate of f. Boston and Jones have suggested that under reasonable hypotheses, one might hope that G has finite index in the full group of automorphisms on an infinite tree corresponding to roots of iterates f^n when f is quadratic. We will show that their conjecture is true over function fields of characteristic 0, and that it would be a consequence of well-known diophantine conjectures over number fields. We will also treat the case of cubic polynomials, where less is known.

    Junyi Xie : Invariant pencils for polynomial selfmaps of the affine plane

    With Jonsson and Wulcan, we classify polynomial selfmaps f of the affine plane of that preserve an irreducible pencil of curves at infinity. More generally, we study a more general classification problem, where the invariant pencil is replaced by more general numerical data at infinity.

  • Conference - Random walks on algebraic structures-On honour of Yves Guivarc'h
    Jun 19, 2017 to Jun 23, 2017

    Rennes, from June 19th to June 23rd

    Organization board: Bachir Bekka, Nizar Demni

    Scientific board: Bachir Bekka, Emmanuel Breuillard, Nizar Demni, Alex Lubotzky

    The conference deals with themes around random walks on algebraic structures. Dynamical sytems on groups, homogeneous spaces, and related structures (graphs, quantum groups , groupoids,...) are rich objects, involving various fields of mathematics (probability, group theory, topology, number theory, operator algebras, Banach space geometry, etc). They are also useful as modelling tools for other sciences (physics, computer science, economics, etc); for instance, expander graphs, which are of interest in theoretical computer science, are graphs for which the associated random walk has special spectral properties and their explicit construction involves sophisticated tools such as Kazhdan's property (T) or Selberg's inequality. The conference aims to give a flavor of the various aspects of the subject by bringing together some experts with major contributions to the field. This conference will also be the opportunity to celebrate the 80th birthday of Yves Guivarc'h, who played a major part in the development of probability on algebraic and geometric structures, a subject with an ever growing interest for the last 30 years.

  • Conference - Dynamics on representation varieties
    Jun 26, 2017 to Jun 30, 2017

    Rennes, from June 26th to June 30th

    Organization board: Ian Biringer, Ludovic Marquis, Juan Souto

    Scientific board: Uri Bader, Jeffrey F. Brock, Jean-Marc Schlenker

    Numerous areas of mathematics are touched by what could be called Dynamics on representation varieties. For instance one could mention ergodic theory, Riemannian geometry, low-dimensional topology, Teichmüller theory, and so on... The aim of this workshop is to bring together graduate students, recent graduates and experts in these different areas, giving everybody ample time for discussions and collaborations. Next to a number of research talks, three mini-courses by Tsachik Gelander, Francois Labourie and Julien Marché will take place.

    We the organizers of this conference affirm that scientific events must be open to everyone, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity. We believe that such events must be supportive, inclusive, and safe environments for all participants. We believe that all participants are to be treated with dignity and respect. Discrimination and harassment cannot be tolerated. We are committed to ensuring that the Conference Dynamics on representation varieties follows these principles. For more information on the Statement of Inclusiveness, see this dedicated web page.

  • School - Analytical aspects of hyperbolic flows
    Jul 3, 2017 to Jul 7, 2017

    Nantes, from July 3rd to July 7th

    Organization board: Sebastien Gouëzel, Laurent Guillopé, Samuel Tapie

    Scientific board: Nalini Anantharaman, Viviane Baladi, Colin Guillarmou, Masato Tsujii

    Hyperbolic flows are dynamical systems with strong chaotic properties, whose study has been started a long time ago, a crucial example being the geodesic flow on negatively curved manifolds. Whereas the qualitative properties of such flows are well understood, their fine quantitative properties (rate of mixing, spectrum...) require more sophisticated tools. They have been studied both from a dynamical point of view (Dolgopyat's techniques) and more analytically: semi-classical methods, initially introduced to study PDEs, have proven very valuable in this context.

    The purpose of this summer school is to make these different techniques accessible to PhD students and young researchers, as well as to give an opportunity for specialists in dynamical systems to learn tools from semi-classical analysis, and conversely. Therefore, the core of this summer school will consist in three introductive mini-courses, completed by a few research talks and question sessions.

  • Conference - Young researcher meeting in dynamics and geometry
    Sep 6, 2017 to Sep 8, 2017

    Rennes, from September 6th to September 8th

    Organization board: Françoise Dal'Bo, Frédéric Paulin, Barbara Schapira, Damien Thomine

    Since its creation the Platon network (GDR National Center for Scientific Research n°3341 http: // / platon/) leads actions towards young researchers in ergodic geometry. The recurrent young researcher meeting is one of the highlights of the year. The goal is to allow about ten PhD students or recent doctors to expose their work and promotes discussions between young and senior researchers. The "Young researcher meeting in dynamics and geometry" follows the spirit of these recurring meetings with an international dimension brought in particular by Swiss and Senegalese networks.

    See also here

5 minutes Lebesgue

Hideyuki Ishi, Analyse sur les cônes homogènes et au-delà

Hideyuki Ishi, de l'université de Nagoya, est invité par le Centre Henri Lebesgue.

Analysis on homogeneous cones and beyond

The set of positive definite real symmetric matrices is a typical example of homogeneous cone. Because of its rich symmetry, a lot of beautiful integral formulas hold over the homogeneous cone. And recently, statisticians find that some of the formulas are valid even for non-homogeneous cones.

Les 5 minutes Lebesgue

22 March 2016
Mathematical field: 
analysis, statistics
fonction gamma, cône, intégrale, homogénéité


H. Ishi "Homogeneous cones and their applications to statistics", pp. 135--154.
G. Letac "Decomposable graphs", pp. 155--196.

Modern methods of multivariate statistics, Travaux en Cours 82, Ed. by P. Graczyk and A. Hassairi, Hermann, Paris, 2014.

File 5min-ishi.mp465.54 MB




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