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Speaker: Bernardi, Olga (Università di Padova)
Title: Birkhoff attractors of dissipative classical and symplectic billiards.
Abstract: We propose a dissipative version both for Birkhoff and symplectic billiards. After discussing their main properties, we study the corresponding Birkhoff attractors in terms of the geometry of the table as well as the straight of the dissipation. We compare the results for the two different dynamics.
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Speaker: Calleja, Renato
(Universidad Nacional Autónoma de México)
Title: Whiskered KAM Tori of Conformally Symplectic Systems.
Abstract: Many physical problems are described by conformally symplectic systems. We study the existence of whiskered tori in a family $f_\mu$ of conformally symplectic maps depending on parameters $\mu$. Whiskered tori are tori on which the motion is a rotation but having as many contracting/expanding directions as allowed by the preservation of the geometric structure.
Our main result is formulated in an a-posteriori format. Given an approximately invariant embedding of the torus for a parameter value $\mu_0$ with an approximately invariant splitting, there is an invariant embedding and invariant splittings for new parameters.
Using the results of formal expansions as the starting point for the a-posteriori method, we study the domains of analyticity of parameterizations of whiskered tori in perturbations of Hamiltonian Systems with dissipation. The proofs of the results lead to efficient algorithms that are quite practical to implement. Whitney regularity will also be discussed.
Joint work with A. Celletti and R. de la Llave.
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Speaker: Chantraine, Baptiste (Université de Nantes)
Title: Twisted product of locally conformally symplectic (lcs) manifolds.
Abstract: In this talk, after motivating the problem through its relations with contact and symplectic geometry, I will talk about a construction of twisted product of lcs manifolds. This construction allows to relates fixed point of Hamiltonian diffeomorphisms to Lagrangian intersections (and this allows us to relate the number of such fixed points to Novikov homology of the Lee class of the lcs structure ). This is a joint work with Kevin Sackel.
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Speaker: Chen, Qinbo (Nanjing University)
Title: A new selection problem for Hamilton-Jacobi equations and its applications.
Abstract: We study a new perturbation problem for first-order convex Hamilton-Jacobi equations, which explores the combined effects of the vanishing discount process and potential perturbations. This framework gives rise to a generalized selection principle that extends the classical vanishing discount approach. As applications, we will introduce a novel solution operator and show how it offers new perspectives on weak KAM theory and Aubry-Mather theory.
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Speaker: Currier, Adrien (Université de Nantes)
Title: Exact Lagrangians in cotangent bundles with locally conformally symplectic structure.
Abstract: Abstract: "First considered by Lee in the 40s, locally conformally symplectic (LCS) geometry appears as a generalization of symplectic geometry which allows for the study of Hamiltonian dynamics on a wider range of manifolds while preserving the local properties of symplectic geometry. After a long period of hibernation (especially as far as the topological aspect is concerned), this topic has recently seen renewed interest. However, to this day, the field of LCS topology remains vastly unexplored.
In this talk, we will introduce the various objects of LCS geometry and their behavior through both definitions and examples. We will then explore some quirks of this generalization through the lens of the nearby Lagrangian conjecture. And, finally, we will provide a partial answer to the question: when is an exact (LCS) Lagrangian homotopy equivalent to the zero-section?
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Speaker: de la Llave, Rafael (Georgia Institute of Technology)
Title: Conformally symplectic dynamics: Interactions between topology, geometry and dynamics.
Abstract: We explore the relations between different categories topology, dynamics, and geometry.
Among the results:
1) The topology of a manifold may restrict the set of conformal factors of maps for a non-exact form. Following Arnaud-Fejoz (2024) we define $$ {\mathcal R}(M) = \{ \eta \in \mathbb{R} | \exists f:M \rightarrow M, \omega \textrm { symplectic not exact} s.t. f^* \omega = \eta \omega\} $$ We show, using topological arguments, that there are manifolds such that $ \mathcal{R} $ is strictly larger that $\{1\}$ but strictly smaller than ${\mathbb R}_+$.
2) We show that a normally hyperbolic manifold for a conformally symplectic map is symplectic if and only if the hyperbolicity rates satisfies some pairing rules involving the conformal factor.
3) For a NHIM, there are several conditions on rates that imply geometric properties for the manifold and conversely geometric properties that imply relations among the rates.
4) For Lagrangian submanifolds (e.g. KAM tori) one can give a description of dynamics in a neighborhood.
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Speaker: Florio, Anna (Université Paris Dauphine PSL)
Title: Genericity of transverse homoclinic points for analytic convex billiards.
Abstract: A celebrated result by Zehnder in the '70s states that a generic analytic area-preserving map of the disk, having the origin as elliptic fixed point, exhibits a transverse homoclinc orbit in every neighborhood of the origin. In an ongoing project with Inmaculada Baldomà, Martin Leguil and Tere Seara, we adapt the strategy of Zehnder and use Aubry-Mather theory for twist maps in order to show that a generic analytic strongly convex billiard has, for every rational rotation number, a periodic orbit with a transverse homoclinic intersection.
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Speaker: Gidea, Marian (Yeshiva University)
Title: Conformally symplectic dynamics: Geometry of homoclinic excursions to a normally hyperbolic invariant manifold.
Abstract: We study conformally symplectic maps that possess a symplectic, normally hyperbolic invariant manifold. Homoclinic excursions can be conveniently studied using the so-called scattering map, which gives the orbit asymptotic in the future as a function of the orbit asymptotic in the past. This map exhibits geometric properties, and can be computed perturbatively (or numerically) in explicit examples. We show that: (i) Even though the dynamics is dissipative, the scattering map is symplectic. We give several proofs of this result, which allow us to apply it in more general situations beyond the conformal symplectic settings. (ii) If the symplectic form is exact, then the scattering map is exact (even if the map is not exact). (iii) When both the symplectic form and the conformally symplectic map are exact, we provide explicit formulas for the primitive function of the scattering map. These formulas have implications for the calculus of variations. This is based on joint work with R. de la Llave and T. M-Seara.
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Speaker: Giralt, Mar (Observatoire de Paris)
Title: Chaotic phenomena to L3 in the Restricted 3-Body Problem.
Abstract: The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, L1,..,L5, called the Lagrange points. The Lagrange point L3 is a saddle-center critical point (collinear with the primaries and beyond the largest one) with a 1-dimensional stable and unstable manifold. When the ratio between the masses of the primaries $\mu$ is small, the modulus of the hyperbolic eigenvalues are weaker, by a factor of order $\sqrt\mu$, than the elliptic ones.
The first result we present is an asymptotic formula for the distance between the stable and unstable manifolds of L3. Due to the rapidly rotating dynamics, this distance is exponentially small with respect to $\sqrt\mu$ and, as a result, classical perturbative methods (i.e the Melnikov-Poincaré method) can not be applied.
The second result studies the family of Lyapunov periodic orbits of L3 with Hamiltonian energy level exponentially close (with respect to $\sqrt\mu$) to that of L3. In particular, we show that there exists a set of periodic orbits whose unstable and stable manifolds intersect transversally. By the Smale-Birkhoff homoclinic theorem, this implies the existence of chaotic motions (Smale horseshoe) exponentially close to L3 and its invariant manifolds. In addition, we also show the existence of a generic unfolding of a quadratic homoclinic tangency which leads to the existence of Newhouse domains for the RPC3BP.
This is a joint work with Inma Baldomá, Maciej J. Capinski and Marcel Guardia.
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Speaker: Gironella, Fabio (CNRS, Nantes Université)
Title: Confoliations in high dimensions.
Abstract: In a pioneering work of 1998, Eliashberg and Thurston proved that, in ambient dimension $3$, $C^2$ foliations can always (besides one example) be $C^0$-approximated by contact structures in the space of plane fields. To do so, they introduce what they called ``confoliations'', that are geometric structures which are mid-way between foliations and contact structures. While these have been studied extensively in dimension $3$ by many authors, the situation in higher dimensions is less clear, and there is not even a consensus on what their definition should be. In this talk, based on a joint work with Robert Cardona, I will give a new definition of high-dimensional (symplectic) confoliation. I will then describe how on one hand it naturally gives a notion of approximation by contact structures that encompasses all previously known ad-hoc examples of deformations of contact structures to foliations with leafwise symplectic-like structures, and on the other hand how they also constitute a good setup, generalizing the contact one, to study symplectic fillability questions.
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Speaker: Haim-Kizlev, Pazit (Institute for Advanced Study)
Title: A counterexample to Viterbo's conjecture.
Abstract: Symplectic capacities are invariants that quantify the size of symplectic manifolds using themes from Hamiltonian dynamics and symplectic topology. Although convexity is not preserved under symplectomorphisms, convex domains still display notable behavior with respect to these capacities. Viterbo's volume-capacity conjecture (2000) suggests that, among convex domains of equal volume, the ball has maximal capacity. By capturing the interplay between convex and symplectic geometries, this simply formulated conjecture has become highly influential in the study of symplectic capacities, prompting extensive research. In this talk, I will present a counterexample to Viterbo's conjecture developed jointly with Yaron Ostrover and discuss follow-up questions.
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Speaker: Humilière, Vincent (Sorbonne Université)
Title: Higher dimensional Birkhoff attractor.
Abstract: The Birkhoff attractor is a closed invariant subsets associated with any dissipative twist map of the annulus (of dimension 2), which was introduced by Birkhoff in 1932. We will see that it can be generalized to higher dimensions using tools from symplectic topology. The construction applies to any conformally symplectic diffeomorphism, including damped Hamiltonian systems. This is based on joint work with Marie-Claude Arnaud and Claude Viterbo.
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Speaker: Sorrentino, Alfonso (University of Rome Tor Vergata)
Title: Some new rigidity properties of Mather's beta function in Billiards and Geometry
Abstract: This talk explores spectral rigidity phenomena appearing in billiard dynamics, with a focus on convex planar billiard-like systems, which include Birkhoff, Symplectic, and Outer billiards. I will present some new rigidity results for the associated action spectrum, which are linked to pointwise properties of Mather's $\beta$-function. Notably, we will observe surprising distinctions in the case of Outer billiards. This work is joint with Misha Bialy and Stefano Baranzini.
Time permitting, I will also discuss analogous results for the stable norm of Riemannian metrics, based on joint work with Anna Florio and Martin Leguil.
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Speaker: Zavidovique, Maxime (Sorbonne Université)
Title: Discounted Hamilton-Jacobi equations with and without monotonicity.
Abstract: We are interested in (viscosity) solutions of Hamilton-Jacobi equations of the form $G( \lambda u_\lambda(x),x,D_x u_\lambda) = cst $ where $u_\lambda : M \to \mathbb{R}$ is a continuous function defined on a closed manifold and $G$ verifies convexity and growth conditions in the last variables. Such solutions cary invariant sets for the contact flow associated to $G$. The parameter $\lambda>0$ is aimed to be sent to $0$. It has been known that when $G$ is increasing in the first variable, $u_\lambda$ exists, is unique and the family converges as $\lambda \to 0$. We will explain that when this hypothesis is dropped, there can be non uniqueness of solutions $u_\lambda$ at $\lambda>0$ fixed. Moreover, there can be coexistence of converging families of solutions $(u_\lambda)_\lambda$ and diverging ones. (Collaboration with Davini, Ni and Yan)
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Speaker: ?? (??)
Title: ??
Abstract:
Short presentations
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Speaker: An, Yujia (Beijing Normal University)
Title: Gevrey KAM equilibria for quasi-periodic Frenkel-Kontorova models
Abstract: We consider the existence of quasi-periodic equilibrium configurations with Gevrey regularity for the Frenkel-Kontorova model with Gevrey-class potentials, extending the analytic results of Su and de la Llave in 2012. The model describes a one-dimensional chain of particles with nearest-neighbor interactions, where equilibrium states correspond to critical points of a formal energy functional. We establish an a posteriori KAM theorem showing that in the Gevrey topology, given an approximate solution of equilibrium equation, which satisfies some appropriate non-degeneracy conditions, then there is a true solution nearby and the solution preserves both the quasi-periodicity and Gevrey regularity. The method of proof is based on a combination of quasi-Newton methods and delicate estimates in spaces of Gevrey functions.
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Speaker: Charfi, Skander (Université Paris Cité)
Title: Recurrent Viscosity Solutions
Abstract: This short talk focuses on a key difference between weak KAM theory in the autonomous and non-autonomous settings. In the autonomous case, A. Fathi showed that the Lax-Oleinik semigroup, which generates viscosity solutions of the Hamilton-Jacobi equation, converges for any initial condition toward a stationary weak KAM solution. However, this convergence generally fails in the time-dependent case. To address this, we turn our attention to the limit viscosity solutions. We note that these are precisely the recurrent viscosity solutions, which emerge as the natural generalization of weak KAM solutions in this broader context.
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Speaker: Deschamps, Lina (Universität Heidelberg)
Title: Periodic Orbits and Contact Type Property in High-Dimensional Magnetic Systems
Abstract: This is joint work with Levin Maier and Tom Stalljohann. We introduce a class of magnetic systems on closed manifolds, called magnetic systems of strong geodesic type, for which there exists at least one null-homologuous embedded periodic orbit on each energy level, of negative energy below the lowest Mané critical value. This result partially answers a conjecture posed by Contreras-Iturriaga-Paternain-Paternain in the early 2000s.
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Speaker: Li, Wenyuan (University of Southern California)
Title: $C^0$-rigidity of Legendrian submanifolds
Abstract: Contact homeomorphisms are points in the closure of the contactomorphism group in the homeomorphism group under the $C^0$-topology. Recently, Dimitroglou Rizell and Sullivan showed that the images of closed Legendrians under contact homeomorphisms are still closed Legendrians if smooth. We will show that for closed Legendrians in cosphere bundles, vanishing of the Maslov class and certain Floer theory invariants is preserved under contact homeomorphisms, and when the conformal factors of contactomorphisms are uniformly bounded, the full Maslov data and Floer theory invariants are preserved. This is joint work in preparation with Tomohiro Asano, Yuichi Ike and Chris Kuo.
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Speaker: Nardi, Alessandra (Università di Padova)
Title: Symplectic billiards: integrability and spectral rigidity.
Abstract: Symplectic billiards were introduced by P. Albers and S. Tabachnikov as a dynamical system where, unlike Birkhoff billiards, the generating function is the area instead of the length. After briefly recalling the definition and main properties of symplectic billiards, this presentation aims to provide an overview of some recent results concerning integrability and (area-)spectral rigidity for this class of billiards. Based on joint work with L. Baracco and O. Bernardi.
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Speaker: Romero Mora, Cesar (Université Paris Dauphine PSL)
Title: Tangent cones of the Aubry set via regularity of sub-solutions for Tonelli Hamiltonians
Abstract: The Aubry set is a compact set that is invariant under the flow of a Tonelli Hamiltonian. The study of this set is central to Aubry-Mather theory and weak KAM theory due to the dynamical information that can be extracted from it. However, this set can be quite complicated from a geometric point of view. Using Green bundles and certain notions of tangent cones, Arnaud (2012) successfully proved results on the regularity of the Aubry set. In addition, she conjectured a more refined estimate, finally proven by Zhang (2020). In this talk, we explore an alternative proof of Zhang's result, based on a local regularization property of Lax-Oleinik operator on sub- solutions (from a global version introduced by Bernard (2007)) and an appropriate link between the Clarke Hessian of such sub-solutions and the cones studied by Arnaud.
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Speaker: Wang, Donghua (Beijing Normal University)
Title: Negative limit of weak KAM solutions for discrete discounted Hamilton-Jacobi equations
Abstract: Firstly, we establish the discrete version of the equations $$-\lambda u+H(x,d_xu)=c(H),$$ where $\lambda>0$ and $c(H)$ is the critical value. Then we show that there exists a minimal backward discrete weak KAM solution and a unique forward discrete weak KAM solution. We define the Aubry set for this system. Under some assumptions on Aubry class, we prove the convergence of the backward discrete weak KAM solutions as discount vanishing.
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Speaker: ?? (??)
Title: ??
Abstract:
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