An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.     

The lagrange intersections for (ℂP n , ℝP n )

Summary If (M, ω) is a compact symplectic manifold andL ⊂M a compact Lagrangian submanifold and if φ is a Hamiltonian diffeomorphism ofM then the V. Arnold conjecture states (possibly under additional conditions) that the number of intersection section points ofL and φ (L) can be estimated by #{Lϒφ (L)}≥ cuplength +1. We shall prove this conjecture for the special case (L, M)=(ℝP ⁿ , ℂP ⁿ ) with the standard symplectic structure.     

Fixed points of non-Hamiltonian symplectomorphisms

In this article we study a version of the Arnold conjecture for symplectic maps that are not Hamiltonian. That is, we give a lower bound for the number of fixed points such a map must have. We achieve the result for symplectic maps with sufficiently small Calabi invariant.     

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.     

Conley conjecture and local Floer homology

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit, the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold, the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.     

Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results. Partially supported by the Swiss National Foundation. Supported by the Swiss National Foundation and the von Roll Research Foundation.     

Surgery on Lagrangian and Legendrian Singularities

Let be a smooth fiber bundle whose total space is a symplectic manifold and whose fibers are Lagrangian. Let L be an embedded Lagrangian submanifold of E. In the paper we address the following question: how can one simplify the singularities of the projection by a Hamiltonian isotopy of L inside E? We give an answer in the case when dim and both L and M are orientable. A weaker version of the result is proved in the higher-dimensional case. Similar results hold in the contact category.?As a corollary one gets an answer to one of the questions of V. Arnold about the four cusps on the caustic in the case of the Lagrangian collapse. As another corollary we disprove Y. Chekanov's conjecture about singularities of the Lagrangian projection of certain Lagrangian tori in . Submitted: January 1998, revised: January 1999.     

Lagrangian isotopy of tori in $${S^2\times S^2}$$ and $${{\mathbb{C}}P^2}$$CP2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space \({{\mathbb{R}}^4}\), the projective plane \({{\mathbb{C}}P^2}\), and the monotone \({S^2 \times S^2}\). The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for \({T^*{\mathbb{T}}^2}\), i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.     

On Anosov energy levels of hamiltonians on twisted cotangent bundles

LetT ^* M denote the cotangent bundle of a manifoldM endowed with a twisted symplectic structure [1]. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by a convex HamiltonianH: T ^* M, and we consider a compact regular energy level ofH, on which this flow admits a continuous invariant Lagrangian subbundleE. When dimM3, it is known [9] that such energy level projects onto the whole manifoldM, and thatE is transversal to the vertical subbundle. Here we study the case dimM=2, proving that the projection property still holds, while the transversality property may fail. However, we prove that in the case whenE is the stable or unstable subbundle of an Anosov flow, both properties hold.     

Engouffrement symplectique et intersections lagrangiennes

Let }L _t{,t ∈ [0, 1], be a path of exact Lagrangian submanifolds in an exact symplectic manifold that is convex at infinity and of dimension ≥6. Under some homotopy conditions, an engulfing problem is solved: the given path }L _t{ is conjugate to a path of exact submanifolds inT ^*L_o. This impliesL _t must intersectL _o at as many points as known by the generating function theory. Our Engulfing theorem depends deeply on a new flexibility property of symplectic structures which is stated in the first part of this work.

     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

Connections on manifolds and new characteristic classes

We give an extension of Maslov-Arnold classes to a certain class of symplectic manifolds. It is proved that any such generalized class of minimal surfaces is equal to zero for a large class of stable minimal surfaces. We describe some applications to pseudo-Riemannian geometry and to the investigation of completely integrable Hamiltonian systems.     

The Hörmander and Maslov Classes and Fomenko's Conjecture

Some functorial properties are studied for the Hörmander classes defined for symplectic bundles. The behavior of the Chern first form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko's conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.     

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.     

Lie solutions of Riemannian transport equations on compact manifolds

Given a couple of smooth positive measures of same total mass on a compact Riemannian manifold, the associated optimal transport equation admits a symplectic Monge-Ampère structure, hence Lie solutions (in a restricted sense, though, still expressing measure-transport). Properties of such solutions are recorded; a structure result is obtained for regular ones (each consisting of a closed 1-form composed with a diffeomorphism) and a quadratic cost-functional proposed for them.     

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

Summary. Generalizing the degenerate KAM theorem under the Rüssmann nondegeneracy and the isoenergetic KAM theorem, we employ a quasilinear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth submanifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the submanifold, we shall show the following: (a) the majority of the unperturbed tori on the submanifold will persist; (b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; (c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a subisoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.     

On the usefulness of an index due to Leray for studying the intersections of Lagrangian and symplectic paths

Using the ideas of Keller, Maslov introduced in the mid-1960's an index for Lagrangian loops, whose definition was clarified by Arnold. Leray extended Arnold results by defining an index depending on two paths of Lagrangian planes with transversal endpoints. We show that the combinatorial and topological properties of Leray's index suffice to recover all Lagrangian and symplectic intersection indices commonly used in symplectic geometry and its applications to Hamiltonian and quantum mechanics. As a by-product we obtain a new simple formula for the Hörmander index, and a definition of the Conley–Zehnder index for symplectic paths with arbitrary endpoints. Our definition leads to a formula for the Conley–Zehnder index of a product of two paths.     

Symplectic integration of Hamiltonian wave equations

Summary The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which the best results can be obtained; in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.     

Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds

We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its “boundary depth,” and establish basic results about how the boundary depths of different Hamiltonians are related. As applications, we prove that certain Hamiltonian symplectomorphisms supported in displaceable subsets have infinitely many nontrivial geometrically distinct periodic points, and we also significantly expand the class of coisotropic submanifolds which are known to have positive displacement energy. For instance, any coisotropic submanifold of contact type (in the sense of Bolle) in any closed symplectic manifold has positive displacement energy, as does any stable coisotropic submanifold of a Stein manifold. We also show that any stable coisotropic submanifold admits a Riemannian metric that makes its characteristic foliation totally geodesic, and that this latter, weaker, condition is enough to imply positive displacement energy under certain topological hypotheses.     

Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.