The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits.
We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action--index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of the n-dimensional complex projective space with exactly n+1 periodic orbits a stronger result holds. Namely, for such a Hamiltonian diffeomorphism, the difference of the action and the mean index on a periodic orbit is independent of the orbit, provided that the symplectic structure on the projective space is normalized to be in the same cohomology class as the first Chern class.
We show that the presence of one non-degenerate, non-contractible periodic orbit of a Hamiltonian on the standard symplectic torus implies the existence of infinitely many simple non-contractible periodic orbits.
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*\T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in \cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*\T^2$. This solves a problem of Arnold in \cite{A}.
We describe a variational approach to a notion of Hamiltonian delay equations. Our delay Hamiltonians are of product form. We consider several examples. For closed symplectically aspherical symplectic manifolds $(M,ω)$ we prove that for generic delay Hamiltonians the number of 1-periodic solutions of the Hamiltonian delay equation is at least the sum of the Betti numbers of $M$, extending the proof of the Arnold conjecture to the case with delay.
Inspired by the techniques of Givental and Théret, we provide a proof with generating functions of a recent result of Ginzburg-Gürel concerning the periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}\text{P}^d$. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely many periodic points.
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 first author introduced a relative symplectic capacity $C$ for a symplectic manifold $(N,ω_N)$ and its subset $X$ which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of $N$ with the annulus $A_R=(R,R)\times\mathbb{R}/\mathbb{Z}$. In the present paper, we give an exact computation of the capacity $C$ of the $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower bound on …
We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping…
We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theorem to study bifurcation of branches of periodic solutions for families of nonlinear Hamiltonian systems.
The unimodality conjecture posed by Tolman in the conference `Moment maps in Various Geometry" in 2005 states that if (M,w) is a 2n-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers is unimodal. Recently, the author and M. Kim proved that the unimodality holds in eight-dimensional cases by using equivariant cohomology theory. In this paper, we generalize the idea in \cite{CK} to an arbitrary dimensional case. Also, we prove the conjecture in arbitrary dimension with an assumption that a m…
In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the Calabi invariant of the locally Hamiltonian equations. Our result improves lower bounds obtained by Lê-Ono and Ono for the least number of nondegenerate locally Hamiltonian symplectic fixed points. Our result also generalizes the homological Arnold conjecture proved by Fukaya-Ono and Liu-Tian.
We study the spectrum of the Schrödinger operators with $n\times n$ matrix valued potentials on a finite interval subject to $θ-$periodic boundary conditions. For two such operators, corresponding to different values of $θ$, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to $θ$ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a …
We consider a natural class of time-periodic infinite-dimensional nonlinear Hamiltonian systems modelling the interaction of a classical mechanical system of particles with a scalar wave field. When the field is defined on a space torus $\mathbb{T}^d=\mathbb{R}^d/(2π\mathbb{Z})^d$ and the coordinates of the particles are constrained to a submanifold $Q\subset\mathbb{T}^d$, we prove that the number of $T$-periodic solutions of the coupled Hamiltonian particle-field system is bounded from below by the $\mathbb{Z}_2$-cuplength of the space of contractible loops in $Q$, provided that the square of…
