Group Of Symplectic Diffeomorphisms Research Articles

Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group

Let $M$ be a closed symplectic manifold with compatible symplectic form and Riemannian metric $g$. Here it is shown that the exponential mapping of the weak $L^{2}$ metric on the group of symplectic diffeomorphisms of $M$ is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the $L^{2}$ metric and Hofer's metric as well as an intriguing difference between the $L^{2}$ geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.

A cocycle on the group of symplectic diffeomorphisms

Abstract We define a cocycle on the group of Hamiltonian diffeomorphisms of a symplectically aspherical manifold and investigate its properties. The main application is an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms.

Hamiltonian pseudo-representations

The question studied here is the behavior of the Poisson bracket under C^0-perturbations. In this purpose, we introduce the notion of pseudo-representation and prove that for a normed Lie algebra, it converges to a representation. An unexpected consequence of this result is that for many non-closed symplectic manifolds (including cotangent bundles), the group of Hamiltonian diffeomorphisms (with no assumptions on supports) has no C^{-1} bi-invariant metric. Our methods also provide a new proof of Gromov-Eliashberg Theorem, it is to say that the group of symplectic diffeomorphisms is C^0-closed in the group of all diffeomorphisms.

New minimal geodesics in the group of symplectic diffeomorphisms

We compute the Hofer distance for a certain class of compactly supported symplectic diffeomorphisms of ℝ2n. They are mainly characterized by the condition that they can be generated by a Hamiltonian flow ϕHt which possesses only constantT-periodic solutions for 0 <T ≤ 1. In addition, we show that on this class Hofer's and Viterbo's distances coincide.

New minimal geodesics in the group of symplectic diffeomorphisms

New minimal geodesics in the group of symplectic diffeomorphisms

Unipotent normal forms for symplectic maps

A normal form theory for symplectic maps with non-diagonalizable linear part is presented. The theory is obtained by working directly with the group of symplectic diffeomorphisms. Using the fact that non-diagonalizable linear symplectic maps have a unique semisimple-unipotent decomposition, the normalization of a symplectic map-with respect to its linear part-proceeds in two steps. First a semisimple normalization, for which the theory is well-established, and second, a unipotent normalization which is presented here. The role of normal form symmetry and reduction of the normalized map is also considered. The theory is applied to two examples of symplectic maps on R 4: first, a non-semisimple multiple -1 eigenvalue, and second, the collision of Floquet multipliers of opposite sign.

The diameter of the symplectomorphism group is infinite

Shnirelman has proved that the group of volume-preversing diffeomorphisms of the cube in R 3 has finite diameter and has announced the result that this is false for the square. Despite the fact that Shnirelman formulated his theorem only for the 3-dimensional cube, his proof can be modified for the case of the group of volume preserving diffeomorphisms of any compact simply-connected Riemannian manifold of dimension > 2. It turns out that the situation with the group of symplectic diffeomorphisms is completely different. We prove that the diameter of the symplectomorphism group of any compact exact symplectic manifold (necessarily with boundary) is infinite

Integrable equations, related with the Poisson algebra

The general r-matrix scheme for the construction of integrable Hamiltonian systems is applied to a Poisson algebra, i.e., the algebra of functions on a symplectic manifold, the commutator in which is defined by the Poisson bracket. Integrable systems of two types are constructed: Hamiltonian systems on the group of symplectic diffeomorphisms, whose Hamiltonians are sums of a leftinvariant kinetic energy and a potential, and first-order systems of equations for two functions of one variable.

A biinvariant metric on the group of symplectic diffeomorphisms and the equations (?/?t)?F={?F, F}

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the symplectic diffeomorphisms group $\mathcal{D}_\omega$ of a symplectic Riemannian manifold $(M,g,\omega)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_\omega$ and calculate the sectional curvature of $\mathcal{D}_\omega (T^n)$.

Equivalence and slice theory for symplectic forms on closed manifolds

In this paper, a study is made of the pullback action of the diffeomorphism group on the totality of symplectic forms on a compact manifold. For this action, the orbit is shown to be a smooth (Banach) manifold consisting of a denumerable union of submanifolds, each lying in a fixed cohomology class. In addition, a precise characterization is given of those symplectic manifolds for which there is a local factorization of the pullback action in the sense of a transverse “slice” of closed 2-forms, invariant under the group of symplectic diffeomorphisms.