Weyl-Titchmarsh theory for symplectic difference systems

In this work, we establish Weyl-Titchmarsh theory for symplectic difference systems. This paper extends classical Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of symplectic difference systems.     

Linearization-preserving self-adjoint and symplectic integrators

This article is concerned with geometric integrators which are linearization-preserving, i.e. numerical integrators which preserve the exact linearization at every fixed point of an arbitrary system of ODEs. For a canonical Hamiltonian system, we propose a new symplectic and self-adjoint B-series method which is also linearization-preserving. In a similar fashion, we show that it is possible to construct a self-adjoint and linearization-preserving B-series method for an arbitrary system of ODEs. Some numerical experiments on Hamiltonian ODEs are presented to test the behaviour of both proposed methods. This work was supported by the Australian Research Council and by the Marsden Fund of the Royal Society of New Zealand.     

Creation of twistless circles in a model of stellar pulsations

In the present paper, we study the Poincaré map associated to a periodic perturbation, both in space and time, of a linear Hamiltonian system. The dynamical system embodies the essential physics of stellar pulsations and provides a global and qualitative explanation of the chaotic oscillations observed in some stars. We show that this map is an area preserving one with an oscillating rotation number function. The nonmonotonic property of the rotation number function induced by the triplication of the elliptic fixed point is superposed on the nonmonotonic character due to the oscillating perturbation. This superposition leads to the co-manifestation of generic phenomena such as reconnection and meandering, with the nongeneric scenario of creation of vortices. The nonmonotonic property due to the triplication bifurcation is shown to be different from that exhibited by the cubic Hénon map, which can be considered as the prototype of area preserving maps which undergo a triplication followed by the twistless bifurcation. Our study exploits the reversibility property of the initial system, which induces the time-reversal symmetry of the Poincaré map.     

Invariant norm quantifying nonlinear content of Hamiltonian systems

Given a Hamiltonian system, one can represent it using a symplectic map. This symplectic map is specified by a set of homogeneous polynomials which are uniquely determined by the Hamiltonian. In this paper, we construct an invariant norm in the space of homogeneous polynomials of a given degree. This norm is a function of parameters characterizing the original Hamiltonian system. Such a norm has several potential applications.     

Analytic smoothing of geometric maps with applications to KAM theory

We show that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in Cr-norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. The symplectic, volume-preserving and contact forms are assumed to be analytic.     

Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.     

Construction of symplectic graphs by using 2-dimensional symplectic non-isotropic subspaces over finite fields

In most extant studies, symplectic graphs are defined by 1-dimensional subspaces and their orthogonality. In this paper, the symplectic graph is defined by 2-dimensional non-isotropic subspaces and their intersection. The symplectic graph is shown to be a 4-Deza graph. While the first subconstituent is shown to be a 4-Deza graph when $\nu ⩾3$ and a 3-Deza graph when $\nu =2$, the second subconstituent is not regular, and the third subconstituent is a 4-Deza graph except in the case $\nu =2$, when it is empty.     

Smoothness of isotopy for symplectic pairs

Let M be a 2n-dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion.     

Frankel's theorem in the symplectic category

We prove that if an -dimensional torus acts symplectically on a -dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel's theorem which says that a Kähler circle action has a fixed point if and only if it is Hamiltonian. The case of is the well-known theorem by McDuff.

     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

The Dirac operator on symplectic spinors

Symplectic spinors were introduced by B. Kostant in [4] in the context of geometric quantization. This paper presents further considerations concerning symplectic spinors. We define the spinor derivative induced by a symplectic covariant derivative. We compute an explicit formula for this spinor derivative and prove some elementary properties. This makes it possible to define the symplectic Dirac operator in a canonical way. In case of a symplectic and torsion-free covariant derivative it turns out to be formally selfadjoint.     

Generalized Bergman kernels on symplectic manifolds

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As a consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut and Lebeau.     

Neumann and Zagier's symplectic relations

We give a new proof of Neumann and Zagier's symplectic relations. This approach was originally suggested by Thurston.     

On the asphericity of a symplectic

C. H. Taubes asked whether a closed (i.e. compact and without boundary) connected oriented three-dimensional manifold whose product with a circle admits a symplectic structure must fiber over a circle. An affirmative answer to Taubes' question would imply that any such manifold either is diffeomorphic to the product of a two-sphere with a circle or is irreducible and aspherical. In this paper, we prove that this implication holds up to connect sum with a manifold which admits no proper covering spaces with finite index. It is pointed out that Thurston's geometrization conjecture and known results in the theory of three-dimensional manifolds imply that such a manifold is a three-dimensional sphere. Hence, modulo the present conjectural picture of three-dimensional manifolds, we have shown that the stated consequence of an affirmative answer to Taubes' question holds.

     

Base subsets of symplectic Grassmannians

Let V and V′ be 2n-dimensional vector spaces over fields F and F′. Let also Ω: V× V→ F and Ω′: V′× V′→ F′ be non-degenerate symplectic forms. Denote by Π and Π′ the associated (2n−1)-dimensional projective spaces. The sets of k-dimensional totally isotropic subspaces of Π and Π′ will be denoted by and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect and by so-called base subsets. We show that every mapping of to sending base subsets to base subsets is induced by a symplectic embedding of Π to Π′.     

An obstruction to quantizing compact symplectic manifolds

We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.

     

Immersions in a symplectic manifold

In this paper we give a homotopy classification of symplectic isometric immersions following Gromov’sh-principle theorem.     

On symplectic transformations

This is an English translation¹ of the Ph.D. thesis ‘Over symplectische transformaties’ that Tonny Albert Springer, ‘born in’s-Gravenhage in 1926’, submitted as thesis for – as is stated on the original frontispiece – the degree of doctor in mathematics and physics at Leiden University on the authority of the rector magnificus Dr. J.H. Boeke, professor in the faculty of law, to be defended against the objections of the Faculty of Mathematics and Physics on Wednesday October 17 1951 at 4 p.m., with promotor Prof. dr. H. D. Kloosterman.     

Lempert mappings and symplectic forms

We use Lempert's version of Riemann mapping to construct non-equivalent symplectic forms on an ellipsoid in .

     

Relative flux homomorphism in symplectic geometry

In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We also show that some quotients of the universal covering of the group of symplectomorphisms are stable under symplectic reduction.