Exotic smooth structures and symplectic forms on closed manifolds

In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.     

The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

KAM theorem of symplectic algorithms for Hamiltonian systems

Summary. In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel (1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system if the system is analytic and the time-step size of the algorithm is s ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system, possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given. Received December 27, 1997 / Revised version received July 15, 1998 / Published online: July 7, 1999     

Pseudotoric structures on toric symplectic manifolds

We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds.     

Bifurcation of a Whitney-Smooth Family of Coisotropic Invariant Tori of a Hamiltonian System under Small Deformations of a Symplectic Structure

We investigate the influence of small deformations of a symplectic structure and perturbations of the Hamiltonian on the behavior of a completely integrable Hamiltonian system. We show that a Whitney-smooth family of coisotropic invariant tori of the perturbed system emerges in the neighborhood of a certain submanifold of the phase space.     

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un)stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et al. (J Diff Equ, 2012), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show that the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.     

Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 71–98, January, 2007.     

Symplectic tori in rational elliptic surfaces

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive 2-dimensional homology class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface . We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.     

An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in {\mathbb C}^2 related to Oh's conjecture

We compute loops integrals on Hamiltonian stationary Lagrangian tori in which are symplectic invariants, then we show an isoperimetric inequality involving these invariants and the area. Finally, we show that the flat torus has least area among Hamiltonian stationary Lagrangian tori of its isotopy class. Received: 4 December 2000; in final form: 18 January 2002 / Published online: 5 September 2002     

Remarks on derived equivalences of Ricci-flat manifolds

After a finite étale cover, any Ricci-flat Kähler manifold decomposes into a product of complex tori, irreducible holomorphic symplectic manifolds, and Calabi–Yau manifolds. We present results indicating that this decomposition is an invariant of the derived category. The main idea to distinguish the derived category of an irreducible holomorphic symplectic manifold from that of a Calabi–Yau manifold is that point sheaves do not deform in certain (non-commutative) deformations of the former, whereas they do for the latter. On the way, we prove a conjecture of C?ld?raru on the module structure of the Hochschild–Kostant–Rosenberg isomorphism for manifolds with trivial canonical bundle as a direct consequence of recent work by Calaque, van den Bergh, and Ramadoss.     

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.     

Preservation and destruction of periodic orbits by symplectic integrators

We investigate what happens to periodic orbits and lower-dimensional tori of Hamiltonian systems under discretisation by a symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous to what is understood for one degree of freedom systems.     

Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: Given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities, we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer-assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to \(\varepsilon =0.9716\)), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.     

Four-Dimensional Simply Connected Symplectic Symmetric Spaces

A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ) belonging to this class is diffeomorphic to Rⁿwith the property that every tensor field on M invariant by the transvection group is constant; in particular, is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rⁿwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.     

Separatrix splitting from the point of view of symplectic geometry

Generally, the invariant Lagrangian manifolds (stable and unstable separatrices) asymptotic with respect to a hyperbolic torus of a Hamiltonian system do not coincide. This phenomenon is called separatrix splitting. In this paper, a symplectic invariant qualitatively describing separatrix splitting for hyperbolic tori of maximum (smaller by one than the number of degrees of freedom) dimension is constructed. The construction resembles that of the homoclinic invariant found by lazutkin for two-dimensional symplectic maps and of Bolotin's invariant for splitting of asymptotic manifolds of a fixed point of a symplectic diffeomorphism. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 890–906, June, 1997. Translated by O. V. Sipacheva     

Contact cuts

We describe a contact analog of the symplectic cut construction [L]. As an application we show that the group of contactomorphisms for certain overtwisted contact structures on lens spaces contains countably many non-conjugate two tori. Supported by the NSF grant DMS-980305.     

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.