Controllability on the group of diffeomorphisms

Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of the identity of the group of diffeomorphisms of M.     

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.     

A two-cocycle on the group of symplectic diffeomorphisms

We investigate the properties of a two-cocycle on the group of symplectic diffeomorphisms of an exact symplectic manifold defined by Ismagilov, Losik, and Michor. We provide both vanishing and nonvanishing results and applications to foliated symplectic bundles and to Hamiltonian actions of finitely generated groups.     

Polynomial diffeomorphisms ofC 2

Partially supported by NSF grant #DMS-8602020     

Minimal heteroclinic geodesics for the n-torus

This paper gives an extension of earlier work of Morse and of Hedlund on minimal heteroclinic geodesics for to the case of provided that an additional geometrical condition is satisfied. It also gives lower bounds on the number of such geodesics.     

The theta group and the continued fraction expansion with even partial quotients

F. Schweiger introduced the continued fraction with even partial quotients. We will show a relation between closed geodesics for the theta group (the subgroup of the modular group generated by z+2 and -1 / z) and the continued fraction with even partial quotients. Using thermodynamic formalism, Tauberian results and the above-mentioned relation, we obtain the asymptotic growth number of closed trajectories for the theta group. Several results for the continued fraction expansion with even partial quotients are obtained; some of these are analogous to those already known for the usual continued fraction expansion related to the modular group, but our proofs are by necessity in general technically more difficult.Supported by The Netherlands Organization for Scientific Research (NWO).     

Equivariant homotopy and deformations of diffeomorphisms

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.     

Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions

In this paper, we use Chas–Sullivan theory on loop homology and Leray–Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite.     

Recursion and group structures of soliton equations

The recursion operator method for nonlinear evolution equations integrable by the inverse spectral transform method is discussed. This method enables us to present the integrable equations in a compact and convenient form and to construct the infinite-dimensional groups of general Bäcklund transformations and the infinite-dimensional symmetry groups for these equations. Adjoint representation of the spectral problems plays a central role in the recursion operator method. Nonlinear integrable equations in 1+1 and 1+2 dimensions are considered.     

On the existence of closed magnetic geodesics via symplectic reduction

Let (M, g) be a closed Riemannian manifold and \(\sigma \) be a closed 2-form on M representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair \((g,\sigma )\) can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle \(T^*E\) of a suitable \(S^1\)-bundle E over M or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of E. We thenstudy the relation between the stability property of energy hypersurfacesin \((T^*M,dp\wedge dq+\pi ^*\sigma )\) and of the corresponding codimension2 coisotropic submanifolds in \((T^*E,dp\wedge dq)\) arising via symplecticreduction. Finally, we reprove the main result of Asselle and Benedetti (J Topol Anal 8(3):545–570, 2016) in this setting.     

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     

Correction and linearization of resonant vector fields and diffeomorphisms

We extend the classical Siegel-Brjuno-Rüssmann linearization theorem to the resonant case by showing that under A. D. Brjuno's diophantine condition, any resonant local analytic vector field (resp. diffeomorphism) possesses a well-defined correction which (1) depends on the chart but, in any given chart, is unique (2) consists solely of resonant terms and (3) has the property that, when substracted from the vector field (resp. when factored out of the diffeomorphism), the vector field or diffeomorphism thus “corrected” becomes analytically linearizable (with a privileged or “canonical” linearizing map). Moreover, in spite of the small denominators and contrary to a hitherto prevalent opinion, the correction's analyticity can be established by pure combinatorics, without any analysis. Received January 7, 1997; in final form April 22, 1997     

Harmonic cohomology of symplectic manifolds

Mathieu (Math. Helv. 70 (1995) 1) introduced a canonic filtration in the de Rham cohomology of a symplectic manifold and proved, that the middle filtration space is the space of harmonic cohomology classes. We give an interpretation of the other filtration spaces, prove a Künneth theorem for harmonic cohomology, prove Poincaré duality for harmonic cohomology and show how surjectivity of certain Lefschetz type mappings is related to properties of the filtration. For a closed symplectic manifold M we also introduce symplectic invariants , and show if M is of dimension 2n with n even.