Issue 1, Volume 38, March 2013

2013

1. Preface Preface Viktor Ginzburg , Eva Miranda , Do Duc Thai , Nguyen Tien Zung PDF

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2. Normalisation of the smooth focus–focus: a simple proof Marc Chaperon PDF

   ▸ Abstract

   We prove that the smooth focus–focus can be normalised. In Appendix B, Jiang Kai shows that the smooth “node–node” can not.

   3-9
3. A sufficient condition for the existence of Hamiltonian bifurcations with continuous isotropy James Montaldi , Miguel Rodríguez-Olmos PDF

   ▸ Abstract

   We present a framework for the study of the local qualitative dynamics of equivariant Hamiltonian flows specially designed for points in phase space with non-trivial isotropy. This is based on the classical construction of structure-preserving tubular neighborhoods for Hamiltonian Lie group actions on symplectic manifolds. This framework is applied to obtaining concrete and testable conditions guaranteeing the existence of bifurcations from symmetric branches of Hamiltonian relative equilibria.

   11-19
4. Coupling symmetries with Poisson structures Camille Laurent-Gengoux , Eva Miranda PDF

   ▸ Abstract

   We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein’s splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered.

   2-32
5. Smooth structures on pseudomanifolds with isolated conical singularities Hông Vân Lê , Petr Somberg , Jiří Vanžura PDF

   ▸ Abstract

   In this note we introduce the notion of a smooth structure on a conical pseudomanifold $M$ in terms of $C^{\infty}$-rings of smooth functions on $M$. For a finitely generated smooth structure $C^{\infty}(M)$ we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of $M$, and the notion of characteristic classes of $M$. We prove the vanishing of a Nash vector field at a singular point for a special class of Euclidean smooth structures on $M$. We introduce the notion of a conical symplectic form on $M$ and show that it is smooth with respect to a Euclidean smooth structure on $M$. If a conical symplectic structure is also smooth with respect to a compatible Poisson smooth structure $C^{\infty}(M)$, we show that its Brylinski–Poisson homology groups coincide with the de Rham homology groups of $M$. We show nontrivial examples of these smooth conical symplectic-Poisson pseudomanifolds.

   33-54
6. Closed Reeb orbits on the sphere and symplectically degenerate maxima Viktor L. Ginzburg , Doris Hein , Umberto L. Hryniewicz , Leonardo Macarini PDF

   ▸ Abstract

   We show that the existence of one simple closed Reeb orbit of a particular type (a symplectically degenerate maximum) forces the Reeb flow to have infinitely many periodic orbits. We use this result to give a different proof of a recent theorem of Cristofaro-Gardiner and Hutchings asserting that every Reeb flow on the standard contact three-sphere has at least two periodic orbits. Our methods are based on adapting the machinery originally developed for proving the Hamiltonian Conley conjecture to the contact setting.

   55-78
7. Geometry of integrable dynamical systems on 2-dimensional surfaces Nguyen Tien Zung , Nguyen Van Minh PDF

   ▸ Abstract

   This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous invariants involved in this classification are the left equivalence classes of period or monodromy functions and the cohomology classes of period cocycles, which can be expressed in terms of Puiseux series. We also study the problem of Hamiltonianization of these integrable vector fields by a compatible symplectic or Poisson structure.

   79-016
8. Smooth normal forms for integrable Hamiltonian systems near a focus–focus singularity San Vũ Ngọc , Christophe Wacheux PDF

   ▸ Abstract

   We prove that completely integrable systems are normalizable in the $C^{\infty}$ category near a focus–focus singularity.

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9. Uniqueness of symplectic structures Dietmar Salamon PDF

   ▸ Abstract

   This survey paper discusses some uniqueness questions for symplectic forms on compact manifolds without boundary.

   123-144
10. Fifteen years of contact circles and contact spheres Hansjörg Geiges , Jesús Gonzalo Pérez PDF

    ▸ Abstract

    Contact circles and contact spheres are geometric structures on 3-manifolds introduced by the authors some fifteen years ago. In this survey we review the main results about these structures and we include two as yet unpublished proofs.

    145-164
11. The Lagrange reduction of the N-body problem, a survey Alain Chenciner PDF

    ▸ Abstract

    In his fundamental “Essai sur le problème des trois corps” (Oeuvres, vol. 6, pp. 229–324, 1772), Lagrange, well before Jacobi’s “reduction of the node”, carries out the first complete reduction of symmetries in this problem. Discovering the so-called homographic motions (Euler had treated only the collinear case), he shows that these motions necessarily take place in a fixed plane, a result which is simple only for the “relative equilibria”. In order to understand the true nature of this reduction—and of Lagrange’s equations—it is necessary to consider the n-body problem in an euclidean space of arbitrary dimension. The actual dimension of the ambient space then appears as a constraint, namely the angular momentum bivector’s degeneracy. I describe in detail the results obtained in a joint paper with Alain Albouy published in French in 1998 (Albouy and Chenciner in Invent. Math. 131:151–184, 1998): for a non-homothetic homographic motion to exist, it is necessary that the motion takes place in an even-dimensional space. Two cases are possible: either the configuration is “central” (that is, we have a critical point of the potential among configurations with a given moment of inertia) and the space where the motion takes place is endowed with a hermitian structure, or it is “balanced” (that is, we have a critical point of the potential among configurations with a given inertia spectrum) and the motion is a new type, quasi-periodic, of relative equilibrium. Only the first type is of Kepler type and hence corresponds to the absolute minimum in Sundman’s inequality. When the space of motion is odd-dimensional, one can look for a substitute to the non-existing homographic motions: a candidate is the family of Hip-Hop solutions, which are “simple” periodic solutions naturally related to relative equilibria through Lyapunov families of quasi-periodic solutions (see Chenciner and Féjoz in Regul. Chaotic Dyn. 14(1):64–115, 2009). Finally, some words are said on the bifurcation of periodic central relative equilibria to quasi-periodic balanced ones.

    165-186
12. The second main theorem for meromorphic mappings into a complex projective space Do Phuong An , Si Duc Quang , Do Duc Thai PDF

    ▸ Abstract

    The main purpose of this article is to show the Second Main Theorem for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(C)$ intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give two unicity theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing few hypersurfaces without counting multiplicity.

    187-205