African Diaspora Journal of Mathematics

On Symplectic Dynamics

Stéphane Tchuiaga

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Abstract

This paper continues to carry out a foundational study of Banyaga's topologies of a closed symplectic manifold $(M,\omega)$ [4]. Our intention in writing this paper is to work out several “symplectic analogues” of some results found in the study of Hamiltonian dynamics. By symplectic analogue, we mean if the first de Rham's group (with real coefficients) of the manifold is trivial, then the results of this paper reduce to some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out an impact of the $L^\infty-$version of Hofer-like length in the investigation of the symplectic nature of the $C^0 -$limit of a sequence of symplectic maps. This yields a symplectic analogue of a result that was proved by Hofer-Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$); then reformulated by Oh-Müller [14] for Hamiltonian diffeomorphisms in general. Furthermore, we show that Polterovich's regularization process for Hamiltonian paths extends over the whole group of symplectic isotopies, and then use it to prove the equality between the two versions of Hofer-like norms. This yields the symplectic analogue of the uniqueness result of Hofer's geometry proved by Polterovich [13]. Our results also include the symplectic analogues of some approximation lemmas found by Oh-Müller [14]. As a consequence of a result of this paper, we prove by other method a result found by McDuff-Salamon [12].

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 20, Number 2 (2017), 69-94.

Dates
First available in Project Euclid: 7 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1496800891

Mathematical Reviews number (MathSciNet)
MR3664476

Zentralblatt MATH identifier
1372.53082

Subjects
Primary: 53D05: Symplectic manifolds, general 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 57R52: Isotopy 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Keywords
flux geometry geodesics Hofer metrics Hofer-like metrics Hodge's theory homotopy isotopies injectivity radius symplectic diffeomorphisms differential forms vector fields

Citation

Tchuiaga, Stéphane. On Symplectic Dynamics. Afr. Diaspora J. Math. (N.S.) 20 (2017), no. 2, 69--94. https://projecteuclid.org/euclid.adjm/1496800891


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