×

On symplectic dynamics. (English) Zbl 1372.53082

The present paper is a continuation of A. Banyaga’s topologies of a closed symplectic manifold \((M,\omega)\) [Contemp. Math. 512, 1–23 (2010; Zbl 1198.53089)]. Two main results are obtained here. The first one states that without appealing to the positivity of the symplectic displacement energy, one can use a \(L^{\infty}\)-version of the Hofer-like length to investigate the symplectic nature of the \(C^0\)-limit of a sequence of symplectic diffeomorphisms. The second main result shows that the Hofer-like geometry is independent on the choice of the Hofer-like norm. The symplectic analogues of some approximation lemmas of [Y.-G. Oh and S. Müller, J. Symplectic Geom. 5, No. 2, 167–219 (2007; Zbl 1144.37033)] are studied here. As a consequence of the present study, a result of D. McDuff and D. Salamon [Introduction to symplectic topology. 2nd ed. New York, NY: Oxford University Press (1998; Zbl 1066.53137)] on the contractibility of the orbits of Hamiltonian loops is proved by an other method.

MSC:

53D05 Symplectic manifolds (general theory)
53D35 Global theory of symplectic and contact manifolds
57R52 Isotopy in differential topology
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
PDF BibTeX XML Cite
Full Text: Euclid