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.


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
Full Text: Euclid