## 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

### Citations:

Zbl 1198.53089; Zbl 1144.37033; Zbl 1066.53137
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