On symplectic dynamics (Q5918215)

The present paper is a continuation of \textit{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 [\textit{Y.-G. Oh} and \textit{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 \textit{D. McDuff} and \textit{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.