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On Symplectic Dynamics

Volume 20, Number 2 (2017), 69 - 94

On Symplectic Dynamics

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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].