On symplectic dynamics

This paper contributes to the foundation of various Hofer-like topologies of a closed symplectic manifold (M,ω). Here, considering an arbitrarily linear section of the natural projection of the space of closed 1-forms onto the first de Rham's group, we study various Hofer-like metrics [2], [7]. The outcome is that different splitting of the space of closed 1-forms lead to similar “symplectic analogues” of some results from Hamiltonian dynamics: Without appealing to the positivity result of any symplectic displacement energy, we point out an impact of the L∞-Hofer-like lengths in the investigation of the symplectic nature of the uniform limits of sequences of symplectic diffeomorphisms isotopic to the identity map: This provides various symplectic analogues of a result that was proved by Hofer-Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on R2n); and reformulated by Oh-Müller [13] for Hamiltonian diffeomorphisms in general. Moreover, we show that Polterovich's regularization method for Hamiltonian paths extends over the whole group of symplectic paths (no matter the choice of the above section), and use this to prove the equality between the two versions of the corresponding Hofer-like metrics defined on the group of time-one maps of all symplectic isotopies: The symplectic analogues of the uniqueness result of Hofer's geometry [14]. This includes some condition(s) that make the latter uniqueness result continues holds in the context of Bus-Leclercq [7], and a short proof of the non-degeneracy of various Hofer-like metrics. Finally, an alternate proof (without appealing to Floer's theory) of a result from flux geometry found by McDuff-Salamon [12] is given, and various symplectic analogues of some approximation results found by Oh-Müller [13] are provided.