On Symplectic Dynamics

TL;DR

This paper explores foundational aspects of Banyaga topologies on symplectic manifolds, establishing analogues of Hamiltonian dynamics results without relying on displacement energy positivity, and proving the uniqueness of Banyaga Hofer-like norms.

Contribution

It introduces symplectic analogues of key Hamiltonian results, extends regularization procedures, and proves the uniqueness of Banyaga Hofer-like norms on symplectomorphism groups.

Findings

Symplectic analogue of Hofer-Zehnder result on $C^0$-limits of symplectic maps.

Equality of two Banyaga Hofer-like norms on the identity component.

Extension of regularization procedures to symplectic isotopies.

Abstract

This paper continues to carry out a foundational study of Banyaga topologies of a closed symplectic manifold [3]. Our intension in writing this paper is to provide several symplectic analogues of some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out the impact of the $L^\infty$ version of Banyaga Hofer-like metric in the investigation of the symplectic nature of the $C^0-$limit of a sequence of symplectic maps. This result is the symplectic analogue of a result that was proved in Hofer-Zehnder [8] (for compactly supported Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$), and then reformulated in Oh-M\"{u}ller [10] for Hamiltonian diffeomorphisms in general. Furthermore, we extend to symplectic isotopies the regularization procedure for Hamiltonian paths introduced in Polterovich [11], and then we use it to prove the equality between the two versions of Banyaga Hofer-like norms defined on the identity component in the group of symplectomorphisms. This result was announced in [2]. It shows the uniqueness of Banyaga Hofer-like geometry, and then yields the symplectic analogue of a result that was proved in Polterovich [11]. Finally, we elaborate the symplectic analogues of some approximation results found in Oh-M\"{u}ller [10], and make some remarks on flux theory.