An enlargement of some symplectic objects

TL;DR

This paper explores the algebraic and topological properties of symplectic homeomorphism groups, investigating potential flux geometries and analogues of symplectic dynamics results, while highlighting open questions and conjectures.

Contribution

It introduces new topological analogues of symplectic dynamics results and discusses the possibility of underlying flux geometries for symplectic homeomorphism groups.

Findings

Proposes topological analogues of symplectic results

Raises questions about flux geometry in symplectic homeomorphisms

Identifies open problems and conjectures in the field

Abstract

The study of algebraic properties of groups of transformations of a manifold gives rise to an interplay between different areas of mathemathics such as topology, geometry, and dynamical systems. Especially, in this paper, we point out some interplays between topology, geometry, and dynamical systems which are underlying to the group of symplectic homeomorphisms. The latter situation can occur when one thinks of the following question. Is there a flux geometry which is underlying to the group of strong symplectic homeomorphisms so that Fathi's Poincare duality theorem continues to hold? We discuss on some possible answers of the above preoccupation, and we elaborate various topological analogues of some well-known results found in the field of symplectic dynamics. We leave several open questions and conjectures.