An enlargement of some symplectic objects

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.