On Cosymplectic Dynamics I

Abstract

This paper is an introduction to cosymplectic topology. Through it, we study the structures of thegroup of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosym-plectic manifold(M,ω,η):(i)−we de ne and present the features of the space of almost cosymplectic vector elds (resp. cosymplectic vector elds);(ii)−we prove by a direct method that the identity component in thegroup of all cosymplectic diffeomorphisms isC0−closed in the group Diff∞(M)(a rigidity result), while in thealmost cosymplectic case, we prove that the Reeb vector eld determines the almost cosymplectic nature oftheC0−limitφof a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A su cient conditionbased on Reeb’s vector eld which guarantees thatφis a cosymplectic diffeomorphism is given (a exibilitycondition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived andthe cosymplectic analogue of a result that was proved by Hofer-Zehnder follows.