  This paper studies the geometry of the group of all co-Hamiltonian
diffeomorphisms of a compact cosymplectic manifold $(M, \omega, \eta)$. The
fix-point theory for co-Hamiltonian diffeomorphisms is studied, and we use
Arnold's conjecture to predict the exact minimum number of fix point that such
a diffeomorphism must have (this minimum number is at least $1$). It follows
that the generating function of any co-Hamiltonian isotopy is a constant
function along it orbits. Therefore, we study the co-Hofer norms for
co-Hamiltonian isotopies, and establish several co-Hamiltonian and almost
co-Hamiltonian analogues of some approximations lemmas and reparameterizations
lemmas found in the theory of Hamiltonian dynamics, we define two
$C^0-$co-Hamiltonian topologies, and use these topologies to define the spaces
of cohameomorphisms, and almost cohameomorphisms. Finally, we raise several
important questions for future studies.
