Hofer-Like Geometry and Flux Theory

This paper meticulously revisit and study the flux geometry of any compact oriented manifold $(M; W)$. We generalize several well-known factorization results, exhibit some orbital conditions for the study of flux geometry, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, show that any smooth path in the kernel of the flux is a vanishing flux path, and show that the kernel of the flux for volume-preserving diffeomorphisms is $C^1$closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map:This recovers several results from symplectic geometry. The fix-points theory does not resist to the above machinery: We prove a general contractibility result with respect to the orbits of the fix-points for volume-preserving diffeomorphisms isotopic to the identity map via vanishing-flux paths, generalize and solve the Arnold conjecture using the Thurston fragmentation property. In the sequel, we use fix-points to: Characterize the flux geometry of certain $C^0$limits of sequences of vanishing-flux paths and volumepreserving diffeomorphisms. Beside this, a $C^0$criterion for the existence of at least one fix-point is given, and a weak version of the generalized $C^0$flux conjecture is solved. Finally, we construct a pseudo right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map, prove several comparison results suitable to the study of the Hofer-like geometry of the group $Ham(N; w)$, of all Hamiltonian diffeomorphisms of a closed symplectic manifold $(N; w)$, derive the equivalence between the Hofer and the Hofer-like metrics on $Ham(N; w)$, and exhibit a computational proof of the non-degeneracy of the Hofer-like energies: Here, an outcome is that the Calabi group controls the Hofer-like geometry of the group $Ham(N; w)$ of any closed symplectic manifold $(N; w)$.