On the geometry of co-Hamiltonian diffeomorphisms

TL;DR

This paper explores the geometric structure of co-Hamiltonian diffeomorphisms on compact cosymplectic manifolds, analyzing fixed points, norms, and topologies, and establishing analogues of Hamiltonian dynamics results.

Contribution

It introduces co-Hamiltonian and almost co-Hamiltonian concepts, defines new topologies, and extends Hamiltonian dynamics lemmas to the co-Hamiltonian setting.

Findings

Minimum fixed points predicted by Arnold's conjecture.

Generation functions are constant along orbits.

Established co-Hofer norms and topologies.

Abstract

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.