A Remark On Hofer-like Geometry

Abstract: We show that Banyaga Hofer's norm-a generalization of the Hofer norm, counting non-Hamiltonian paths and flux-is precisely the classical Hofer norm when limited to Hamiltonian diffeomorphisms on compact symplectic manifolds. This result proves a conjecture of Banyaga and fills the gap between Hofer and Hofer-like geometries: the refined Hofer-like structure degenerates to standard Hofer geometry within the Hamiltonian subgroup. We show that while the flux group plays an important role in symplectic geometry in general, it does not shorten the paths connecting Hamiltonian maps and hence intuitively confirms that minimal energy paths for Hamiltonian diffeomorphisms are inherently Hamiltonian. Such an equivalence creates great practical possibilities, since it allows the straightforward extension of essential results from Hofer geometry-namely, non-degeneracy, bounds on displacement energies, and infinite diameters-to the Hofer-like setting of Hamiltonian maps. This work provides a refined understanding of the interplay between symplectic topology, Hofer-like geometry, and Hamiltonian dynamics.