Integrable Natural Hamiltonian Systems on the Suspensions of Toric Automorphism

Abstract

This paper concerns two aspects: integrability of natural Hamiltonian systems on a Riemannian manifold and the topological entropy of the Hamiltonian flow. First we characterize the integrable Hamiltonian systems on the suspension of any finite dimensional toric automorphism. Second we prove that for the integrable Hamiltonian system having its Hamiltonian H with total energy no less than eH there exists a subset \({\Omega\subset{\mathcal D}:=\{e\in \mathbb R;\,e\ge e_H \}}\) of Lebesgue measure zero such that the Hamiltonian flow restricted to each energy surface {H = e} with \({e\in{\mathcal D}\backslash\Omega}\) has a positive topological entropy.