Generic Hamiltonian Dynamical Systems are neither Integrable nor Ergodic

2n-manifold action-angle coordinates action-angle variables analytic Hamiltonian B₁ B₂ B₂x Baire space C-topology canonical coordinates classical mechanics compact manifold compact symplectic manifold completely integrable corresponding cotangent bundle cotangent bundle T*M countable defined degenerate periodic orbits dense in H dense set dF¹ diffeomorphism dynamical system earlier paper 11 eigenvalues elliptic critical point energy manifold ergodic Hamiltonians exists an open flow dH H₁ H₂ Hamiltonian dynamical Hamiltonian flow Hamiltonian function Hamiltonian matrix Hamiltonian system Hamiltonian vector fields integrable Hamiltonians integral curve invariant tori LEMMA level H linear space main theorem meager in H metric neighborhood noncompact symplectic manifolds nondegenerate action-angle nonempty open set nonergodic nonsingular normalized Hamiltonians O₁ O₂ open and dense open interval open set P₁ Poisson bracket positive measure real analytic real differentiable function regular and consists regular value rotation numbers submanifold symplectic manifold M²n tangent space topology torus Y₁ θα ан