Invariant fibrations of geodesic flows (Q556230)

Let \(\Sigma\) be a \({\mathbb T}^{n+1}\) bundle over \({\mathbb T}^n\) with an \(\mathbb R\)-split, free abelian monodromy group. Let \(\Psi\) be the basis of a root system of a simple Kac-Moody Lie algebra. The following result is shown: For each \(\Sigma\) and \(\Psi\) there is an integrable Hamiltonian system on \(T^*\Sigma\) with positive topological entropy. The list of topological entropy is also given. Topological entropy is applied to show that the flows associated to nondual Toda lattices are typically topologically nonconjugate via an energy-preserving homeomorphism. For degree higher than 3, a Gelfond conjecture on algebraic numbers is used to classify Hamiltonian systems by energy-preserving conjugacies. For degree 3, this result follows from a well-known Gelfond result on algebraic numbers.