Smooth Integrable Geodesic Flows With Positive Topological Entropy

In this paper we construct manifolds of arbitrarily finite dimensions with integrable geodesic flows which have positive topological entropy. Traditionally, it is thought that completely integrable dynamical systems are relatively simple. As we know, the topological entropy is used to measure the complexity of dynamical systems. Generally speaking, a system is complicated if it has positive topological entropy. Our examples show that a smooth integrable Hamiltonian system can have complicated dynamical phenomenae.