proceedings of the american mathematical society Volume 113, Number 3, November 1991 ENTROPY AND COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS GABRIEL PATERNAIN (Communicated by Kenneth R. Meyer) Abstract. Let H be a Hamiltonian on a four-dimensional symplectic man- ifold. Suppose the system is completely integrable and on some nonsingular compact level surface Q the integral is such that the connected components of the set of critical points form submanifolds. Then we prove that the topologi- cal entropy of the system restricted to Q is zero. As a corollary we deduce the nonexistence of completely integrable geodesic flows by means of integrals as described above for compact surfaces with negative Euler characteristic. 1. Introduction Let M be a four-dimensional symplectic manifold and H a Hamiltonian on M. Denote by sgrad H the symplectic gradient of H. Let Q be a nonsingular compact level surface of H. Suppose the system is completely integrable; that is, suppose there exists an additional function on M which is independent of H (almost everywhere) and is in involution with H (such a function is called an integral). Restricting this integral to Q gives a smooth function /. Denote by htop the topological entropy of the Hamiltonian flow restricted to Q. In this article we want to prove the following: Theorem 1. Let M be a smooth symplectic manifold and let sgrad// be a Hamiltonian field on M4 . Suppose that the system is completely integrable and that on some nonsingular compact level surface Q the integral f verifies either of the following conditions: (a) / is real analytic. (b) The connected components of the set of critical points of f form sub- manifolds. Then hlo=0. top Received by the editors March 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 58F07, 58F11; Secondary 58F17. Key words and phrases. Completely integrable Hamiltonians, topological entropy, geodesic flow, homoclinic orbits, integral. © 1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 871 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use