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Entropy and completely integrable Hamiltonian systems

Author: Gabriel Paternain
Journal: Proc. Amer. Math. Soc. 113 (1991), 871-873
MSC: Primary 58F17; Secondary 58F05
MathSciNet review: 1059632
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Abstract: Let $ H$ be a Hamiltonian on a four-dimensional symplectic manifold. 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 topological 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.

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Keywords: Completely integrable Hamiltonians, topological entropy, geodesic flow, homoclinic orbits, integral
Article copyright: © Copyright 1991 American Mathematical Society