Entropy and completely integrable Hamiltonian systems

Author:
Gabriel Paternain

Journal:
Proc. Amer. Math. Soc. **113** (1991), 871-873

MSC:
Primary 58F17; Secondary 58F05

DOI:
https://doi.org/10.1090/S0002-9939-1991-1059632-7

MathSciNet review:
1059632

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Abstract: Let be a Hamiltonian on a four-dimensional symplectic manifold. Suppose the system is completely integrable and on some nonsingular compact level surface 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 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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DOI:
https://doi.org/10.1090/S0002-9939-1991-1059632-7

Keywords:
Completely integrable Hamiltonians,
topological entropy,
geodesic flow,
homoclinic orbits,
integral

Article copyright:
© Copyright 1991
American Mathematical Society