10/20/2005

Nancy Hingston
College of New Jersey

Periodic solutions of Hamilton's equations on Tori

Let the torus T^2n carry the standard symplectic structure, and a Hamiltonian function H of period 1 in the time variable. By the Arnold Conjecture, proved for the torus by Conley and Zehnder, the Hamiltonian flow has at least 2n+1 orbits of period 1. Conley and Zehnder also proved, under the additional assumption that all period 1 orbits are nondegenerate: If there are only finitely many orbits of period 1, then there are orbits of arbitrarily large minimal (integer) period. We prove this statement also holds in the degenerate case. Thus there are always infinitely many orbits of integer period. This settles a conjecture of Conley for the torus; this conjecture is still open for other compact symplectic manifolds.