For contact Hamiltonian systems satisfying a monotonicity condition, the omega-limit set of every positive globally minimizing orbit is contained in the Mane set of semi-static orbits.
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Dynamics of globally minimizing orbits in contact Hamiltonian systems
For contact Hamiltonian systems satisfying a monotonicity condition, the omega-limit set of every positive globally minimizing orbit is contained in the Mane set of semi-static orbits.