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Variational Construction of Asymptotic Orbits in Contact Hamiltonian Systems

For a Tonelli contact Hamiltonian flow on the 1-jet space of a closed manifold -- one satisfying fiberwise convexity and superlinearity but WITHOUT any monotonicity assumption on the contact variable -- this is the technique of building semi-infinite action-minimizing orbits asymptotic to the action-minimizing invariant sets N_u attached to weak KAM solutions u of the associated Hamilton-Jacobi equation, and heteroclinic orbits running between N_u and N_v for two distinct solutions. The construction extends the method of characteristics to the solution semigroup of the Hamilton-Jacobi equation, uses the calibration properties of weak KAM solutions, and exhibits the resulting Mane sets and their asymptotic/heteroclinic structure in the phase space stratified by these solutions.