| Connecting Orbits Between Static Classes For Generic Lagrangian Systems (2007) | |||||||||||||||
Abstract | |||||||||||||||
| . Let L be a C 1 convex superlinear Lagrangian on a closed manifold M . We show that if the number of static classes is nite, then there exist chains of semistatic orbits that connect any two given static classes. Using this property we show that if there is only one static class, then the homoclinic orbits to the set of static orbits generate over R the relative homology of the pair (M; U ), where U is a suÆciently small neighborhood of the set of static orbits in M . We show that generically in the sense of Ma~ne [10] the set of semistatic orbits coincides with the support of a uniquely minimizing measure, therefore generically, the homoclinic orbits to the support of the minimizing measure generate over R the relative homology of the pair (M; U ), where U is a suÆciently small neighborhood of the projection of the support of the measure to M . This last result was obtained-with a dierent proof- by S. Bolotin in [1, 2] assuming the existence of a C 1+Lip function f : M ! R su... | |||||||||||||||
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