Connecting orbits between static classes for generic Lagrangian systems

Let L be a C∞ convex superlinear Lagrangian on a closed manifold M. We show that if the number of static classes is finite, 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 sufficiently small connected neighborhood of the set of static orbits in M. We show that generically in the sense of Mañé (in: F. Ledrappier, J. Lewowicz, S. Newhouse (Eds.), International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Mañé), Pitman Research Notes in Mathematics, Vol. 362, 1996, pp. 120–131 (reprinted in Bol. Soc. Bras. Mat. 28(2) (1997) 141–157) 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 sufficiently small connected neighborhood of the projection of the support of the measure to M. This last result was obtained—with a different proof—by Bolotin (Proceedings of the International Congress of Mathematics, Vol. 1,2, Zürich, 1994, Birkhäuser, Basel, 1995, pp. 1169–1178; in: V.V. Kozlov (Ed.), Dynamical Systems in Classical Mechanics, American Mathematical Society Translation Series 2, Vol. 168, American Mathematical Society, Providence, RI, 1995, pp. 21–90) assuming the existence of a C1+Lip function f : M→R such that L+c−df⩾0, where c is the critical value of L. Finally, we obtain two consequences. The first one says that if M is a closed manifold with first Betti number ⩾2 then there exists a generic set O⊂C∞(M, R) such that if ψ∈O the Lagrangian L+ψ has a unique minimizing measure and this measure is uniquely ergodic. When this measure is supported on a periodic orbit, this orbit is hyperbolic and the stable and unstable manifolds have transverse homoclinic intersections. The second consequence says that if M is a closed manifold with first Betti number different from zero and if L is a symmetric Lagrangian, then there exists a generic set O⊂C∞(M, R) such that if ψ∈O, then L+ψ has a unique minimizing measure and this measure is supported on a hyperbolic fixed point whose stable and unstable manifolds have transverse homoclinic intersections.