Connecting orbits of autonomous Lagrangian systems

In contrast to the time-dependent case, the time-t-section of Mañé set of autonomous Lagrangian systems is independent of time, thus, it is nowhere disconnected. This causes some difference in the study of dynamics, for instance, Mather's c-equivalence cannot exist among different cohomology classes if they are not in a flat of the α-function (cf (Bernard 2002 Ann. Inst. Fourier 52 1533–68.)). In this paper, we show how to construct connecting orbits in autonomous systems, and propose a modified notion of c-equivalence. We also apply the result to construct diffusion orbits in an energy surface