In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian.
"On the Conley decomposition of Mather sets." Rev. Mat. Iberoamericana 26 (1) 115 - 132, March, 2010.