Systèmes lagrangiens et fonction $β$ de Mather

We review the author's results on Mather's $β$ function : non-strict convexity of $β$ when the configuration space has dimension two, link between the size of the Aubry set and the differentiability of $β$, correlation between the rationality of the homology class and the differentiability of $β$, equality of the Mather set and the Aubry set for a large number of cohomology classes when the configuration space has dimension two, link beween the differentiability of $β$ and the integrability of the system. Mañé's conjectures are discussed in Chapters 6 and 7. A short list of open problems is given at the end of each chapter. In Appendix A we prove a theorem which extends Theorem 5 of reference [Mt09]. In Appendix B we discuss a geometrical problem which arises from Chapter 3, but may be of independant interest.

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