Syst\`emes lagrangiens et fonction β de Mather
Abstract
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. Ma\~n\'e'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.