  We review the author's results on Mather's $\beta$ function : non-strict
convexity of $\beta$ when the configuration space has dimension two, link
between the size of the Aubry set and the differentiability of $\beta$,
correlation between the rationality of the homology class and the
differentiability of $\beta$, 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 $\beta$ 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.
