Syst\`emes lagrangiens et fonction $\beta$ de Mather

TL;DR

This paper reviews key results on Mather's beta function, exploring its convexity, differentiability, and links to system integrability and Aubry sets, especially in two-dimensional configuration spaces.

Contribution

It provides a comprehensive review of the author's results on Mather's beta function, including new links between convexity, differentiability, and system properties in low dimensions.

Findings

Non-strict convexity of beta in 2D configuration spaces

Correlation between homology rationality and beta differentiability

Equality of Mather and Aubry sets for many cohomology classes in 2D

Abstract

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.