{
  "id": "1102.1264",
  "version": "v1",
  "published": "2011-02-07T10:23:02.000Z",
  "updated": "2011-02-07T10:23:02.000Z",
  "title": "Systèmes lagrangiens et fonction $β$ de Mather",
  "authors": [
    "Daniel Massart"
  ],
  "comment": "M\\'emoire d'habilitation, 53 pages, 5 figures",
  "categories": [
    "math.DS",
    "math.AP",
    "math.DG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-07T10:23:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "systèmes lagrangiens",
      "configuration space",
      "aubry set",
      "differentiability",
      "cohomology classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.1264M"
    }
  }
}