Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 989-1007.

Dans cet article, on étudie les mesures minimisantes de Hamiltoniens de Tonelli. Plus précisément, on explique quelles relations existent entre les fibrés de Green et différentes notions comme :•les exposants de Lyapunov des mesures minimisantes ;•les solutions KAM faibles. On en déduit par exemple que si tous les exposants de Lyapunov dʼune mesure minimisante μ sont nuls, alors le support de cette mesure est C 1 -régulier en μ-presque tout point.

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as:•the Lyapunov exponents of minimizing measures;•the weak KAM solutions. In particular, we deduce that the support of every minimizing measure μ, all of whose Lyapunov exponents are zero, is C 1 -regular μ-almost everywhere.

DOI : 10.1016/j.anihpc.2012.04.007
Classification : 37J50, 35D40, 37C40, 34D08, 35D65
Keywords: Minimizing orbits and measures, Lyapunov exponents, Weak KAM theory, Green bundles, Regularity of solutions to Hamilton–Jacobi equations
Mots-clés : Orbites et mesures minimisantes, Exposants de Lyapunov, Théorie KAM faible, Fibrés de Green, Régularité des solutions de lʼéquation de Hamilton–Jacobi 
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     title = {Green bundles, {Lyapunov} exponents and regularity along the supports of the minimizing measures},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Arnaud, M.-C. Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 989-1007. doi : 10.1016/j.anihpc.2012.04.007. http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.007/

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