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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 -regular μ-almost everywhere.
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 -régulier en μ-presque tout point.
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
@article{AIHPC_2012__29_6_989_0,
author = {Arnaud, M.-C.},
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},
pages = {989--1007},
publisher = {Elsevier},
volume = {29},
number = {6},
year = {2012},
doi = {10.1016/j.anihpc.2012.04.007},
mrnumber = {2995103},
zbl = {1269.37031},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.04.007/}
}
TY - JOUR AU - Arnaud, M.-C. TI - Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 989 EP - 1007 VL - 29 IS - 6 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.04.007/ DO - 10.1016/j.anihpc.2012.04.007 LA - en ID - AIHPC_2012__29_6_989_0 ER -
%0 Journal Article %A Arnaud, M.-C. %T Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 989-1007 %V 29 %N 6 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.04.007/ %R 10.1016/j.anihpc.2012.04.007 %G en %F AIHPC_2012__29_6_989_0
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
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