//img.uscri.be/pth/c3e2a100f54a73ce0669aa2912b368b4fe1f9c66
Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

The tiered Aubry set for autonomous Lagrangian functions

De
28 pages
The tiered Aubry set for autonomous Lagrangian functions M.-C. ARNAUD ? March 5, 2008 Abstract Let L : TM ? R be a Tonelli Lagrangian function (with M compact and connected and dimM ≥ 2). The tiered Aubry set (resp. Man˜e set) AT (L) (resp. N T (L)) is the union of the Aubry sets (resp. Man˜e sets) A(L + ?) (resp. N (L + ?)) for ? closed 1-form. Then : 1. the set N T (L) is closed, connected and if dimH1(M) ≥ 2, its intersection with any energy level is connected and chain transitive; 2. for L generic in the Man˜e sense, the sets AT (L) and N T (L) have no interior; 3. if the interior of AT (L) is non empty, it contains a dense subset of periodic points. Then, we give an example of an explicit Tonelli Lagrangian function satisfying 2 and an example proving that when M = T2, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different. Resume Soit L : TM ? R un lagrangien de Tonelli (avec M compacte et connexe et dimM ≥ 2).

  • ?universite d'avignon et des pays de vaucluse

  • hamiltonian function

  • dual mather

  • tonelli lagrangian

  • avignon

  • function satisfying

  • lagrangien de tonelli

  • function


Voir plus Voir moins
The
tiered
Aubry
set for autonomous functions
M.-C. ARNAUD
March 5, 2008
Abstract
Lagrangian
LetL:T MRbe a Tonelli Lagrangian function (withMcompact and connected and dimM).Thetie2(tespserAderyrbu)eta˜.Mesn´AT(L) (resp.NT(L)) is the unionoftheAubrysets(resp.Man˜e´sets)A(L+λ) (resp.N(L+λ)) forλclosed 1-form. Then : 1. the setNT(L) is closed, connected and if dimH1(M)2, its intersection with any energy level is connected and chain transitive; 2. forLntciMahegerines,ehtseten˜e´essnAT(L) andNT(L) have no interior; 3. if the interior ofAT(L) is non empty, it contains a dense subset of periodic points. Then, we give an example of an explicit Tonelli Lagrangian function satisfying 2 and an example proving that whenM=T2, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.
R´esum´e
SoitL:T MRun lagrangien de Tonelli (avecMcompacte et connexe et dimM2).LensembledAubry(resp.deMa˜ne´)e´tag´eAT(L) (resp.NT(Lnue´noies))artl desensemblesdAubry(resp.deMan˜e´)A(L+λ) (resp.N(L+λ)) pourλ1-forme ferm´ee.Onmontre: 1.NT(L et si dim e) t ´ fH1(M)2, sa trace avec chaque niveau es erme, connex d´energieestconnexeettransitiveparchaˆıne; ysenAnalredatoiGte´riee´naenoilA(Eietr´eomsrtie´dUinevestdysPaigAvnenoL,esrobaaVedulcu 2151), F-84 018Avignon, France. e-mail: Marie-Claude.Arnaud@univ-avignon.fr
1
2. siLe´eeng´stiruqaesunedsMe˜an´e,lesensemblesAT(L) etNT(L)tnosindert´urie vide; 3.silint´erieurdeAT(Liqods.uespntri´epeiosndeeiedaptrtunetienlconde,iivnontse) On donne ensuite un exemple explicite satisfaisant 2 et un exemple montrant que si M=T2,AT(LsKre.M.A.ralenue´dnoiotseadhtdelnced´ereteerueˆtreneid´)p
Contents
1
2
3
Introduction
Peierlsbarrier,Ma˜n´epotential,AubryandMa˜n´esets,proofof proposition 1
Radially transformed set and Aubry set, proof of proposition 4
3
8
14
4 Green bundles, conjugate points and proofs of of theorem 2 and corol-lary 3 18
5
Proof of proposition 6
2
25