The tiered Aubry set for autonomous Lagrangian functions

Marie-Claude Arnaud

Description

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

Sujets

• Arnaud
• Hamiltonian mechanics
• Avignon
• Function

Informations

• Publié par: pefav
• Nombre de lectures: 27
• Langue: English

Extrait

The

tiered

Aubry

set for autonomous functions

M.-C. ARNAUD ∗

March 5, 2008

Abstract

Lagrangian

LetL:T M→Rbe a Tonelli Lagrangian function (withMcompact and connected and dimM≥).Thetie2(tespserAderyrbu)eta˜.Mesn´A T (L) (resp.N T (L)) is the unionoftheAubrysets(resp.Man˜e´sets)A(L+λ) (resp.N(L+λ)) forλclosed 1-form. Then : 1. the setN T (L) is closed, connected and if dimH 1 (M)≥2, its intersection with any energy level is connected and chain transitive; 2. forLntciMahegerines,ehtseten˜e´essnA T (L) andN T (L) have no interior; 3. if the interior ofA T (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=T 2 , 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 M→Run lagrangien de Tonelli (avecMcompacte et connexe et dimM≥ 2).L’ensembled’Aubry(resp.deMa˜ne´)e´tag´eA T (L) (resp.N T (Lnue´noies))artl desensemblesd’Aubry(resp.deMan˜e´)A(L+λ) (resp.N(L+λ)) pourλ1-forme ferm´ee.Onmontre: 1.N T (L et si dim e) t ´ fH 1 (M)≥2, sa trace avec chaque niveau es erme, connex d’´energieestconnexeettransitiveparchaˆıne; ∗ ysenAnalred’atoiGte´riee´naenoilA(Eietr´eomsrtie´’dUinevestdysPaigAvnenoL,esrobaaVedulcu 2151), F-84 018Avignon, France. e-mail: [e-mail]

1

2. siLe´eeng´stiruqaesunedsMe˜an´e,lesensemblesA T (L) etN T (L)tnosind’ert´urie vide; 3.sil’int´erieurdeA T (Liqods.uespntri´epeiosndeeiedaptrtunetienlconde,iivnontse) On donne ensuite un exemple explicite satisfaisant 2 et un exemple montrant que si M=T 2 ,A T (LsKre.M.A.ralenue´dnoiotse’adhtdelnced´ereteerueˆtreneid´ff)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