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Closing Aubry sets I

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Niveau: Supérieur, Doctorat, Bac+8
Closing Aubry sets I A. Figalli? L. Rifford† July 7, 2011 Abstract Given a Tonelli Hamiltonian H : T ?M ? R of class Ck, with k ≥ 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set x¯, and a critical viscosity subsolution u, such that u is a C1 critical solution in an open neighborhood of the positive orbit of x¯. Suppose further that u is “C2 at x¯”. Then there exists a Ck potential V : M ? R, small in C2 topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) If M is two dimensional, (1) holds replacing “C1 critical solution + C2 at x¯” by “C3 critical subsolution”. These results can be considered as a first step through the attempt of proving the Man˜e's conjecture in C2 topology. In a second paper [27], we will generalize (2) to arbitrary dimension. Moreover, such an extension will need the introduction of some new techniques, which will allow us to prove in [27] the Man˜e's density Conjecture in C1 topology. Our proofs are based on a combination of techniques coming from finite dimensional control theory and Hamilton-Jacobi theory, together with some of the ideas which were used to prove C1-closing lemmas for dynamical systems.

  • hamiltonian viewpoint

  • following man˜e

  • nonempty compact

  • aubry-mather theory

  • c1-closing lemmas

  • ?? al

  • c1 critical


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July 7, 2011
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Closing Aubry sets I A. FigalliL. Rifford
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Introduction 1.1 Aubry-Mather theory from the Lagrangian viewpoint . . . . . . . . . . . . . . . . 1.2 Aubry-Mather theory from the Hamiltonian viewpoint . . . . . . . . . . . . . . . 1.3TheMan˜e´Conjecture.................................
3 Connecting Hamiltonian orbits by potentials 12 3.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Controlling the action by potentials 23 4.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0resappsuauthorthrpehargoetrotybdoBbocaJ-nooe´htteiAMeKrijocemrP0--7AtRN-036BLANmilt1,Ha faible”. AF is also supported by NSF Grant DMS-0969962. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin TX 78712, USA (ude.tu.hsaxelialat@migf) itopil,soShpainAedeNice-iversit´nUce08NiVclaP,ra0,16oresMR,U´enn2166RSCN-.J.obaLodueiD.A Cedex 02, France (.fcerfirdrofinu@)
Contents
Abstract Given a Tonelli HamiltonianH:TMRof classCk, withk2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set ¯ d a critical vi ity subsolutionu, such thatuis aC1critical solution in an open x scos, an neighborhood of the positive orbit of ¯x. Suppose further thatuis “C2atx¯”. Then there exists aCkpotentialV:MR, small inC2topology, for which the Aubry set of the new HamiltonianH+V Ifis either an equilibrium point or a periodic orbit. (2)M is two dimensional, (1) holds replacing “C1critical solution +C2at ¯x” by “C3critical subsolution. These results can be considered as a first step through the attempt of proving the Man˜e´sconjectureinC2 a second paper [27], we will generalize (2) to arbitrarytopology. In dimension. Moreover, such an extension will need the introduction of some new techniques, whichwillallowustoprovein[27]theMa˜ne´sdensityConjectureinC1topology. Our proofs are based on a combination of techniques coming from finite dimensional control theory and Hamilton-Jacobi theory, together with some of the ideas which were used to proveC1-closing lemmas for dynamical systems.
2 3 5 8
2 Statement of the results
5 Proof of Theorem 2.1 30 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Preliminary steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Closing the Aubry set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 Control of the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.5 Construction of a critical subsolution . . . . . . . . . . . . . . . . . . . . . . . . . 42
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Proof of Theorem 2.4 47 6.1 Preliminary step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2 Modification of the potential and conclusion . . . . . . . . . . . . . . . . . . . . . 48 6.3 Construction of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7 Final comments
A Conventions and standing notation
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B Controllability of nonlinear control systems 52 B.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 B.2 Singular controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 B.3 Application to partial controllability I . . . . . . . . . . . . . . . . . . . . . . . . 55 B.4 Application to partial controllability II . . . . . . . . . . . . . . . . . . . . . . . . 56 B.5 Application to partial controllability III . . . . . . . . . . . . . . . . . . . . . . . 57
C Quantitative Inverse Function Theorem
D The Mai Lemma
E
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Proofs of Lemmas 3.3, 4.3 and 6.1 59 E.1 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 E.2 Proof of Lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 E.3 Proof of Lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
References
1 Introduction
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Let (M gbe a smooth compact Riemannian manifold without boundary of dimension) n2. GivenH:TMRonhTotmomiHalielsaeconjectureintlnoai,nhtMe˜a´nCktopology (with VCk˜ k2) asserts that, for generic potentials (M), the projected Aubry setA(H+V) associated to the HamiltonianH+Vis either an equilibrium point or a periodic orbit. This paper is the first of a series of articles where we plan to make progress toward a proofoftheMan˜e´ConjectureinC2topology. The aim of this first paper is to show how to provethedensitypartoftheM˜ne´ConjectureinC2topology under the following assumptions a (Theorem 2.1): there exist a recurrent point of the projected Aubry set ¯x, and a critical viscosity subsolutionu, such thatuis aC1critical solution in an open neighborhood of the positive orbit ofx¯, anduis “C2atx¯”. Then, in two dimensions we show how to replace the above assumption by replacing “C1critical solution +C2at the point” with “C3critical subsolution” (Theorem 2.4). In a second paper we will perform the extension of this last result to arbitrary dimension [27, Theorem 1.1]. Moreover, the proof of this last result will involve the introduction of some newideasandtechniques,whichwillallowustoprovethe(densitypartofthe)Ma˜n´eConjecture inC1topology [27, Theorem 1.2].
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Before describing our results in detail, we first introduce the Aubry-Mather theory from both the Lagrangian and the Hamiltonian points of view. Some conventions and standing notation are gathered in Appendix A.
1.1 Aubry-Mather theory from the Lagrangian viewpoint LetL:T MRbe aCkTonelli Lagrangian, that is, a Lagrangian of classCk(withk2) satisfying the two following assumptions:
(L1)Superlinear growth:For everyK0, there is a finite constantC(K) such that
L(x v)Kkvkx+C(K)(x v)T M (L2)Strict convexity:For every (x v)T M, the second derivative along the fibers2v2(x v) is positive definite.
Thecritical valueofLis defined as c[L] :=Tin>f0T1Aγ; [0 T]|γC1[0 T] M γ(0) =γ(T)(1.1) whereAγ; [0 T]denotes theactionof theC1curveγ: [0 T]Mon the time interval [0 T], that is, Aγ; [0 T]:=Z0TLγ(t)˙γ(t)dt By the assumptions onL, the critical valuec[L] is necessarily finite, and satisfies
inf (xv)T ML(x v)≤ −c[L]xinfML(x0)To each closed curveγCp1er[0 T] M, we can associate a probability measureγonT M by f d fγ(t)˙γ(t)dtfC0(T MR)ZT Mγ:=T1Z0T FollowingMa˜ne´[35],wecallholonomic probability measureany element in the set H:=nγ|T >0 γC1per[0 T] Mowhere the closure is taken with respect to the weak- Definetopology on the space of measures. theaction functional
AL:P(T M)−→R∪ {+∞} 7AL() :=RT MLd
By construction, we have infAL()|∈ H=c[L]The setHis a (nonempty) closed convex subset ofP(T M), which is not compact (with respect to the weak- thanks to (L1), the settopology). However,H0:=H ∩ {AL≤ −c(L) + 1}is a compact convex subset ofH. This implies thatALattains a minimum onH, that is, c[L] =minH{AL()}
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The measures∈ Hsuch thatAL() =c[L] are calledminimizing measures. It can be shown that they are invariant under the Euler-Lagrange flowφtL[35], and they minimize the functional ALamong all Borel probability measures onT Mwhich are invariant underφL. t TheMather setofLis the nonempty compact subset ofT Mdefined as ˜[Supp()M(L) := AL()=c[L] and theprojected Mather setM(L)Mis given by M(L) :=πM(L)˜ In [37], Mather proved the following result: ˜ Mather’s Graph Theorem I.The setM(L)T Mis invariant underφtL. Moreover the ˜ mapπ|M˜(L):M(L)Mis injective, and π|M(˜L1:M(L)→ M(˜L) )
is Lipschitz.
Following Mather [38], for everyT >0 we define the functionhT:M×MRas hT(x y) := infnAγ; [0 T]|γC1[0 T] M γ(0) =x γ(T) =yoThePeierls barrierassociated withLis the functionh:M×MRdefined by h(x y) := lTim+infnhT(x y) +c[L]ToIt is immediately seen that the following inequalities hold for allT >0, for everyx y zM:
h(x z)h(x y) +hT(y z) +c[L]T  h(x z)hT(x y) +c[L]T+h(y z)In particular, we deduce that the following “triangle inequality” holds:
h(x z)h(x y) +h(y z)x y zM
By compactness ofMand (1.1), it is not difficult to prove that there is at least one pointxM such thath(x x Hence the above triangle inequality shows that) = 0.his finite everywhere on M×M. Theprojected Aubry setA(L) is then defined as the nonempty compact set given by A(L) :=nxM|h(x x) = 0o(1.2)
We observe that for everyx∈ A(L) there exist a sequence{Tk}kNof real numbers tending to +, and a sequence{γk}kNofC1curvesγk: [0 Tk]M, such thatγk(0) =γk(Tk) and k→∞Aγk; [0 Tk]+c[L]Tk= 0lim ApplyingtheArzela`-AscoliTheorem,itcanbeshownthatthesequence{γ˜k}of curvesγk γ˙k: [0 Tk]T Mis relatively compact, so that for each integerl >0 the sequence of curves ]7˜γ˜γkk((tT)k+tfi)iftt<00 t[l l 4
admits, up to a subsequence, a uniform limit. Then, one can show that such limit curve is uniquely determined [38], and deduce that to eachx∈ A(L) it can be associated in a unique way aCk1curveγx:RM, withγx(0) =x, which solves the Euler-Lagrange equation ddtvLγx(t) γ˙x(t)=xLγx(t)˙γx(t)tRThen, theAubry setofLis the compact subset ofT Mdefined by A(˜L) :=nγx(t) γ˙x(t)|x∈ A(L) tRo˜ ˜ It can be proved that Aubry setA(L) contains the Mather setM(L). Moreover, in [38] Mather showed the following result:
Mather’s Graph Theorem II.The setA˜(L)T Mis invariant underφLt the. Moreover ˜ mapπ|A˜(L):A(L)Mis injective, its image coincides withA(L), and A(L)1: ˜ π|˜A(L)→ A(L)
is Lipschitz.
˜ ˜ In other terms, Mather’s Graph Theorems state thatM(L)⊂ A(L) are contained in the graph of a Lipschitz section ofT M.
1.2 Aubry-Mather theory from the Hamiltonian viewpoint TheTonelli HamiltonianH:TMRassociated toLby Legendre-Fenchel duality is defined as vTxMnp(v)L(x v)o(x p)TxM H(x p) := max Thanks to our assumptions onL, it is well-known thatHis of classCkand satisfies both properties of superlinear growth and strict convexity inTM: (H1)Superlinear growth:For everyK0, there is a finite constantC(K) such that H(x p)Kkpkx+C(K)(x p)TM (H2)Strict convexity:For every (x p)TM, the second derivative along the fibers2p2(x p) is positive definite.
Under the above assumptions, the Hamiltonian flowφtHofHis of classCk1, and is conjugated with the Euler-Lagrange flowφLtofL. Thecritical valueora˜Mvlaceulace´nitirofHis defined as
c[H] :=c[L]while the Aubry set “seen inTM” is defined as ˜A˜(L)A(H) :=L
(1.3)
whereL:T MTM Bydenotes the Legendre transform (see Appendix A). construction A˜(H) is a nonempty compact subset ofTMwhich is invariant underφtH a series of papers. In [16, 17, 18], Fathi established a deep link between the concept of Aubry sets and the concept of viscosity solutions of the Hamilton-Jacobi associated withH, which we now describe.
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A continuous functionu:MRis called aviscosity subsolutionof the Hamilton-Jacobi equation Hx du(x)=cxM(1.4) if, for everyC1functionφ:MRsuch thatφuand everyzM, the following holds:
φ(z) =u(z) =H(z dφ(z))c
This is equivalent to asking that uγ(b)uγ(a)γ(t) γ˙ ZbaL(t)dt+c(ba) (1.5) for everyC1curveγ: [a b]M. A continuous functionu:MRis called aviscosity solutionof (1.4) if, for everyC1 functionφ:MRsuch thatφuand everyzM, the following holds1:
φ(z) =u(z) =H(z dφ(z)) =c
As shown by Fathi, a continuous functionu:MRsolution of (1.4) if and only ifis a viscosity it is a viscosity subsolution of (1.4) and, for eachxM, there is aCk1curveγx: (−∞0]M such that uxuγx(T)=Z0TLγx(t)˙γx(t)dt+cTT0(1.6) In [16], Fathi proved the following result:
Fathi’s Weak KAM Theorem.The critical Hamilton-Jacobi equation Hx du(x)=c[H]xM
admits at least one viscosity solution.
(1.7)
Let us recall that, by the compactness ofM,c[H] is the only value ofcfor which the Hamilton-Jacobi equation (1.4) admits a viscosity solution. Indeed, if a continuous function u:MRis a viscosity subsolution of (1.4) for somecR, then for everyC1curve γ: [0 T]Mone has ukuγ(T)uγ(0)γ(t) γ˙ (t)dt+cT  2kZ0TL 1We notice that the definitions of viscosity subsolution and viscosity solution given here are equivalent to the usual definitions: usually, a continuous functionu:MRis called aviscosity solutionof the first-order partial differential equation F`x u(x) du(x)´= 0xM if it satisfies the two following properties: (i) (uissupersolution) For everyC1functionφ:MRsuch thatφuand everyzM, it holds φ(z) =u(z) =F(z φ(z) dφ(z))c (ii) (uissubsolution) For everyC1functionφ:MRsuch thatφuand everyzM, it holds
φ(z) =u(z) =F(z φ(z) dφ(z))c
SinceHis convex in thepsublevel sets, the above definitions are equivalent to the onevariable with bounded given in the paper.
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