Dynamics of the dominant Hamiltonian, with applications to Arnold diffusion
DDynamics of the dominant Hamiltonian, withapplications to Arnold diffusion
V. Kaloshin ˚ , K. Zhang : October 19, 2018
Abstract
It is well known that instabilities of nearly integrable Hamiltonian systemsoccur around resonances. Dynamics near resonances of these systems is wellapproximated by the associated averaged system, called slow system . Eachresonance is defined by a basis (a collection of integer vectors). We introduce aclass of resonances whose basis can be divided into two well separated groupsand call them dominant . We prove that the associated slow system can be wellapproximated by a subsystem given by one of the groups, both in the sense ofthe vector field and weak KAM theory.One of crucial ingredients of proving Arnold diffusion is understanding thestructure of invariant (Aubry) sets of nearly integrable systems. As an importantapplication we construct a diffusion path for a generic nearly integrable systemsuch that invariant (Aubry) sets along this path have a "simple" structure similarto the structure of Aubry-Mather sets of twist maps. This is a crucial ingredientin proving Arnold diffusion for convex Hamiltonians in any number of degrees offreedom.
Contents ˚ University of Maryland at College Park ( vadim.kaloshin gmail.com ) : University of Toronto ( kzhang math.utoronto.edu ) a r X i v : . [ m a t h . D S ] J a n The choice of basis and averaging 16
A Diffusion path with dominant structure 51
A.1 Diffusion path for Arnold diffusion . . . . . . . . . . . . . . . . . . . . 51A.2 Nondegeneracy conditions for Arnold diffusion . . . . . . . . . . . . . . 54A.3 Properties of the nondegeneracy condition . . . . . . . . . . . . . . . . 56A.4 Construction of a diffusion path and surgery of resonant manifolds . . . 58A.4.1 An initial step of the induction . . . . . . . . . . . . . . . . . . 59A.4.2 Step 2 of the induction . . . . . . . . . . . . . . . . . . . . . . . 62A.4.3 Step s ` B Diffusion mechanism and AM property 70C Normally hyperbolic invariant manifolds 74
C.1 Normally hyperbolic invariant manifolds via isolation block . . . . . . . 74C.2 Existence of Lipschitz invariant manifolds . . . . . . . . . . . . . . . . 76C.3 NHIC for the dominant system . . . . . . . . . . . . . . . . . . . . . . 772
Introduction
Consider a nearly integrable system with n degrees of freedom H ε p θ, p, t q “ H p p q ` εH p θ, p, t q , θ P T n , p P R n , t P T . (1.1)We will restrict to the case where the integrable part H is strictly convex, more precisely,we assume that there is D ą D ´ Id ď B pp H p p q ď D Idas quadratic forms, where Id denotes the identity matrix.The main motivation behind this work is the question of Arnold diffusion, thatis, topological instability for the system H ε . Arnold provided the first example in[Arn64b], and asks ([Arn63; Arn64a; Arn94]) whether topological instability is “typical”in nearly integrable systems with n ě n “
1, due to lowdimensionality).It is well known that the instabilities of nearly integrable systems occurs alongresonances. Given an integer vector k “ p ¯ k, k q P Z n ˆ Z with ¯ k ‰
0, we define theresonant submanifold to be Γ k “ t p P R n : k ¨ p ω p p q , q “ u , where ω p p q “ B p H p p q .More generally, we consider a subgroup Λ of Z n ` which does not contain vectors ofthe type p , ¨ ¨ ¨ , , k q , called a resonance lattice . The rank of Λ is the dimension of thereal subspace containing it. Then for a rank d resonance lattice Λ, we defineΓ Λ “ č t Γ k : k P Λ u “ d č i “ Γ k i , where t k , ¨ ¨ ¨ , k d u is any linear independent set in Λ. We call such Γ Λ a d ´ resonancesubmanifold ( d ´ resonance for short), which is a co-dimension d submanifold of R n , andin particular, an n ´ resonant submanifold is a single point. We say that Λ is irreducible ifit is not contained in any lattices of the same rank, or equivalently, span R Λ X Z n ` “ Λ.We now focus on the diffusion that occurs along a connected net of p n ´ q´ resonances,with each p n ´ q´ resonance being a curve in R n . Let us first consider diffusion alonga single p n ´ q´ resonance Γ. It is shown in [BKZ11] that generically, diffusion indeedoccur along Γ, except for a finite subset of n ´ resonances (called the strong resonances )which divides Γ into disconnected components. A strong resonance can be viewed asthe intersection of Γ with a transveral 1 ´ resonance manifold Γ k (see Figure 1).The main obstacle to proving diffusion along Γ reduces to whether the diffusioncan “cross” the strong resonances. In a more general diffusion path that contains twointersecting p n ´ q´ resonances Γ and Γ , the intersection is an n ´ resonance whichby definition is strong. The question is then whether one can travel along Γ and then“switch” to Γ at the intersection. Solution to either problem requires an understandingof the system near an n ´ resonance.For an n ´ resonance t p u “ Γ Λ , we assume that Λ is irreducible, and B “ r k , ¨ ¨ ¨ , k n s is an ordered basis over Z . The study of diffusion near p reduces to the study3 Γ k Figure 1: Diffusion path and essential resonances in n “
3. The hollow dots requirescrossing, while the grey dots requires switchingof a particular slow system defined on T n ˆ R n , denoted H sp , B . More precisely, inan O p? ε q´ neighborhood of p , the system H ε admits the normal form (see [KZ13],Appendix B) H sp , B p ϕ, I q ` ? εP p ϕ, I, τ q , ϕ P T n , I P T n , τ P ? ε T , where ϕ i “ k i ¨ p θ, t q , ď i ď n, p p ´ p q{? (cid:15) “ ¯ k I ` ¨ ¨ ¨ ¯ k n I n . Therefore, H (cid:15) is conjugate to a fast periodic perturbation to H sp , B . While it is possibleto give a basis free definition of the slow system, we opt to choose a particular basis B , and rendering our setup basis dependent . Such averaged systems were studied in[Mat08].When n “
2, the slow system is a 2 degrees of freedom mechanical system, thestructure of its (minimal) orbits is well understood. This fact underlies the results onArnold diffusion in two and half degrees of freedom (see [Mat03], [Mat08], [Mat11],[Che13], [KZ13],[GK14b], [KMV04], [Mar12a], [Mar12b]). This is no longer the casewhen n ą
2, which is a serious obstacle to proving Arnold diffusion in higher degrees offreedom. In [KZ14] it is proposed that we can sidestep this difficulty by using dimensionreduction : using existence of normally hyperbolic invariant cylinders (NHICs) to restrictthe system to a lower dimensional manifold. This approach only works when the slowsystem has a particular dominant structure , which is the topic of this paper.In order to make this idea specific it is convenient to define the slow system for any p and any d ´ resonance d ď n . For p P R n , an irreducible rank d resonance lattice Λ,and its basis B “ r k , ¨ ¨ ¨ , k d s , the slow system is H sp , B p ϕ, I q “ K p , B p I q ´ U p , B p ϕ q , ϕ P T d , I P T d . (1.2)Suppose the fourier expansion of H is ř k P Z n ` h k p p q e πik ¨p θ,t q , then K p , B p I q “ B pp H p p qp I ¯ k ` ¨ ¨ ¨ ` I d ¯ k d q ¨ p I ¯ k ` ¨ ¨ ¨ ` I d ¯ k d q , (1.3)4 p , B p ϕ , ¨ ¨ ¨ , ϕ d q “ ´ ÿ l P Z d h l k `¨¨¨ l d k d p p q e πi p l ϕ `¨¨¨` l d ϕ d q . (1.4)The system H sp , B is only dynamically meaningful when p P Γ Λ . However, the moregeneral set up allows us to embed the meaningful slow systems into a nice space.We say that the resonance lattice Λ admits a dominant structure if it contains anirreducible lattice Λ st of rank m ă d , such that M p Λ | Λ st q : “ min k P Λ z Λ st | k | " max k P B st | k | , (1.5)where | k | “ sup i | k i | is the sup-norm. Given the relation Λ st Ă Λ, one can choosean adapted basis B “ r k , ¨ ¨ ¨ , k d s of Λ, meaning that B st “ r k , ¨ ¨ ¨ , k m s is a properlyordered basis of Λ st .In this case we have two systems H sp , B st and H sp , B , which we will call the strongsystem and slow system respectively. When the lattices have a dominant structure(see (1.5)), the slow system H sp , B inherits considerable amount of information from thestrong system. Indeed, let us denote H sp , B st “ K st p I , ¨ ¨ ¨ , I m q ´ U st p ϕ , ¨ ¨ ¨ , ϕ m q ,H sp , B “ K s p I , ¨ ¨ ¨ , I d q ´ U s p ϕ , ¨ ¨ ¨ , ϕ d q , under (1.5) and after choosing an appropriate adapted basis, we will show that K st p I , ¨ ¨ ¨ , I m q “ K s p I , ¨ ¨ ¨ , I m , , ¨ ¨ ¨ , q , } U s ´ U st } C ! } U st } C , (1.6)which indicates H sp , B can be approximated by H sp , B st . The variables ϕ i , I i , 1 ď i ď m are called the strong variables , while ϕ i , I i , m ` ď i ď d are called the weak variables .Recall that for each convex Hamiltonian H , we can associate a Lagrangian L “ L H ,and the Euler-Lagrange flow is conjugate to the Hamiltonian flow. In particular, when H “ K p I q ´ U p ϕ q , and K p I q is a quadratic form, ϕ “ v, v “ ddt pB I K p I qq “ pB II K q B ϕ U p ϕ q . whose vector field is denoted the Euler-Lagrange vector field. Denote by X st Lag and X sLag the Euler-Lagrange vector fields associated to the Hamiltonians H sp , B st and H sp , B .Since the system for X st Lag is only defined for the strong variables p ϕ i , v i q , 1 ď i ď m ,we define a trivial extension of X st Lag by setting ϕ i “ v i “ m ` ď i ď d .We show that after choosing a proper adapted basis, and a suitable rescalingtransformation in the weak variables, the transformed vector field X sLag converges tothat of X st Lag in some sense. In particular, if X st Lag admits a normally hyperbolic invariantcylinder (NHIC), so does X sLag . In a separate direction, we also obtain a limit theoremon the weak KAM solutions by variational arguments. We now formulate our mainresults in loose language, leaving the precise version for the next section.5 ain Result. Assume that r ą n ` p d ´ m q ` . Given a fixed lattice Λ st of rank m with a fixed basis B st , for each rank d, m ď d ď n irreducible lattice Λ Ą Λ st , thereexists an adapted basis B such that:1. (Geometrical) As M p Λ | Λ st q Ñ 8 , the projection of X sLag to the strong variables p ϕ i , v i q , ď i ď m converges to X st Lag uniformly. Moreover, by introducing acoordinate change and rescaling affecting only the weak variables p ϕ i , v i q , m ` ď i ď d , the transformed vector field of X sLag converges to a trivial extension of thevector field of X st Lag . As a corollary, we obtain that if X st Lag admits an NHIC, thenso does X sLag for sufficiently large M p Λ | Λ st q .2. (Variational) As M p Λ | Λ st q Ñ 8 , the weak KAM solution of H sp , B (of properlychosen cohomology classes) converges uniformly to a trivial extension of a weakKAM solution of H sp , B st , considered as functions on R d . We also obtain corollariesconcerning the limits of Mañe, Aubry sets, rotation number of minimal measure,and Peierl’s barrier function. The precise definitions of these objects will be givenlater. The combination of the persistence of NHIC, and limits of the weak KAM solutionsallows us to localize and restrict Aubry sets. As a demonstration of our theory, inAppendix A we show that one can construct a connected net of p n ´ q´ resonances,such that all strong resonances have the dominant structure. We then show that most Aubry sets with a chosen rational homology class are contained in three dimensionalNHICs, and the topology of such Aubry sets resembles the Aubry-Mather sets fortwist maps. This is closely related to the AM property we introduce in Appendix A.The relation between the property of the Aubry sets and diffusion mechanism is givenin Appendix B. In [KZ13] and [KZ14] using these structures we prove existence ofArnold diffusion. This net of diffusion paths also can be chosen to be γ ´ dense for anypre-determined γ ą
0. We show that for a “typical” H ε (for any n ą H sp , B st approximates H sp , B is related to the classic result ofpartial averaging (see for example [AKN06]). The statement min k P Λ z Λ st | k | " max k P Λ st | k | says that the resonances in Λ st is much stronger than the rest of the resonances in Λ.Partial averaging says that the weaker resonances contributes to smaller terms in anormal form.However, our treatment of the partial averaging theory is quite different from theclassical theory. By looking at the rescaling limit, we study the property of the averagingindependent of the small parameter ε . The theory is far from a simple corollary of (1.6),with the main difficulty coming from the fact that as M p Λ | Λ st q Ñ 8 , the quadraticpart of the system H sp , B becomes unbounded.In [Mat08], John Mather developed a theory of (partial) averaging for a nearlyintegrable Lagrangian system L ε p θ, v, t q “ L p v q ` εL p θ, v, t q , where p θ, v, t q P T n ˆ R n ˆ T . See definition of the AM property in Appendix A.1
6n particular, it is shown that the slow system relative to a resonant lattice can bedefined on the tangent bundle of a sub-torus T d ˆ R d , d ă n . Quantitative estimateson the action of minimizing orbits of the original system versus the slow system areobtained. Our variational result is related to [Mat08], but different in many ways. Wework with the scaling limit system, and the small parameter ε does not show up inour analysis. We also avoid quantitative estimates (in the statement of the theorem)and obtain a limit theorem in weak KAM solutions. This allows us to take consecutivelimits, which is very useful for our construction of diffusion path in higher dimensions(see Appendix A).The formulation of the limit theorem in weak KAM solution requires special care. Anatural candidate is Tonelli convergence (convergence of Lagrangian within the Tonellifamily, see [Ber10]). In our setup, H sp , B st and H sp , B are defined on different spaces, weneed to consider the trivial extension of H sp , B st to a higher dimensional space. Theextended Lagrangian is then degenerate and obviously not Tonelli. Moreover, thestandard C norm of the Lagrangian becomes unbounded in the limit process. Wenevertheless obtain the convergence of weak KAM solutions.While this paper is mainly motivated by Arnold diffusion, we hope our treatment ofpartial averaging is of independent interest.The plan of the paper is as follows. The rigorous formulation of the results will bepresented in section 2. The choice of the basis is handled in section 3, and the estimatesof the vector fields, including the geometrical result is in section 4. The variationalaspect is more involved, and occupies sections 5 and 6, with some technical estimatesdeferred to section 7 . As we already mentioned Appendices A and B are devoted toapplication of dominant systems to Arnold diffusion. In Appendix C we prove Theorem2.3 about existence of normally hyperbolic invariant cylinders stated in section 2.4. Recall that a slow system is H sp , B p ϕ, I q “ K p , B p I q ´ U p , B p ϕ q , ϕ P T d , I P R d , defined for a rank- d irreducible lattice Λ with ordered basis B “ r k , ¨ ¨ ¨ , k d s , and apoint p P R n . Let Q p p q “ B pp H p p q P Sym p n q , where Sym p n q denote the space of n ˆ n symmetric matrices. Define Q p p q “ „ Q p p q
00 0 P Sym p n ` q , (2.1)then (1.3) becomes K p , B p I , ¨ ¨ ¨ , I d q “ Q p p qp I k ` ¨ ¨ ¨ ` I d k d q ¨ p I k ` ¨ ¨ ¨ ` I d k d q . (2.2)7et us also denote Z B p ϕ , ¨ ¨ ¨ , ϕ d , p q “ ÿ l P Z d h l k `¨¨¨ l d k d p p q e πi p l ϕ `¨¨¨` l d ϕ d q , (2.3)where H p θ, p, t q “ ř k P Z d ` h k p p q e πik ¨p θ,t q , then (1.4) becomes U p , B p ϕ q “ ´ Z B p ϕ, p q . We now fix a rank- m irreducible resonant lattice Λ st , called the strong lattice, andstudy all irreducible lattices Λ Ą Λ st of rank d . Fix a basis B st “ r k , ¨ ¨ ¨ , k m s of thestrong lattice Λ st , we extend it to an adapted basis B “ r k , ¨ ¨ ¨ , k d s of Λ. The extendedbasis is of course non-unique, and our first theorem concerns the choice of a “nice”basis. When B is an adapted basis, the corresponding slow system is H sp , B is denoted H sp , B st , B wk , where B st “ r k , ¨ ¨ ¨ , k m s “ r k st1 , ¨ ¨ ¨ , k st m s , B wk “ r k m ` , ¨ ¨ ¨ , k d s “ r k wk1 , ¨ ¨ ¨ , k wk d ´ m s . Denote ϕ st “ p ϕ , ¨ ¨ ¨ , ϕ m q , ϕ wk “ p ϕ m ` , ¨ ¨ ¨ , ϕ d q ,I st “ p I , ¨ ¨ ¨ , I m q , I wk “ p I m ` , ¨ ¨ ¨ , I d q , we write H sp , B p ϕ, I q “ H sp , B st , B wk p ϕ st , ϕ wk , I st , I wk q“ K p , B st , B wk p I st , I wk q ´ U p , B st p ϕ st q ´ U wk p , B st , B wk p ϕ st , ϕ wk q , where U p , B st p ϕ st q “ ´ Z B st p ϕ st , p q , U wk p , B st , B wk p ϕ st , ϕ wk q “ ´p Z B ´ Z B st qp ϕ st , ϕ wk , p q . Note that the slow system to Λ st is H sp , B st p ϕ st , I st q “ K p , B st p I st q ´ U p , B st p ϕ st q“ K p , B st , B wk p I st , q ´ U p , B st p ϕ st q , where the second line follows directly from (2.2). In other words, H sp , B st p ϕ st , I st q canbe obtained from H sp , B st , B wk by setting I wk “ U wk p , B st , B wk “ U wk p , B st , B wk can be madearbitrarily small. Moreover, we can further decompose U wk p , B st , B wk to obtain preciseestimates on its dependence on each of the weak angles ϕ wk j . To do this we define, foreach 1 ď i ď d , B i “ r k , ¨ ¨ ¨ , k i s an ordered basis and Λ i the corresponding rank i lattice. Then B m “ B st Ă ¨ ¨ ¨ Ă B d “ B . For m ă i ď d , we write U wk p , B i ´ , B i “ ´p Z B i ´ Z B i ´ qp ϕ , ¨ ¨ ¨ , ϕ i , p q“ ´p Z B i ´ Z B i ´ qp ϕ st , ϕ wk1 , ¨ ¨ ¨ , ϕ wk i ´ m , p q . H sp , B st , B wk “ K p , B ´ U p , B st ´ U wk p , B m , B m ` ´ ¨ ¨ ¨ ´ U wk B d ´ , B d . (2.4) Theorem 2.1.
Let Λ st Ă Λ be irreducible resonance lattices of rank m and d resp.,where m ă d . Fix an ordered basis B st “ r k st1 , ¨ ¨ ¨ , k st m s of Λ st . Suppose H is C r with r ą n ` p d ´ m q` , and } H } C r “ . Then there exists a constant κ “ κ p H , B st , n q ą ,integer vectors B wk “ r k wk1 , ¨ ¨ ¨ , k wk d ´ m s with r B st , B wk s forming an adapted basis, suchthat the following hold.1. For any ď i ă j ď d ´ m , | k wk i | ă κ p ` | k wk j |q .2. For ď j ď d ´ m , we have } U wk p , B j ` m ´ , B j ` m } C ď } Z B j ` m ´ Z B j ` m ´ } C ď κ | k wk j | ´ r ` n ` p d ´ m q` . Remark.
Item 2 implies that as M p Λ | Λ st q Ñ 8 , for the specifically chosen basis, wehave } U wk p , B st , B wk } C Ñ .Item 1 says that vectors in B wk are approximately in an increasing order. This,when combined with item 2, implies that the norm of the weak potentials U p , B j ` m ´ , B j ` m are approximately in an decreasing order. This theorem is proven in section 3.We will call any Hamiltonian that satisfy the conclusions of Theorem 2.1 a dominantHamiltonian . In the next section, we define an abstract space of dominant Hamiltonians.
We start with the following data:1. A C function Q : R n Ñ Sym p n q , where Sym p n q denote the space of n ˆ n symmetric matrices. Define as before Q p p q “ „ Q p p q
00 0 , and assume D ´ Id ď Q ď D Id.2. An irreducible resonance lattice Λ st Ă Z n ` of rank 1 ď m ă n , and a basis B st “ r k st1 , ¨ ¨ ¨ , k st m s .3. Constants κ ą q ą p k , ¨ ¨ ¨ , k m q “ p k st1 , ¨ ¨ ¨ , k st m q and p k m ` , ¨ ¨ ¨ , k d q “p k wk1 , ¨ ¨ ¨ , k wk d ´ m q , and apply the same convention to the variables ϕ and I . DefineΩ m,d : “ p Z n ` q d ˆ R n ˆ C p T m q ˆ C p T m ` q ˆ ¨ ¨ ¨ ˆ C p T d q , H s : Ω m,d Ñ C p T d ˆ R d q , p B st “ r k st1 , ¨ ¨ ¨ , k st m s , B wk “ r k wk1 , ¨ ¨ ¨ , k wk d ´ m s , p , U st , U wk “ t U wk1 , ¨ ¨ ¨ , U wk d ´ m uq ÞÑ H s p B st , B wk , p , U st , U wk q “ K p , B st , B wk p I q´ U st p ϕ , ¨ ¨ ¨ , ϕ m q´ d ´ m ÿ j “ U wk j p ϕ , ¨ ¨ ¨ , ϕ j ` m q , where K p , B st , B wk p I q “ Q p p qp k I ` ¨ ¨ ¨ k d I d q ¨ p k I ` ¨ ¨ ¨ ` k d I d q . We equip Ω m,d with the product topology, with discrete topology on k wk j and thestandard norms on other components. H s is smooth in p , U st , U wk1 , ¨ ¨ ¨ , U wk d ´ m . LetΩ m,d p B st q be the subset of Ω m,d with fixed B st .We define Ω m,dκ,q p B st q Ă Ω m,d p B st q to be the tuple p B wk , p , U st , U wk q satisfying thefollowing conditions:1. For any 1 ď i ă j ď d ´ m , | k wk i | ď κ p ` | k wk j |q .2. For each 1 ď j ď d ´ m , } U wk j } C ď κ | k wk j | ´ q .Each element in H s p Ω m,dκ,q q is called an p m, d q´ dominant Hamiltonian with constants p κ, q q . Define µ p B wk q “ min ď j ď d ´ m | k wk j | , then in Ω m,dκ,q p B st q , we have } U wk j } ď κµ p B wk q ´ q , i.e. the weak potential U wk : “ ř d ´ mj “ U wk j Ñ µ p B wk q Ñ 8 .We now restate Theorem 2.1 using the formal definition. Theorem (Theorem 2.1 restated) . Under the assumptions of Theorem 2.1, there existsa constant κ “ κ p H , B st , n q ą , integer vectors B wk “ r k wk1 , ¨ ¨ ¨ , k wk d ´ m s with B st , B wk forming an adapted basis, such that p B wk , p, U p , B st , p U p , B m , B m ` , ¨ ¨ ¨ , U p , B d ´ , B d qq P Ω m,dκ,r ´ n ´ p d ´ m q´ p B st q . The strong Hamiltonian is defined by the mapping H st : R n ˆ C p T m q Ñ C p T m ˆ R m q , H st p p , U st q “ K p , B st , B wk p I st , q ´ U st p ϕ st q . We extend the definition to Ω m,d by writing H st p B st , B wk , p , U st , U wk q “ H st p p , U st q .We will prove all our limit theorems in the space Ω m,dκ,q p B st q . We fix B st , κ ą p B wk , p, U st , U wk q P Ω m,dκ,q p B st q . Denote H s “ H s p B st , B wk , p, U st , U wk q , H st “ H st p p, U st q . H s p ϕ, I q “ K p I q ´ U st p ϕ st q ´ U wk p ϕ st , ϕ wk q , H st p ϕ st , I st q “ K p I st , q ´ U st p ϕ st q , where U wk “ ř d ´ mj “ U wk j . As µ p B wk q Ñ 8 , we have } U wk } C Ñ
0. However, K p I st , I wk q is not a small perturbation of K p I st , q , in fact, as µ p B wk q Ñ 8 , K p I st , I wk q becomesunbounded (since each | k wk j | Ñ 8 , see also (2.2)) .We write B II K “ „ A BB T C , A “ B I st I st K, B “ B I st I wk K, C “ B I wk I wk K, (2.5)then p A q ij “ p k st i q T Qk st j , p B q ij “ p k st i q T Qk wk j , p C q ij “ p k wk i q T Qk wk j . (2.6)Note in particular that A “ B I st I st H st . The Hamiltonian equation for H s reads ϕ st “ AI st ` BI wk , I st “ B ϕ st U, ϕ wk “ B T I st ` CI wk , I wk “ B ϕ wk U, where U “ U st ` U wk . Then the Lagrangian vector field is ϕ st “ v st , v st “ A B ϕ st U ` B B ϕ wk U, ϕ wk “ v wk , v wk “ B T B ϕ st U ` C B ϕ wk U, (2.7)which will be compared to the Lagrangian vector field of H st ϕ st “ v st , v st “ A B ϕ st U st , (2.8)denoted X st . To show that the projection of (2.7) converges to (2.8), we only need toshow } B B ϕ wk U } Ñ I wk variable. Introduce the coordinate change p ϕ st , v st , ϕ wk , v wk q ÞÑ p ϕ st , v st , ϕ wk , I wk q . (2.9)This is a “half Lagrangian” setting in the sense that p ϕ st , v st q is remain the Lagrangiansetup, while p ϕ wk , I wk q is in the Hamiltonian format. Using v st “ AI st ` BI wk , v wk “ B T I wk ` CI wk , we get v wk “ B T A ´ v st ´ ˜ CI wk , where ˜ C “ C ´ B T A ´ B is an invertible symmetric matrix. Then the half-Lagrangianequation writes ϕ st “ v st , v st “ A B ϕ st U ` B B ϕ wk U, ϕ wk “ B T A ´ v st ´ ˜ CI wk , I wk “ B ϕ wk U. (2.10)11e denote by X s p ϕ st , v st , ϕ wk , I wk q the vector field of (2.10), defined on the universalcover R m ˆ R m ˆ R d ´ m ˆ R d ´ m .Consider the trivial lift of the strong Lagrangian vector field X st , defined on theuniversal cover ϕ st “ v st , v st “ A B ϕ st U, ϕ wk “ , I wk “ , (2.11)whose vector field we denote by X st L p ϕ st , v st , ϕ wk , I wk q . We show that X st L is a rescalinglimit of X s .Given 1 ě σ ě ¨ ¨ ¨ ě σ d ´ m ą
0, let Σ “ diag t σ , ¨ ¨ ¨ , σ d ´ m u . We define a rescalingcoordinate change Φ Σ : R d Ñ R d byΦ Σ : p ϕ st , v st , ϕ wk , I wk q ÞÑ p ϕ st , v st , ˜ ϕ wk , ˜ I wk q : “ p ϕ st , v st , Σ ´ ϕ wk , Σ I wk q . (2.12)The rescaled vector field for X s is˜ X s : “ p Φ Σ q ´ X s ˝ Φ ´ , ˜ X s p ϕ st , v st , ϕ wk , I wk q “ p Φ Σ q ´ X s p ϕ st , v st , Σ ´ ϕ wk , Σ I wk q , (2.13)while X st L is unchanged under the rescaling. Theorem 2.2.
Fix B st and κ ą . Assume that q ą . Then there exists a constant M “ M p B st , Q, κ, q, d ´ m q ą , such that for p B wk , p, U st , U wk q P Ω m,dκ,q p B st q and H s “ H s p B st , B wk , p, U st , U wk q , H st “ H st p p, U st q , such that the following hold.For the rescaling parameter σ j “ | k wk j | ´ q ` , uniformly on R m ˆ R d ´ m ˆ R m ˆ R d ´ m we have } Π p ϕ st ,v st q p ˜ X s ´ X st L q} C ď M µ p B wk q ´p q ´ q , } D ˜ X s ´ DX st L } C ď M µ p B wk q ´ q ´ . In particular, Theorem 2.2 implies that as µ p B wk q Ñ 8 , the vector field ˜ X s convergesto X st L in the C topology over compact sets. Theorem 2.2 is proven in section 4.1. Our main application for Theorem 2.2 is to prove persistence of normally hyperbolicinvariant cylinders (NHICs).Let W be a manifold. For R ą
0, let B lR Ă R l denote the ball of radius R at theorigin. A 2 l ´ cylinder Λ is defined by Λ “ χ p T l ˆ B lR q , where χ : T l ˆ B lR Ñ W is anembedding.Let φ t be a C flow on W , and Λ Ă W be a cylinder. We say that Λ is normallyhyperbolic (weakly) invariant cylinder (NHWIC) if there exists t ą φ t is tangent to Λ at every z P Λ.12 For each z P Λ a , there exists a splitting T z M “ E c p z q ‘ E s p z q ‘ E u p z q , where E c p z q “ T z Λ , weakly invariant in the sense that Dφ t p z q E σ p z q “ E σ p φ t z q , if z, φ t z P Λ and σ “ c, s, u. • There exists 0 ă α ă β ă C Riemannian metric g called the adaptedmetric on a neighborhood of Λ such that whenever z, φ t z P Λ, } Dφ t p z q| E s } , }p Dφ p z q| E u q ´ } ă α, }p Dφ t p z q| E c q ´ } , } Dφ t p z q| E c } ą β, where the norms taken is with respect to the metric g .The cylinder is called normally hyperbolic (fully) invariant if it satisfies the aboveconditions, and both Λ and B Λ are invariant under φ t . A more common definition ofnormally hyperbolic (fully) invariant cylinders assumes a spectral radius condition, butour definition is equivalent, see e.g. [BS02] Prop.5.2.2.Moreover:• If the parameters α, β satisfies the bunching condition α ă β , then the bundles E s , E u are C smooth.• When E s , E u are smooth, we can always choose the adapted metric g such that E s , E u and E c are orthogonal.Recall that X st Lag , X sLag denotes the Lagrangian vector fields. Suppose X st Lag admits anormally hyperbolic (fully) invariant cylinder Λ st , we claim that X sLag admits an weaklyinvariant cylinder diffeomorphic to Λ st ˆ p T d ´ m ˆ R d ´ m q . Theorem 2.3.
Consider a strong lattice B st , a strong potential U st P C p T m q , κ ą , a ą , and q ą .Assume that the Euler-Lagrange vector field X st Lag of H st “ H st p B st , p , U st q admitsa l ´ dimensional cylinder Λ st a “ χ st p T l ˆ B l ` a q that is normally hyperbolic (fully)invariant, with the parameters ă α ă β ă .Then there exists an open set V Ą Λ st0 such that for any δ ą , there exists M ą , such that for any p B st , B wk , p , U st q P Ω κ,qm,d p B st q , H s “ H s p B st , B wk , p , U st q , thefollowing hold.There exists a C embedding η s “ p η st , η wk q : p T l ˆ B l q ˆ p T d ´ m ˆ R d ´ m q Ñ p T m ˆ B m q ˆ p T d ´ m ˆ R d ´ m q , such that Λ s “ η s pp T l ˆ B l q ˆ p T d ´ m ˆ R d ´ m qq is a p l ` d ´ m q´ dimensional NHWICunder X sLag . Moreover, we have } η st ´ χ st } C ă δ, and any X sLag ´ invariant set contained in V ˆ p T d ´ m ˆ R d ´ m q is contained in Λ s . α ă β is not necessary, and is assumed for simplicity of theproof. Nevertheless, the assumption is satisfied in our intended application and in mostperturbative settings. The proof is presented in Appendix C. We will develop a similar perturbation theory for the weak KAM solutions of thedominant Hamiltonian. The weak KAM solution is closely related to some importantinvariant sets of the Hamiltonian system, known as the Mather, Aubry and Mañe sets.•
Preliminaries in weak KAM solutions
In this section we give only enough concepts to formulate our theorem. A moredetailed exposition will be given in Section 5.1. Let H : T d ˆ R d Ñ R be a C Hamiltonian satisfying the condition D ´ Id ď B II H p ϕ, I q ď D Id. Theassociated Lagrangian L : T d ˆ R d Ñ R is given by L H p ϕ, v q “ sup I P R n t I ¨ v ´ H p ϕ, I qu . Let c P R d » H p T d , R q , we define Mather’s alpha function to be α H p c q “ ´ inf ν "ż p L H ´ c ¨ v q dν * , where the infimum is taken over all Borel probability measures on T d ˆ R d that isinvariant under the Euler-Lagrange flow of L H .A continuous function u : T d Ñ R is called a (backward) weak KAM solution to L H ´ c ¨ v if for any t ą
0, we have u p x q “ inf y P T d ,γ p q“ y,γ p t q“ x ˆ u p y q ` ż t p L H p γ p t q , γ p t qq ´ c ¨ γ p t q ` α H p c qq dt ˙ , where γ : r , t s Ñ T d is absolutely continuous. Weak KAM solutions exist andare Lipschitz (see [Fat08], [Ber10]).• The relation between Lagrangians
We now turn to the weak KAM solutionsof dominant Hamiltonians. Fix B st and consider p B wk , p, U st , U wk q P Ω m,d p B st q and write H s “ H s p B st , B wk , p, U st , U wk q , H st “ H st p p, B st , U st q . Note H s p ϕ st , ϕ wk , I st , I wk q “ K p I st , I wk q ´ U st p ϕ st q ´ U wk p ϕ st , ϕ wk q , U wk “ ř d ´ mj “ U wk j , and H st p ϕ st , I st q “ K p I st , q ´ U st p ϕ st q . Denote L s “ L H s and L st “ L H st , we have L s p ϕ st , ϕ wk , v st , v wk q “ L s p v st , v wk q ` U st p ϕ st q ` U wk p ϕ st , ϕ wk q ,L st p ϕ st , v st q “ L st0 p v st q ` U st p ϕ st q , where L s , L st0 are quadratic functions with pB vv L s q “ pB II K q ´ and pB v st v st L st0 q “pB I st I st K q ´ as matrices.Given c “ p c st , c wk q P R m ˆ R d ´ m “ R d , we show that the weak KAM solution of L s ´ c ¨ v is related to the weak KAM solution of L st ´ ¯ c ¨ v st , where ¯ c is computedusing an explicit formula. More precisely, we define¯ c “ c st ` A ´ Bc wk , where B II K “ „ A BB T C as in (2.6). Then (we refer to section 4.2 for details) L s p ϕ st , ϕ wk , v st , v wk q ´ p c st , c wk q ¨ p v st , v wk q “ L st p ϕ st , v st q ´ ¯ c ¨ v st ` p v wk ´ B T A ´ v st ´ ˜ Cc wk q ¨ ˜ C ´ p v wk ´ B T A ´ v st ´ ˜ Cc wk q` c wk ¨ ˜ Cc wk ` U wk p ϕ st , ϕ wk q , where ˜ C “ C ´ B T A ´ B . The above computation suggests a connection betweenthe Lagrangian L s ´ c ¨ v and L st ´ ¯ c ¨ v st . Indeed, in Proposition 5.5 we show α H s p c q ´ } U wk } C ď α H st p ¯ c q ` c wk ¨ ˜ Cc wk ď α H s p c q ` } U wk } C . • Semi-continuity of weak KAM solutions
We now state our main variational results. We consider a sequence of domi-nant Hamiltonians with µ p B wk q Ñ 8 , and cohomology classes c i such that thecorresponding ¯ c i converge. Then the weak KAM solutions has a converging subse-quence, and the limit point is the weak KAM solution of the strong Hamiltonian.This is sometimes referred to as upper semi-continuity. Theorem 2.4.
Fix B st and κ ą . Assume that q ą p d ´ m q .For p P R n , U st0 P C p T m q and ¯ c P R m , we consider a sequence p B wk i , p i , U st i , U wk i q P Ω m,dκ,q p B st q , c i “ p c st i , c wk i q P R m ˆ R d ´ m , nd let u i be a weak KAM solution of L H s p B st , B wk i ,p i ,U st i , U wk i q ´ c i ¨ v. Denote K i “ K p i , B st , B wk i , and A i “ B I st I st K i , B i “ B I st I wk K i , C i “ B I wk I wk K i . Assume: – µ p B wk i q Ñ 8 , p i Ñ p , U st i Ñ U st0 . – c st i ` A ´ i B i c wk i Ñ ¯ c .Then:1. The sequence t u i u is equi-continuous. In particular, the sequence t u i p¨q ´ u i p qu is pre-compact in the C topology.2. Let u be any accumulation point of the sequence u i p¨q ´ u i p q . Then thereexists u st : T m Ñ R such that u p ϕ st , ϕ wk q “ u st p ϕ st q , i.e, u is independent of ϕ wk .3. u st is a weak KAM solution of L H st p p ,U st0 q ´ ¯ c ¨ v st . The proof of Theorem 2.4 occupies sections 4 and 5, with some technical statementsdeferred to section 7.
Remark.
Theorem 2.1 implies that by choosing a good basis, we can express a slowsystem as a dominant system with parameters κ, q , where q “ r ´ n ´ p d ´ m q ´ .For Theorem 2.2 we need q ą , and for Theorem 2.4 we need q ą p d ´ m q . Thereforefor application to nearly integrable systems, we need r ą n ` p d ´ m q ` as stated inour main result. Using the point of view in [Ber10], the semi-continuity of the weak KAM solution isclosely related to the semi-continuity of the Aubry and Mañe sets. These propertieshave important applications to Arnold diffusion. In section 6 we develop an analog ofthese results for the dominant Hamiltonians.
In this section we prove Theorem 2.1. The proof consists of two parts: the choice ofthe basis and estimates on the norms. 16 .1 The choice of the basis
Recall that we have a fixed irreducible lattice Λ st Ă Z n ` of rank m ă n , and a fixedbasis B st “ t k , ¨ ¨ ¨ , k m u for Λ st . The following proposition describes the choice of theadapted basis for any irreducible Λ Ą Λ st . Proposition 3.1.
Let Λ st Ă Z n ` be an irreducble lattice of rank m ă n , and fix a basis k , ¨ ¨ ¨ , k m . Let Λ Ą Λ st be an irreducible lattice of rank m ă d ď n , then there exists k m ` , ¨ ¨ ¨ , k d P Z n ` such that k , ¨ ¨ ¨ , k d form a basis of Λ , and the following hold.1. For each m ă j ď d , | k j | ď ¯ M ` p d ´ m q M j , where ¯ M “ | k | ` ¨ ¨ ¨ ` | k m | , Λ j “ span Z t k , ¨ ¨ ¨ , k j u , M j “ M p Λ j | Λ j ´ q .
2. For each m ă i ă j ď d , | k i | ď ¯ M ` p d ´ m q| k j | . We now describe the choice of the vectors k m ` , ¨ ¨ ¨ , k d . We define k i “ k i for1 ď i ď m , and define k i with i ą m inductively using the following procedure. Suppose k , ¨ ¨ ¨ , k i are defined, letΛ i “ span R t k , ¨ ¨ ¨ , k i u X Λ , M i ` “ min t| k | : k P Λ z Λ i u . We define k i ` to be a vector reaching the minimum in the definition of M i ` , i.e | k i ` | “ M i ` . We have | k i | “ M i , m ă i ď d, | k j | ď | k i | , m ă j ă i ď d, but k , ¨ ¨ ¨ , k d may not form a basis. We turn them into a basis using the followingprocedure (see [Sie89]).For each j “ , ¨ ¨ ¨ , m , define c j “ min t s j : s j, k ` ¨ ¨ ¨ ` s j,j ´ k j ´ ` s j k j P Λ , s j P R ` , s j,i P R ` Y t uu . (3.1)We define c j,j ´ using a similar minimization given the value c j : c j,j ´ “ min t s j,j ´ : s j, k ` ¨ ¨ ¨ ` s j,j ´ k j ´ ` c j k j , s j,i P R ` Y t uu . We now define c j,i for 1 ď i ď j ´ c j,i , ¨ ¨ ¨ , c j,j ´ are all defined, then c j,i ´ “ min t s j,i ´ : s j, k ` ¨ ¨ ¨ ` s j,i ´ k i ´ ` c j,i k i ` ¨ ¨ ¨ ` c j,j ´ k j ´ ` c j k j P Λ ,s j, , ¨ ¨ ¨ , s j,i ´ P R ` Y t uu . k j “ c j, k ` ¨ ¨ ¨ ` c j,j ´ k j ´ ` c j k j . We have the following lemma from the geometry of numbers.
Lemma 3.2 (see [Sie89]) . Let Λ Ă Z n ` be a lattice of rank d ď n and let k , ¨ ¨ ¨ , k d be any linearly independent set in Λ . Let k j “ c j, k ` ¨ ¨ ¨ ` c j,j ´ k j ´ ` c j k j . be defined using the procedure above. Then1. For each ď j ď d , k , ¨ ¨ ¨ , k j form a basis of span R t k , ¨ ¨ ¨ , k j u X Λ over Z . Inparticular, k , ¨ ¨ ¨ , k d form a basis of Λ .2. For ď j ă d and ď i ď j ´ , we have ď c j,i ă , ă c j ď .
3. If for some ď m ď d , k , ¨ ¨ ¨ , k m already form a basis of span R t k , ¨ ¨ ¨ , k m u X Λ over Z , then k “ k , ¨ ¨ ¨ , k m “ k m .Proof. For proof of item 1, we refer to [Sie89], Theorem 18. Item 2 and 3 follow fromdefinition and item 1 as we explain below.For item 2, note that for any k j “ c j, k ` ¨ ¨ ¨ ` c j,j ´ k j ´ ` c j k j P Λ, we can alwayssubtract an integer from any c j,i or c j and remain in Λ. If the estimates do not hold,we can get a contradiction by reducing c j,i or c j .For item 3, if k , ¨ ¨ ¨ , k m is a basis (over Z ) of span R t k , ¨ ¨ ¨ , k m u X Λ, then allcoefficients of k j “ c j, k ` ¨ ¨ ¨ ` c j,j ´ k j ´ ` c j k j P Λ for j ď m must be integers. Thenthe constraints of item 2 implies c j,i “ c j “
1, namely k j “ k j . Proof of Proposition 3.1.
We choose the basis k , ¨ ¨ ¨ , k d as described. Lemma 3.2implies k j “ k j for 1 ď j ď m . Using0 ă c j ` ď , ď c j ` ,i ă , we get | k j | ď | k | ` ¨ ¨ ¨ ` | k j | “ | k | ` ¨ ¨ ¨ ` | k m | ` M m ` ` ¨ ¨ ¨ ` M j . Since M m ` ď ¨ ¨ ¨ ď M d , and ¯ M “ | k | ` ¨ ¨ ¨ ` | k m | , we get | k j | ď ¯ M ` p j ´ m q M j ď ¯ M ` p d ´ m q M j . Moreover, for i ă j , we have | k i | ď ¯ M ` p d ´ m q M i ă ¯ M ` p d ´ m q M j ď ¯ M ` p d ´ m q| k j | . We note that the basis, as chosen in Proposition 3.1, satisfies item 1 of Theorem 2.1for κ ě max t ¯ M , d ´ m u . 18 .2 Estimating the weak potential In this section we prove the second item in Theorem 2.1 and conclude its proof. Assumethat H P C r p T n ˆ R n ˆ T q with r ą n ` d ´ m `
4. Let the basis k , ¨ ¨ ¨ , k n bechosen as in Proposition 3.1. We show that there exists κ “ κ p B st , Q, n q ą m ă i ď d , } U wk p , B i ´ , B i } C ď } Z B i ´ Z B i ´ } C ď κ | k i | ´ r ` n ´ m ` . For a lattice Λ let r H s Λ p θ, p, t q “ ÿ k P Λ h k p p q e πik ¨p θ,t q , then we have p Z B i ´ Z B i ´ qp k ¨ p θ, t q , ¨ ¨ ¨ , k i ¨ p θ, t q , p q “ pr H s Λ i ´ r H s Λ i ´ qp θ, p, t q , and the norm of r H s Λ i ´ r H s Λ i ´ can be estimated using a standard estimates of theFourier series. Lemma 3.3 (c.f. [BKZ11], Lemma 2.1, item 3) . Let H p θ, p, t q “ ř k P Z n ` h k p p q e πik ¨p θ,t q satisfy } H } C r “ , with r ě n ` . There exists a constant C n depending only on n ,such that for any subset ˜Λ Ă Z n ` with min k P ˜Λ | k | “ M ą , we have } ÿ k P ˜Λ h k p p q e πik ¨p θ,t q } C ď C n M ´ r ` n ` . Since min k P Λ i z Λ i ´ | k | “ M i , we apply Lemma 3.3 to Λ i z Λ i ´ to get }r H s Λ i ´ r H s Λ i ´ } C ď C n M ´ r ` n ` i . (3.2)To estimate Z B i ´ Z B i ´ , we apply a linear coordinate change. Given k , ¨ ¨ ¨ , k i , wechoose ˆ k i ` , ¨ ¨ ¨ , ˆ k n ` P Z n ` to be coordinate vectors (unit integer vectors) so that P i : “ “ k ¨ ¨ ¨ k i ˆ k i ` ¨ ¨ ¨ ˆ k n ` ‰ is invertible. We extend p Z B i ´ Z B i ´ qp ϕ , ¨ ¨ ¨ , ϕ i q trivially to a function of p ϕ , ¨ ¨ ¨ , ϕ n ` q ,then p Z B i ´ Z B i ´ q ˆ P Ti „ θt ˙ “ pr H s Λ i ´ r H s Λ i ´ qp θ, t q . We get } Z B i ´ Z B i ´ } C ď p ` } P ´ i }qp ` }p P Ti q ´ }q}r H s Λ i ´ r H s Λ i ´ } C . We apply the following lemma in linear algebra:19 emma 3.4.
Given ď s ď n ` , let P “ “ k ¨ ¨ ¨ k s ‰ be an integer matrix withlinearly independent columns. Then there exists c n ą depending only on n such that min } v }“ } P v } “ min } v }“ p v T P T P v q “ }p P T P q ´ } ´ ě c ´ n | k | ´ ¨ ¨ ¨ | k m | ´ . In particular, if s “ n ` , then } P ´ } “ }p P T q ´ } ď c n | k | ¨ ¨ ¨ | k n ` | .Proof. We only estimate }p P T P q ´ } . Let a ij “ p P T P q ij and b ij “ p P T P q ´ ij , then usingCramer’s rule and the definition of the cofactor, we have | b ij | ď p P T P q ÿ σ ź s ‰ i a sσ p s q , where σ ranges over all one-to-one mappings from t , ¨ ¨ ¨ , m uzt i u to t , ¨ ¨ ¨ , m uzt j u .Since P is a nonsingular integer matrix, we have det p P T P q ě
1. Moreover, a ij “ k Ti k j ď n | k i || k j | . Therefore | bij | ď ÿ σ ź s ‰ i | k s || k σ p s q | ď c n p ź s ‰ i | k s |qp ź s ‰ j | k s |q , where c n is a constant depending only on n . Using the fact that the norm of a matrix isbounded by its largest entry, up to a factor depending only on dimension, by changingto a different c n , we have }p P T P q ´ } ď c n sup i,j | B ij | ď c n sup i,j p ź s ‰ i | k s |qp ź s ‰ j | k s |q ď c n p m ź s “ | k s |q . If s “ n `
1, then } P ´ } “ }p P T P q ´ } “ }p P P T q ´ } “ }p P T q ´ } .Using Lemma 3.4, there exists a constant c n ą n such that } P ´ i } “ }p P Ti q ´ } ď c n | k | ¨ ¨ ¨ | k i || ˆ k i ` | ¨ ¨ ¨ | ˆ k n ` | . We have | k | , ¨ ¨ ¨ , | k m | ď ¯ M , | ˆ k i ` | “ ¨ ¨ ¨ “ | ˆ k n ` | “
1, and from Lemma 3.2, | k m ` | , ¨ ¨ ¨ , | k i | ď ¯ M ` p d ´ m q M i . Hence there exists a constant c n, ¯ M ą } P ´ i } “ }p P Ti q ´ } ď c n, ¯ M M i ´ mi . Combine with (3.2), we get for κ “ κ p n, ¯ M q , } Z B i ´ Z B i ´ } C ď κM ´ r ` n ` ` p i ´ m q i ď κM ´ r ` n ` ` p d ´ m q i ď κ | k i | ´ r ` n ` d ´ m ` . This implies item 2 of Theorem 2.1. The proof is complete.20
Strong and slow systems of dominant Hamiltoni-ans
In this section we study the relation between Hamiltonians and the correspondingLagrangians for dominant systems. We start by comparing the Hamitonian vector fieldsand then compare their Lagrangians.
In this section we expand on section 2.3 and prove Theorem 2.2. Fix B st , κ ą p B wk , p, U st , U wk q P Ω m,dκ,q p B st q , we recall the notations H s “ H s p B st , B wk , p, U st , U wk q , H st “ H st p p, U st q . Then H s p ϕ, I q “ K p I q ´ U st p ϕ st q ´ U wk p ϕ st , ϕ wk q , H st p ϕ st , I st q “ K p I st , q ´ U st p ϕ st q . Recall from (2.5) that B II K “ „ A BC D , then p A q ij “ p k st i q T Qk st j , p B q ij “ p k st i q T Qk wk j , p C q ij “ p k wk i q T Qk wk j . The vector field X s p ϕ st , v st , ϕ wk , I wk q defined on the universal cover R m ˆ R m ˆ R d ´ m ˆ R d ´ m is obtained from the Lagrangian vector field via the coordinate change˜ CI wk “ B T A ´ v st ´ v wk (see (2.9)). The vector field X st L p ϕ st , v st , ϕ wk , I wk q is defined asa trivial extension of the Lagrangian vector field of H st , also defined on the universalcover. More explicitly (see (2.10), (2.11)) X s “ »——– v st A B ϕ st U ` B B ϕ wk UB T A ´ v st ´ ˜ CI wk B ϕ wk U fiffiffifl , X st L “ »——– v st A B ϕ st U st fiffiffifl . (4.1)Given 1 ě σ ě ¨ ¨ ¨ ě σ d ´ m ą
0, let Σ “ diag t σ , ¨ ¨ ¨ , σ d ´ m u . The rescaling isΦ Σ : R d Ñ R d , given by (2.12). We denote by ˜ X s p ϕ st , v st , ˜ ϕ wk , ˜ I wk q the rescaled X s .Using (4.1), we have˜ X s ´ X st L “ p Φ Σ q ´ X s ˝ Φ ´ ´ X st “ »——– p A B ϕ st U wk ` B B ϕ wk U wk qp ϕ st , Σ ´ ˜ ϕ wk q Σ B T A ´ v st ´ Σ ˜ C Σ ˜ I wk Σ ´ B ϕ wk U wk p ϕ st , Σ ´ ˜ ϕ wk q fiffiffifl (4.2)21oting that U st is independent of ϕ wk , so B ϕ wk U “ B ϕ wk U wk . Furthermore D p ˜ X s ´ X st L q “ p Φ Σ q ´ DX s ˝ p Φ Σ q ´ ´ X st L “ »——– A B ϕ st ϕ st U wk ` B B ϕ st ϕ wk U wk B B ϕ wk ϕ wk U wk Σ ´
00 Σ B T A ´ C ΣΣ ´ B ϕ st ϕ wk U wk ´ B ϕ wk ϕ wk U wk Σ ´ fiffiffifl . (4.3)The quantities in (4.2) and (4.3) are estimated as follows. Lemma 4.1.
Fix B st , κ ą . Assume q ą . Then there exists a constant M “ M p B st , Q, κ, q, d ´ m q such that for the parameters σ j “ | k wk j | ´ q ` , uniformly over R m ˆ R m ˆ R d ´ m ˆ R d ´ m , the following hold.1. For any ď i ď m and ď j ď d ´ m , }B ϕ wk j U wk } C , }B ϕ st i ϕ wk j U wk } C ď M | k wk j | ´ q ;for any ď i, j ď d ´ m , }B ϕ wk i ϕ wk j U wk } C ď M sup t| k wk i | ´ q , | k wk j | ´ q u .2. } A B ϕ wk U wk } C ď M sup j t| k wk j | ´ q u , } A B ϕ st ϕ st U wk } C ď M sup j t| k wk j | ´ q u .3. } B B ϕ wk U wk } C ď M sup j t| k wk j | ´p q ´ q u , } B B ϕ st ϕ wk U } C ď M sup j t| k wk j | ´p q ´ q u .4. } B B ϕ st ϕ wk U Σ ´ } C ď M sup j t| k wk j | ´ q ´ u .5. } Σ ´ B ϕ wk ϕ wk U Σ ´ } C ď M sup j t| k wk j | ´ q ´ u .6. } Σ B T A ´ } C ď M sup j t| k wk j | ´ q ´ u .7. } Σ ˜ C Σ } C ď M sup j t| k wk j | ´ q ´ u . We first prove Theorem 2.2 using our lemma.
Proof of Theorem 2.2.
Noting that Π p ϕ st ,v st q p ˜ X s ´ X st L q is the first and third line of (4.2),using item 2 and 3 of Lemma 4.1 we get } Π p ϕ st ,v st q p ˜ X s ´ X st L q} ď M sup j t| k wk j | ´p q ´ q u “ M ˚ µ p B wk q ´p q ´ q , for any constant M ě M ˚ , where M is from Lemma 4.1.Since D p ˜ X s ´ X st L q is bounded, up to a universal constant, the sum of the norms ofall the non-zero blocks in (4.3), using Lemma 4.1 items 4-8, we get } D ˜ X s ´ DX st L } ď M sup j t| k wk j | ´ q ´ u “ M ˚ µ p B wk q ´ q ´ , where M depends only on M . 22he rest of the section is dedicated to proving Lemma 4.1. Proof of Lemma 4.1.
Denote ¯ M “ | k st1 | ` ¨ ¨ ¨ ` | k st m | , which depends only on B st . Item 1 . We have }B ϕ wk j U wk } C ď d ´ m ÿ l “ }B ϕ wk j U wk l } C ď ÿ l ě j }B ϕ wk j U wk l } C ď κ ÿ l ě j | k wk l | ´ q ď p d ´ m q κ q ` | k wk j | ´ q , where the second inequality is due to U wk l depending only on p ϕ wk1 , ¨ ¨ ¨ , ϕ wk l q , and thelast two inequalities uses the definition of Ω m,dκ,q , see section 2.2. By the same reasoning,we have }B ϕ st i ϕ wk j U wk } ď ÿ l ě j } U wk l } C ď p d ´ m q κ q ` | k wk j | ´ q , }B ϕ wk i ϕ wk j U wk } ď ÿ l ě sup t i,j u } U wk l } C ď p d ´ m q κ q ` sup t| k wk i | ´ q , | k wk j | ´ q u the second and third estimate follows. Item 2 .We have |p A B ϕ st U wk q j | “ | ÿ i p k st i q T Qk st j B ϕ st j U wk | ď m ¯ M } Q }}B ϕ st j U }ď m p d ´ m q ¯ M } Q } κ q ` | k wk j | ´ q , where the last line is due to item 1. Similarly, |p A B ϕ st ϕ st U wk q ij | ď ¯ M } Q }}B ϕ st i ϕ st j U } ď p d ´ m q ¯ M } Q } κ q ` | k wk j | ´ q . Since the vector or matrix norm is bounded by the supremum of all matrix entries, upto a constant depending only on dimension, item 2 follows. In the sequel, we apply thesame reasoning and only estimate the supremum of matrix/vector entries.
Item 3 . Similar to item 2, |p B B ϕ wk U q j | “ | ÿ i p k st i q T Qk wk j B ϕ wk j U wk | ď p d ´ m q ¯ M } Q }| k wk j |}B ϕ wk j U wk }ď p d ´ m q ¯ M } Q }p d ´ m q κ q ` | k wk j | ´p q ´ q , while |p B B ϕ st ϕ wk U wk q ij | “ |p k st i q T Qk wk j B ϕ st i ϕ wk j U wk | ď p d ´ m q κ q ` ¯ M } Q }| k wk j | ´p q ´ q . tem 5 . |p B B ϕ wk ϕ wk U Σ ´ q ij | “ | ÿ l p k st i q T Qk wk l B ϕ wk l ϕ wk j U σ ´ j | ď ¯ M } Q } ÿ l ě j | k wk l | σ ´ j |B ϕ wk l ϕ wk j U |ď ¯ M } Q }p d ´ m q κ q ` | k wk j || k wk j | ´ q | k wk j | q ` “ ¯ M } Q }p d ´ m q κ q ` | k wk j | ´ q ´ , where the inequality of the second line uses | k wk l | ď κ | k wk j | , item 1 and the choice of σ j . Item 6 . Using item 1 and choice of σ j , we have |p Σ ´ B ϕ wk ϕ wk U wk Σ ´ q ij | “ | σ ´ i B ϕ wk i ϕ wk j U wk σ ´ j |ď p d ´ m q κ q ` σ ´ i σ ´ j sup t| k wk i | ´ q , | k wk j | ´ q uď p d ´ m q κ q ` sup t| k wk i | ´ q ´ , | k wk j | ´ q ´ u . Item 7 . We have |p Σ B T q ij | “ | σ i p k wk i q T Qk st j | ď ¯ M } Q } sup j t| k wk j | σ j u “ ¯ M } Q } sup j t| k wk j | ´ q ´ u and uses } Σ B T A ´ } ď } Σ B T }} A ´ } , noting that } A ´ } depends only on Q and B st . Item 8 . Recall ˜ C “ C ´ B T A ´ B . We have |p Σ C Σ q ij | “ | σ i p k wk i q T Qk wk j σ j | ď p sup j σ j | k wk j |q } Q } ď } Q } sup j t| k wk j | ´ q ´ u . Suppose S , S are positive definite symmetric matrices with S ě S , for any v P R d ´ m , v T S v “ v T p S ´ S ` S q v ě v T S v, we obtain } S } ě } S } . Since C ´ B T A ´ B ě
0, we have Σ C Σ ´ Σ B T A ´ B Σ ě C Σ and Σ B T A ´ B Σ we get } Σ ˜ C Σ } ď } Σ C Σ } ` } Σ B T A ´ B Σ } ď } Σ C Σ } . Item 8 follows.
We derive the special form of the slow Lagrangian described in section 2.5. We fix B st , κ ą p B wk , p, U st , U wk q P Ω m,dκ,q p B st q . Denote H s “ H s p B st , B wk , p, U st , U wk q , H st “ H st p p, U st q and the associated Lagrangian is denoted L s and L st .As before we write H s p ϕ, I q “ K p I q ´ U st p ϕ st q ´ U wk p ϕ st , ϕ wk q , H st p ϕ st , I st q “ K p I st , q ´ U st p ϕ st q , L s p ϕ, v q “ L s p v q ` U st p ϕ st q ` U wk p ϕ st , ϕ wk q , L st p ϕ st , v st q “ L st0 p v st q ` U st p ϕ st q , where B vv L s “ pB II K q ´ , B v st v st L st0 “ pB I st I st K q ´ for v “ B I K and v st “ B I st K . Recallthe notation B II K “ „ A BB T C , A “ B I st I st K, B “ B I st I wk K, C “ B I wk I wk K. Lemma 4.2.
With the above notations we have1. L s p v, ϕ q “ L st p ϕ st , v st q` p v wk ´ B T A ´ v st q ¨ ˜ C ´ p v wk ´ B T A ´ v st q ` U wk p ϕ st , ϕ wk q , (4.4) where ˜ C “ C ´ B T A ´ B.
2. Let c “ p c st , c wk q P R m ˆ R d ´ m . We denote ¯ c “ c st ` A ´ Bc wk , w wk “ v wk ´ B T A ´ v st , (4.5) then L s p v, ϕ q ´ c ¨ v “ L st p ϕ st , v st q ´ ¯ c ¨ v st ` p w wk ´ ˜ Cc wk q ¨ ˜ C ´ p w wk ´ ˜ Cc wk q ´ c wk ¨ ˜ Cc wk ` U wk p ϕ wk , ϕ st q . (4.6) Proof.
We have the following identity in block matrix inverse, which can be verified bya direct computation. „ A BB T C ´ “ „ A ´
00 0 ` „ ´ A ´ B Id ˜ C ´ “ ´ B T A ´ Id ‰ . Then L s p v st , v wk q “ “ p v st q T p v wk q T ‰ ˆ„ A ´
00 0 ` „ Id ´ A ´ B ˜ C ´ “ Id ´ B T A ´ ‰˙ „ v st v wk “ v st ¨ A ´ v st ` p v wk ´ B T A ´ v st q ¨ ˜ C ´ p v wk ´ B T A ´ v st q“ L st0 p v st q ` p v wk ´ B T A ´ v st q ¨ ˜ C ´ p v wk ´ B T A ´ v st q , We stress here that no coordinate change is performed: w wk is simply an abbreviation for v wk ´ B T A ´ v st . L s ´ p c st , c wk q ¨ p v st , v wk q“ L st0 p v st q ´ p c st ` A ´ Bc wk q ¨ v st ` w wk ¨ ˜ C ´ w wk ´ c wk ¨ v wk ` A ´ Bc wk ¨ v st “ L st0 p v st q ´ ¯ c ¨ v st ` w wk ¨ ˜ C ´ w wk ´ c wk ¨ p v wk ´ B T A ´ v st q“ L st0 p v st q ´ ¯ c ¨ v st ` w wk ¨ ˜ C ´ w wk ´ p ˜ Cc wk q ¨ ˜ C ´ w wk “ L st0 p v st q ´ ¯ c ¨ v st ` p w wk ´ ˜ Cc wk q ¨ ˜ C ´ p w wk ´ ˜ Cc wk q ´ c wk ¨ ˜ Cc wk . We obtain (4.6).The Euler-Lagrange flow of L s satisfies the following estimates. Lemma 4.3.
Fix B st , κ ą . Assume that q ą , L s “ L H s p B st , B wk ,p,U st , U wk q , with p B wk , p, U st , U wk q P Ω m,dκ,q p B st q . Let γ “ p γ st , γ wk q : r , T s Ñ T d satisfy the Euler-Lagrange equation of L s .1. There exists a constant M “ M p B st , Q, κ, q q such that } : γ st ´ A B ϕ st U st p γ st q} C ď M p µ p B wk qq ´p q ´ q .
2. There exists a constant M “ M p B st , Q, κ, q, } U st }q such that } : γ st } C ď M . Proof.
Observe that the p ϕ st , v st q component of the Euler-Lagrange vector field of L s isprecisely the vector field Π ϕ st ,v st ˜ X s in Theorem 2.2. The Euler-Lagrange equation of L st (which is X st in Theorem 2.2) is : ϕ st “ A B ϕ st U st . Hence item 1 is a rephrasing ofthe first conclusion of in Theorem 2.2.Since } A B ϕ st U st } ď } A }} U st } , and } A } depends only on B st and Q , item 2 followsdirectly from item 1. In this section, we provide some basic information about the weak KAM solution of thedominant system.In section 5.1, we give an overview on the relevant weak KAM theory. Recall thatin section 4.2, we derive the relation between the slow Lagrangian and the strongLagrangian. In section 5.2, we obtain a compactness result for the strong component ofa minimizing curve. In section 5.3 to 5.5, we prove Theorem 2.4 with some technicalstatements deferred to section 7. 26 .1 Weak KAM solutions of Tonelli Lagrangian
For an extensive exposition of the topic, we refer to [Fat08].
Tonelli Lagrangian.
The Lagrangian function L “ L p ϕ, v q : T d ˆ R d Ñ R is calledTonelli if it satisfies the following conditions.1. (smoothness) L is C r with r ě B vv L is strictly positive definite.3. (superlinearity) lim } v }Ñ8 | L p x, v q|{} v } “ 8 .The Lagrangians considered in this paper are Tonelli. Minimizers.
An absolutely continuous curve γ : r a, b s Ñ T d is called minimizing for theTonelli Lagrangian L if ż ba L p γ, γ q dt “ min ξ ż ba L p ξ, ξ q dt, where the minimization is over all absolutely continuous curves ξ : r a, b s Ñ T d with b ą a , such that ξ p a q “ γ p a q , ξ p b q “ γ p b q . The functional A p γ q “ ż ba L p γ, γ q dt is called the action functional . The curve γ is called an extremal if it is a critical pointof the action functional. A minimizer is extremal, and it satisfies the Euler-Lagrangeequation ddt pB v L p γ, γ qq “ B ϕ L p γ, γ q . Tonelli Theorem and a priori compactness.
By the Tonelli Theorem (c.f [Fat08],Corollary 3.3.1), for any r a, b s Ă R with b ą a , ϕ, ψ P T d , there always exists a C r minimizer. Moreover, there exists D ą b ´ a suchthat } γ } ď D ([Fat08] Corollary 4.3.2). This property is called the a priori compactness. The alpha function and minimal measures.
A measure µ on T d ˆ R d is called a closedmeasure (see [Sor10], Remark 4.40) if for all f P C p T d q , ż df p ϕ q ¨ v dµ p ϕ, v q “ . This notion is equivalent to the more well known notion of holonomic measure definedby Mañe ([Mañ97]).For c P H p T d , R q » R d , the alpha function α L p c q “ ´ inf ν ż p L p ϕ, v q ´ c ¨ v q dν p ϕ, v q , L “ L H we also use the notation α H p c q . A measure µ is called a c ´ minimizing if it reachesthe infimum above. A minimizing measure always exists, and is invariant under theEuler-Lagrange flow (c.f [Mañ97; Ber08]). Hence this definition of the alpha functionis equivalent to the one given in section 2.5, where the minimization is over invariantprobability measures. Rotation number and the beta function.
The rotation number ρ of a closed measure µ isdefined by the relation ż p c ¨ v q dµ p ϕ, v q “ c ¨ ρ, for all c P H p T d , R q . For h P H p T d , R q » R d , the beta function is β L p h q “ inf ρ p ν q“ h ż L p ϕ, v q dν p ϕ, v q . When L “ L H we use the notation β H p h q . The alpha function and beta function areLegendre duals: β L p h q “ sup c P R d t c ¨ h ´ α L p c qu . The Legendre-Fenichel transform.
Define the Legendre-Fenichel transform associated tothe beta function LF β : H p T d , R q Ñ the collection of nonempty, compact convex subsets of H p T d , R q , (5.1)defined by LF β p h q “ t c P H p T n , R q : β L p h q ` α L p c q “ c ¨ h u . Domination and calibration.
For α P R , a function u : T d Ñ R is dominated by L ` α if for all r a, b s Ă R and piecewise C curves γ : r , T s Ñ T d , we have u p γ p b qq ´ u p γ p a qq ď ż ba L p γ, γ q dt ` α p b ´ a q . A piecewise C curve γ : I Ñ R defined on an interval I Ă R is called p u, L, α q -calibratedif for any r a, b s Ă I , u p γ p b qq ´ u p γ p a qq “ ż ba L p γ, γ q dt ` α p b ´ a q . Weak KAM solutions.
A function u : T d Ñ R is called a weak KAM solution of L ifthere exists α P R such that the following hold.28. u is dominated by L ` α .2. For all ϕ P T d , there exists a p u, L, α q -calibrated curve γ : p´8 , s Ñ T d with γ p q “ ϕ .This definition of the weak KAM solution is equivalent to the one given in section 2.5(see [Fat08], Proposition 4.4.8), and the constant α “ α L p q , where α L is the alphafunction. Peierls’ barrier.
For T ą
0, we define the function h TL : T d ˆ T d Ñ R by h TL p ϕ, ψ q “ min γ p q“ ϕ,γ p T q“ ψ ż T p L p γ, γ q ` α L q dt. Peierls’ barrier is h L p ϕ, ψ q “ lim T Ñ8 h TL p ϕ, ψ q . The limit exists, and the function h L isLipschitz in both variables. Denote h L,c “ h L ´ c ¨ v . Mather, Aubry and Mañe sets . These sets are defined by Mather (see [Mat93]). Herewe only introduce the projected version. Define the projected Aubry and Mañe sets as A L p c q “ t x P T d : h L,c p x, x q “ u , N L p c q “ " y P T d : min x,z P A L p c q p h L,c p x, y q ` h L,c p y, z q ´ h L,c p x, z qq “ * . The Mather set is ˜ M L p c q “ Ť µ supp p µ q is the closure of the support of all c ´ minimalmeasures. Its projection π ˜ M p c q “ M p c q onto T d is called the projected Mather set.Then M L p c q Ă A L p c q Ă N L p c q . When L “ L H we also use the subscript H to identify these sets. Static classes.
For any ϕ, ψ P A L p c q , Mather defined the following equivalence relation: ϕ „ ψ if h L,c p ϕ, ψ q ` h L,c p ψ, ϕ q “ . The equivalence classes defined by this equivalence condition are called the static classes.The static classes are linked to the family of weak KAM solutions, in particular, if thereis only one static class, then the weak KAM solution is unique up to a constant.In this section, we provide a few useful estimates in weak KAM theory, and proveTheorem 2.4. In section 5.2, we prove a projected version of the a priori compactnessproperty. We then introduce an approximate version of Lipschitz property and use itto prove Theorem 2.4. 29 .2 Minimizers of strong and slow Lagrangians, their a prioricompactness
We prove a version of the a priori compactness theorem for the strong component.
Proposition 5.1.
Fix B st , κ ą . For any R ą , there exists M “ M p B st , Q, R, κ q such that the following hold. For any p B wk , p, U st , U wk q P Ω m,dκ,q p B st q X t} U st } C ď R u and L s “ L H s p B st , B wk ,p,U st , U wk q , let T ě , c “ p c st , c wk q P R m ˆ R d ´ m and γ “ p γ st , γ wk q : r , T s Ñ T d be a minimizerof L s ´ c ¨ v . Then for ¯ c “ c st ` A ´ Bc wk , we have } γ st ´ A ¯ c } ď M. We first state a lemma on the strong component of the action and relate minimizersof the slow system with those of the strong one.
Lemma 5.2.
In the notations of Proposition 5.1 for T ě and c P R d , let γ “p γ st , γ wk q : r , T s Ñ T d be a minimizer for the lagrangian L s ´ c ¨ v . Then ż T p L st ´ ¯ c ¨ v st qp γ st , γ st q dt ď min ζ ż T p L st ´ ¯ c ¨ v st qp ζ, ζ q dt ` T } U wk } C , where the minimization is over all absolutely continuous ζ : r , T s Ñ T m with ζ p q “ γ st p q , ζ p T q “ γ st p T q .Proof. Let γ st0 : r , T s Ñ T m be such that ż T p L st ´ ¯ c ¨ v st qp γ st0 , γ st0 q dt “ min ζ ż T p L st ´ ¯ c ¨ v st qp ζ, ζ q dt with ζ p q “ γ st p q , ζ p T q “ γ st p T q . Define γ “ p γ st0 , γ wk0 q : r , T s Ñ T d , by γ wk0 p t q “ γ wk p t q ´ A ´ Bγ st p t q ` A ´ Bγ st0 p t q . Note that γ wk0 p q “ γ wk p q , γ wk0 p T q “ γ wk p T q , γ wk0 ´ A ´ B γ st0 “ γ wk ´ A ´ B γ st . (5.2)Using (4.6) and (5.4), we have L s ´ c ¨ v ` c wk ¨ ˜ C ´ c wk “ L st ´ ¯ c ¨ v st ` p v wk ´ B T A ´ v st ´ ˜ C wk q ¨ ˜ C p v wk ´ B T A ´ v st ´ ˜ C wk q ` U wk (5.3)30ince γ is a minimizer for L s ´ c ¨ v , ż T p L s ´ c ¨ v qp γ, γ q dt ď ż T p L s ´ c ¨ v qp γ , γ q dt. By (5.3), we have ż T p L st ´ ¯ c ¨ v st qp γ st , γ st q dt ` ż T U wk p γ p t qq dt ` p γ wk ´ B T A ´ γ st ´ ˜ C wk q ¨ ˜ C p γ wk ´ B T A ´ γ st ´ ˜ C wk qď ż T p L st ´ ¯ c ¨ v st qp γ st0 , γ st0 q dt ` ż T U wk p γ p t qq dt ` p γ wk0 ´ B T A ´ γ st0 ´ ˜ C wk q ¨ ˜ C p γ wk0 ´ B T A ´ γ st0 ´ ˜ C wk q . By (5.2), the second and fourth line of the above inequality cancels, therefore ż T p L st ´ ¯ c ¨ v st qp γ st , γ st q dt ď ż T p L st ´ ¯ c ¨ v st qp γ st0 , γ st0 q dt ` T } U wk } C . Proof of Proposition 5.1.
First, observe that any segments of a minimizer is still aminimizer. By dividing the interval r , T s into subintervals, it suffice to prove ourproposition for T P r , q .We first produce an upper bound formin ζ ż T p L st ´ ¯ c ¨ v st `
12 ¯ c ¨ A ¯ c qp ζ, ζ q dt. By completing the squares as in Lemma 4.2, we have L st ´ ¯ c ¨ v st `
12 ¯ c ¨ A ¯ c “ p v st ´ A ¯ c q ¨ A ´ p v st ´ A ¯ c q ` U st p ϕ st q . (5.4)We then take ζ p t q “ γ st p q ` tA ¯ c ` tT y where y P r , q d is such that ζ p q ` T A ¯ c ` y “ γ st p T q mod Z m . We then have ζ ´ A ¯ c “ T y , so ż T p L st ´ ¯ c ¨ v st `
12 ¯ c ¨ A ¯ c qp ζ , ζ q dt ď T } A ´ }} y } ` T } U st } C ď d } A ´ } ` } U st } C using T P r , q and } y } ď d . 31sing Lemma 5.2, and adding ¯ c ¨ A ¯ c to the Lagrangian to both sides, we obtain ż T p L st ´ ¯ c ¨ v st `
12 ¯ c ¨ A ¯ c qp γ st , γ st q dt ď T } U wk } ` d } A ´ } ` } U st } C ď d } A ´ } ` } U st } C ` } U wk } C since T P r , q .We now use the above formula get an L estimate on p γ st ´ A ¯ c q and use the Poincaréestimate to conclude. Using the above formula and (5.4), we have ż T p γ st ´ A ¯ c q ¨ A ´ p γ st ´ A ¯ c q dt ď d } A ´ } ` } U st } C ` } U wk } C . Using the fact that A ´ is strictly positive definite, we get } γ st ´ A ¯ c } L ď } A }p d } A ´ } ` } U st } C ` } U wk } C q “ : M . Then ›››› T ż T p γ st ´ A ¯ c q dt ›››› ď T ż T } γ st ´ A ¯ c } dt ď M . (5.5)Moreover, from Lemma 4.3, } : γ st } ď M p B st , Q, κ, q, R q . The Poincaré estimate gives, for some uniform constant D ą ›››› p γ st ´ A ¯ c q ´ T ż T p γ st ´ A ¯ c q dt ›››› L ď } : γ st } L ď DM . Combine with (5.5) and we conclude the proof.
The weak KAM solutions of the slow Hamiltonian is Lipshitz, however, it is not clearif the Lipschitz constant is bounded as µ p B wk q Ñ 8 . To get uniform estimates, weconsider the following weaker notion. Definition.
For
D, δ ą , a function u : R d Ñ R is called p D, δ q approximatelyLipschitz if | u p x q ´ u p y q| ď D } x ´ y } ` δ, x, y P R d . For u : T d Ñ R , the approximate Lipschitz property is defined by its lift to R d . In Proposition 5.3 and 5.4 we state the approximate Lipschitz property of a weakKAM solution in weak and strong angles.32 roposition 5.3.
Fix B st , κ ą . Assume that q ą p d ´ m q . For R ą , there existsa constant M “ M p B st , Q, κ, q, R q ą , such that for all p B wk , p, U st , U wk q P Ω m,dκ,q p B st q X t} U st } ď R u , and δ p B wk q “ M µ p B wk q ´p q ´ d ` m q , let u “ u p ϕ st , ϕ wk q : T m ˆ T d ´ m Ñ R be a weak KAM solution of L H s p B st , B wk , p, U st , U wk q ´ c ¨ v. Then for all ϕ st P T m , the function u p ϕ st , ¨q is p δ, δ q approximately Lipschitz. Proposition 5.4.
There exists a constant M “ M p B st , Q, κ, q, R q ą , let δ p B wk q “ M p µ p B wk qq ´p q ´ d ` m q , and u be the weak KAM solution described in Proposition 5.3.Then for all ϕ wk P T d ´ m , the function u p¨ , ϕ wk q is p M , δ q approximately Lipschitz. The proof of these statements are deferred to section 7.
In this section we provide a few useful estimates in weak KAM theory and proveTheorem 2.4 using Propositions 5.3 and 5.4. Recall that the notations c “ p c st , c wk q , ¯ c “ c st ` A ´ Bc wk . Proposition 5.5.
We have ˇˇˇˇ α H s p c q ´ α H st p ¯ c q ` p ˜ Cc wk q ¨ c wk ˇˇˇˇ ď } U wk } C , Proof.
Let µ be a minimal measure for L s ´ c ¨ v . Let π denote the natural projectionfrom p ϕ st , ϕ wk , v st , v wk q to p ϕ st , v st q . By Lemma 4.2 we have ´ α H s p c q “ ż p L s ´ c ¨ v q dµ “ ż p L st ´ ¯ c ¨ v st q dµ ˝ π ´ c wk ¨ ˜ Cc wk ` ż ˆ p w wk ´ ˜ Cc wk q ¨ ˜ C ´ p w wk ´ ˜ Cc wk q ` U wk ˙ dµ ě ´ α H st p ¯ c q ´ } U wk } C ´ c wk ¨ ˜ Cc wk . (5.6)On the other hand, let µ st be an ergodic minimal measure for L st ´ ¯ c ¨ v st . For an L st ´ Euler-Lagrange orbit ϕ st p t q in the support of µ st , and any ϕ wk0 P T d ´ m , define ϕ wk p t q “ ϕ wk0 ` B T A ´ ϕ st p t q ` ˜ Cc wk t, t P R (5.7)33nd write γ “ p γ st , γ wk q . We take a weak- ˚ limit point µ s of the probability measures T p γ, γ q| r ,T s as T Ñ `8 . Then µ s is a closed measure (see section 5.1).Since on the support of µ s , v wk ´ B T A ´ v st ´ ˜ Cc wk “
0, we have ´ α H s p c q ď ż p L s ´ c ¨ v q dµ s “ ż p L st ´ ¯ c ¨ v st q dµ st ` ż U wk dµ ´
12 ˜ Cc wk ¨ c wk ď ´ α H st p ¯ c q ` } U } C ´
12 ˜ Cc wk ¨ c wk . The following proposition establishes relations between rotation numbers of minimalmeasures of the slow and strong systems.
Proposition 5.6.
Let µ s be an ergodic minimal measure of L s ´ c ¨ v , and let p ρ st , ρ wk q denote its rotation number. Then ď p ˜ C p ρ wk ´ B T A ´ ρ st ´ ˜ Cc wk qq ¨ p ρ wk ´ B T A ´ ρ st ´ ˜ Cc wk q ď } U wk } C and ď α H st p ¯ c q ` β H st p ρ st q ´ ¯ c ¨ ρ st ď } U wk } C . Proof.
Using (5.6) and the conclusion of Proposition 5.5, we have } U wk } C ě ż p L st ´ ¯ c ¨ v st ` α H st p ¯ c qq dµ s ˝ π ` ż p ˜ C ´ p w ´ ˜ Cc wk qq ¨ p w ´ ˜ Cc wk q dµ s ˝ π. (5.8)Note the first of the two integrals is non-negative by definition, we obtain0 ď ż p w wk ´ ˜ Cc wk q ¨ ˜ C ´ p w wk ´ ˜ Cc wk q dµ s ď } U wk } C . Denote ¯ w wk : “ ş w wk dµ s “ ρ wk ´ B T A ´ ρ st , and rewrite the left hand side of the lastformula as12 p ˜ C ´ p ¯ w wk ´ ˜ Cc wk qq ¨ p ¯ w wk ´ ˜ Cc wk q ` ż ˜ C ´ p ¯ w wk ´ ˜ Cc wk q ¨ p w wk ´ ¯ w q dµ s ` ż p ˜ C ´ p w wk ´ ¯ w wk qq ¨ p w wk ´ ¯ w wk q dµ s . Note that the second term vanishes and the third term is non-negative. Therefore12 p ˜ C ´ p ¯ w wk ´ ˜ Cc wk qq ¨ p ¯ w wk ´ ˜ Cc wk q ď } U wk } C } U wk } C ě ż p L st ´ ¯ c ¨ v st ` α H st p ¯ c qq dµ ˝ π “ ż L st dµ ˝ π ´ ¯ c ¨ ρ st ` α H st p ¯ c q . Using ş L st dµ ˝ π ě β H st p ρ st q we get the upper bound of the second conclusion. Thelower bound holds by definition. We now prove Theorem 2.4. Fix B st and κ ą p B wk i , p i , U st i , U wk i q P Ω m,dκ,q p B st q and c i “ p c st i , c wk i q be a sequence satisfying theassumption of the theorem, namely µ p B wk i q Ñ 8 , p i Ñ p , U st i Ñ U st0 in C , and c st ` A ´ i B i c wk i Ñ ¯ c . Item 1 . Let u i be the weak KAM solution to L si “ L H s p B st , B wk i ,p i ,U st i , U wk i q ´ c i ¨ v . Wefirst show the sequence t u i u is equi-continuous.Let M ˚ be a constant larger than the constants in both Proposition 5.3 and 5.4.Using both propositions, for any ϕ “ p ϕ st , ϕ wk q , ψ “ p ψ st , ψ wk q , | u i p ϕ st , ϕ wk q ´ u i p ψ st , ψ wk q| ď M ˚ } ϕ st ´ ψ st } ` δ i } ϕ wk ´ ψ wk } ` δ i , where δ i “ M ˚ p µ p B wk i qq ´ q ´ d ` m .Since δ i Ñ i Ñ 8 , for any 0 ă ε ă M ą i ą M , 3 δ i ă ε . It follows that if } ϕ ´ ψ } ă ε D ă
1, then | u i p ϕ st , ϕ wk q ´ u i p ψ st , ψ wk q| ă ε. Since t u i u i ď M is a finite family, it is equi-continuous. In particular, there exist σ ą | u i p ϕ q ´ u p ψ q| ă ε, if 1 ď i ď M, } ϕ ´ ψ } ă σ. This proves equi-continuity. Moreover, since u i are all periodic, u i ´ u i p q are equi-bounded, therefore Ascoli’s theorem applies and the sequence is pre-compact in uniformnorm. Item 2.
Let u be any accumulation point of u i ´ u i p q , without loss of generality,we assume u i ´ u i p q converges to u uniformly. Proposition 5.3 implies thatlim i Ñ8 sup ϕ st p max u i p ϕ st , ¨q ´ min i u i p ϕ st , ¨qq ď i Ñ8 δ i “ , therefore u is independent of ϕ wk . Item 3.
From item 2, there exists u st p ϕ st q “ lim i Ñ8 u i p ϕ st , ϕ wk q . We show u st is aweak KAM solution of L st0 ´ ¯ c ¨ v st “ L H st p p ,U st0 q ´ ¯ c ¨ v st . Denote L st i “ L H st p p i ,U st i q , wehave L st i Ñ L st0 in C . 35e first show that u st is dominated by L st0 ´ ¯ c ¨ v st . Let ξ st : r , T s Ñ T m be anextremal curve of L st0 . In the same way as (5.7) in the proof of Proposition 5.5, wedefine ξ i “ p ξ st i , ξ wk i q : r a, b s Ñ T m such that ξ st i p a q “ ξ st p a q , ξ st i p b q “ ξ st p b q and ξ st i ´ B Ti A ´ i ξ ´ ˜ Cc wk i “
0. Since for ¯ c i “ c st i ` A ´ i B i c wk i , u i are dominated by L si ´ c i ¨ v ` α H si p c i q , we have u i p ξ i p b qq ´ u i p ξ i p a qq ď ż ba p L si ´ c i ¨ v s ` α H si p c i qqp ξ i , ξ i q dt “ ż ba p L st i ´ ¯ c i ¨ v st qp ξ st i , ξ st i q dt ` ż ba p U wk i p ξ i q ` α H si p c i q ´
12 ˜ C i c wk i ¨ c wk i q dt, where the equality is due to ξ st i ´ B Ti A ´ i ξ i ´ ˜ Cc wk i “
0. Using the fact that } U wk i } C Ñ L st i Ñ L st0 , and from Proposition 5.5, α H si ´ ˜ C i c wk i ¨ c wk i Ñ α H st p ¯ c q as i Ñ 8 , we get u st p ξ st p b qq ´ u st p ξ st p a qq ď ż ba p L st0 ´ ¯ c ¨ v st ` α H st p ¯ c qq dt. (5.9)Therefore u st is dominated by L st0 ´ ¯ c ¨ v st .Secondly, we show that for any ϕ st P T m , there exists a p u st , L st0 , ¯ c q -calibrated curve γ st : p´8 , s Ñ T m with γ st p q “ ϕ st . Because u i are weak KAM solutions, for each i there exists a p u i , L si ´ c i ¨ v, α p H si qq -calibrated curve γ i “ p γ st i , γ wk i q : p´8 , s Ñ T d .By Proposition 5.1, all γ st i are uniformly Lipschitz, so there exists a subsequence thatconverges in C loc pp´8 , s , T d q . Assume without loss of generality that γ st i Ñ γ st , since γ i “ p γ st i , γ wk i q is extremal for L si , we have : γ st i “ ddt p A i I st ` B i I wk q “ A i B ϕ st U st i ` B i B ϕ st U wk i . By our assumption, as i Ñ 8 , A i Ñ A : “ B v st v st L st0 , and by Lemma 4.3 } B i }} U wk i } C Ñ : γ st “ A B ϕ st U st p γ st q , (5.10)which is the Euler-Lagrange equation for L st0 .On the other hand, since γ i are p u i , L si ´ c i ¨ v, α p H si qq calibrated, for any r a, b s Ăp´8 , s , u i p γ i p b qq ´ u i p γ i p a qq “ ż ba p L si ´ c i ¨ v s ` α H si p c i qqp γ i , γ i q dt ě ż ba p L st ´ ¯ c i ¨ v st qp γ st i , γ st i q dt ` ż ba p U wk i ` α H si p c i q ´
12 ˜ C i c wk i ¨ c wk i qp γ i , γ i q dt Take limit again to get u st p γ st p b qq ´ u st p γ st p a qq ě ż ba p L st0 ´ ¯ c ¨ v st ` α H st p ¯ c qqp γ st , γ st q dt. Because γ st is an L st extremal curve (see (5.10)), (5.9) hold for γ st . Combining withlast displayed formula, (5.9) becomes an equality. Then γ st is a calibrated curve for L st ´ ¯ c ¨ v st ` α H st p ¯ c q , and u st is a weak KAM solution.36 The Mañe and the Aubry sets and the barrierfunction
We prove the following result.
Proposition 6.1.
Fix B st and κ ą . Assume that p B wk i , p i , U st i , U wk i q satisfies the as-sumptions of Theorem 2.4. Denote H si “ H si p B st , B wk i , p i , U st i , U wk i q , H st0 “ H st p p , U st0 q , L si “ L H si and L st0 “ L H st0 .1. Any limit point of ϕ i P N H si p c i q is contained in N H st0 p ¯ c q ˆ T d ´ m .2. If A H st0 p ¯ c q contains only finitely many static classes, then any limit point of ϕ i P A H si p c i q is contained in A H st0 p ¯ c q ˆ T d ´ m .3. Assume that A H st p ¯ c q contains only one static class. Let ϕ i “ p ϕ st i , ϕ wk i q P A H si p c i q be such that ϕ st i Ñ ϕ st P A H st0 p ¯ c q . Then for any ψ “ p ψ st , ψ wk q P T d , lim i Ñ8 h L si ,c i p ϕ, ψ q “ h L st0 , ¯ c p ϕ st , ψ st q .
4. Let p ρ st i , ρ wk i q be the rotation number of any c i ´ minimal measure of L si . Then wehave lim i Ñ8 p ρ wk i ´ B Ti A ´ i ρ st i ´ ˜ C i c wk i q “ , and any accumulation point ρ of ρ st i is contained in the set B α H st p ¯ c q . The proof of item 2 requires additional discussion and is presented in Section 6.2.In Section 6.1 we prove item 1, 3 and 4.
We first state an alternate definition of the Aubry and Mañe sets due to Fathi (see also[Ber08]). Let u be a weak KAM solution for the Lagrangian L . We define G p L, u q tobe the set of points p ϕ, v q P T d ˆ R d such that there exists a p u, L, α L q -calibrated curve γ : p´8 , s Ñ T d , such that p ϕ, v q “ p γ p q , γ p qq . Let φ t denote the Euler-Lagrangeflow of L , then˜ I p L, u q “ č t ď φ t p G p L, u qq , ˜ A L “ č u ˜ I p L, u q , ˜ N L “ ď u ˜ I p L, u q , (6.1)where the union and intersection are over all weak KAM solutions of L . The Aubry setand Mañe set of c P H p T d , R q is defined as˜ A L p c q “ ˜ A L ´ c ¨ v , ˜ N L p c q “ ˜ N L ´ c ¨ v . The projected Aubry and Mañe sets are the projection of these sets to T d .We now turn to the setting of Proposition 6.1. Let L si , L st0 , c i , ¯ c be as in theassumption. The strategy of the proof is similar to the one in [Ber10].37 emma 6.2. Let u i be a weak KAM solution of L si ´ c i ¨ v . Assume that ˜ ϕ i “ p ϕ i , v i q P ˜ I p L si ´ c i ¨ v, u i q satisfies ˜ ϕ i Ñ ˜ ϕ “ p ϕ, v q “ p ϕ st , ϕ wk , v st , v wk q , and u i p ϕ st , ϕ wk q Ñ u st p ϕ st q . Then p ϕ st , v st q P ˜ I p L st ´ ¯ c ¨ v st , u st q . Proof.
We first show that p ϕ i , v i q P G p L si ´ c i ¨ v, u i q implies p ϕ st , v st q P G p L st ´ ¯ c ¨ v st , u st q .Indeed, there exists γ i : p´8 , s Ñ T d , each p u i , L si ´ c i ¨ v, α L si p c i qq -calibrated, with p γ i , γ i qp q “ p ϕ, v q . We follow the same line as proof of item 3 in Theorem 2.4(section 5), then by restricting to a subsequence, γ st i converges in C loc pp´8 , s , T d q to a p u st , L st0 ´ ¯ c ¨ v st , α H st0 p ¯ c qq -calibrated curve γ st . In particular p γ st i , γ st i q Ñ p γ st , γ st q , whichimplies p ϕ st , v st q P G p L st ´ ¯ c ¨ v st , u st q .Let φ it denote the Euler-Lagrange flow of L si , and φ st t the flow for L st . Let π denotethe projection to the strong components p ϕ st , v st q , then from Lemma 4.3 πφ it Ñ φ st t uniformly. As a result for a fixed T ą p ϕ i , v i q P ˜ I p L st i ´ c i ¨ v, u i q , we have p ϕ i , v i q “ p ϕ st i , ϕ wk i , v st i , v wk i q P φ i ´ T ` G p L si ´ c i ¨ v, u i q ˘ , hence p ϕ st i , v st i q Ñ p ϕ st , v st q P ϕ st ´ T ` G p L st ´ ¯ c ¨ v st , u st q ˘ . Since T ą p ϕ st , v st q P ˜ I p L st0 ´ ¯ c ¨ v st , u st q . Proof of Proposition 6.1, part I.
We first prove item 1. Suppose ˜ ϕ i P ˜ N H si p c i q , thenthere exists weak KAM solutions u i of L si ´ c i ¨ v , such that p ϕ i , v i q P ˜ I p L si ´ c i ¨ v, u i q .By Theorem 2.4, after restricting to a subsequence, we have u i p ϕ st , ϕ wk q Ñ u st p ϕ st q . ByLemma 6.2, p ϕ st i , v st i q Ñ p ϕ st , v st q implies p ϕ st , v st q P ˜ I p L st0 ´ ¯ c ¨ v st , u st q Ă ˜ N H st0 p ¯ c q .For item 3, suppose ϕ i “ p ϕ st i , ϕ wk i q P A H si p c i q satisfies ϕ st i Ñ ϕ st P A H st0 p ¯ c q . Then h L si ,c i p ϕ i , ¨q is a weak KAM solution of L si ´ c ¨ v (see [Fat08], Theorem 5.3.6). ByTheorem 2.4, by restricting to a subsequence, there exists a weak KAM solution u st of L st0 ´ ¯ c ¨ v st such thatlim i Ñ8 h L si ,c i p ϕ i , ψ st , ψ wk q ´ h L si ,c i p ϕ i , , q “ u st p ψ st q . We may further assume that h L si ,c i p ϕ i , , q Ñ C P R . Since A H st0 p ¯ c q has only one staticclass, there exists a constant C ą u st p ψ st q ` C “ h L st , ¯ c p ϕ st , ψ st q . Using the fact that ϕ i P A H si p c i q , we get h L si ,c i p ϕ i , ϕ i q “
0. Taking the limit, u st p ϕ st q “ ´ C “ h L st , ¯ c p ϕ st , ϕ st q ´ C “ ´ C. Therefore lim i Ñ8 h L si ,c i p ϕ st i , ϕ wk i , ψ st , ψ wk q “ h L st , ¯ c p ϕ st , ψ st q . Item 4: Let ρ i “ p ρ st i , ρ wk i q be the rotation number of minimal measures of L si ´ c i ¨ v ,then from Proposition 5.6,lim i Ñ8 ρ wk i ´ B Ti A ´ i ρ st i ´ ˜ C i c wk i “ . ρ st i Ñ ρ st P R m , then by taking limit in the second conclusion ofProposition 5.6, we get α H st0 p ¯ c q ` β H st0 p ρ st q ´ ¯ c ¨ ρ st “ , using Fenchel duality, ρ st is a subdifferential of the convex function α H st0 at ¯ c . Our strategy of the proof mostly follow [Ber10].Given a compact metric space X , a semi-flow φ t on X , and ε, T ą
0, an p ε, T q´ chainconsists of x , ¨ ¨ ¨ , x N P X and T , ¨ ¨ ¨ , T N ´ ě T , such that d p φ T i x i , x i ` q ă ε . We saythat x C X y if for any ε, T ą
0, there exists an p ε, T q´ chain with x “ x and x N “ y .The relation C X is called the chain transitive relation (see [Con88]).The family of maps ¯ φ t “ φ t defines a semi-flow on the set G p L ´ c ¨ v, u q , andtherefore defines a chain transitive relation. Given ϕ, ψ P T d and a weak KAM solution u of L ´ c ¨ v , we say that ϕ C u ψ if there exists ˜ ϕ “ p ϕ, v q , ˜ ψ “ p ψ, w q P T d ˆ R d suchthat ˜ ϕ C X ˜ ψ, where X “ G p L ´ c ¨ v, u q . Item 1 in the following Proposition is due to Mañe, and item 2 is due to Mather.The version presented here is contained in [Ber10].
Proposition 6.3.
Let L be a Tonelli Lagrangian, then:1. Let ϕ P A L p c q and u be a weak KAM solution of L ´ c ¨ v , we have ϕ C u ϕ .2. Suppose A L p c q has only finitely many static classes, and there exists a weak KAMsolution u such that ϕ C u ϕ . Then ϕ P A L p c q . Proposition 6.3 implies that, when A L p c q has finitely many static classes, the Aubryset coincides with the set t ϕ : ϕ C u ϕ u . We will prove semi-continuity for this set. Definition.
Let X be a compact metric space with a semi-flow φ t . A family of piecewisecontinuous curves x i : r , T i s Ñ X is said to accumulate locally uniformly to p X , φ t q iffor any sequence S i P r , T i s , the curves x i p t ` S i q has a subsequence which convergesuniformly on compact sets to a trajectory of φ t . Lemma 6.4. [Ber10] Suppose x i : r , T i s Ñ X accumulates locally uniformly to p X, φ t q , x i p q Ñ x and x i p T i q Ñ y , then x C X y .Proof of Proposition 6.1, part II. We prove item 2. Let ϕ i “ p ϕ st i , ϕ wk i q P A H si p c i q and ϕ st i Ñ ϕ st , we show that ϕ st P A H st0 p ¯ c q . According to Proposition 6.3, ϕ i C u ϕ i . Let ˜ ϕ i be the unique point in ˜ A H si p c i q projecting to ϕ i , then there exists weak KAM solutions u i of L si ´ c i ¨ v , such that ˜ ϕ i C ˜ ϕ i in G p L si ´ c i ¨ v, u i q . Fix ε i Ñ M i Ñ 8 , then foreach i , there exists T i, ă ¨ ¨ ¨ ă T i,N i , T i,j ` ´ T i,j ą M i , and a piecewise C curve γ i “ p γ st i , γ wk i q : r , T i s Ñ T d , satisfying39. γ i |p T i,j , T i,j ` q satisfies the Euler-Lagrange equation of L si ;2. d ` pp γ i p T i,j ´q , γ i p T i,j ´qq , pp γ i p T i,j `q , γ i p T i,j `qq ˘ ă ε i . Using Lemma 4.3, the projection of the Euler-Lagrange flow of L si to p ϕ st , v st q convergesuniformly over compact interval to the Euler-Lagrange flow of L st0 . This, combined withitem 2 and Lemma 6.2, implies that p γ st i , γ st i q accumulates locally uniformly to p G p L st0 ´ ¯ c ¨ v st , u st q , φ st ´ t q where φ st t is the Euler-Lagrange flow of L st0 . Therefore ϕ st i Ñ ϕ st impies ϕ st C u ϕ st . UsingProposition 6.3 again, we get ϕ st P A H st0 p ¯ c q . In this section we prove Proposition 5.3 and 5.4. For p B wk , p, U st , U wk q P Ω m,dκ,q p B st q Xt} U st } ď R u , recall the notations H s “ H s p B st , B wk , p, U st , U wk q , H st “ H st p p, U st q , L s “ L H s , L st “ L H st . In this section we show that Proposition 5.3 implies Proposition 5.4. Proposition 5.3 isproven in the next two sections.We first state a lemma of action comparison between an extremal curve and its“linear drift”.
Lemma 7.1.
Let L : T d ˆ R d Ñ R be a Tonelli Hamiltonian, T ě , and γ : r , T s Ñ T d be an extremal curve. Then for any ď i ď d , h ą , and a unit vector f P R d , ż T L p γ ` thT f, γ ` hT f q dt ´ ż T L p γ, γ q dt ď pB v L p γ p T q , γ p T qq ¨ f q h ` ˆ } f ¨ pB vv L q f } T ` } f ¨ pB ϕv L q f } ` T } f ¨ pB ϕϕ L q f } ˙ h . Proof.
We compute L p γ ` thT f, γ q ´ L p γ, γ ` hT f q ď B ϕ L p γ, γ q ¨ thT f ` B v L p γ, γ q hT f } f ¨ pB vv L q f } h T ` } f ¨ pB ϕv L q f } th T ` } f ¨ pB ϕϕ L q f } t h T . It follows from the Euler-Lagrange equation that B ϕ L p γ, γ q ¨ thT ` B v L p γ, γ q hT “ ddt ˆ B v L thT ˙ , and our estimate follows from direct integration.40he following lemma establishes a relation between “approximate semi concavity”with approximate Lipschitz property. Lemma 7.2.
For
D, δ ą , assume that u : T d Ñ R satisfies that for all ϕ P T d , thereexists l P R d such that u p ϕ ` y q ´ u p ϕ q ď l ¨ y ` D } y } ` δ, y P R d , Then } l } ď ? d p D ` δ q , and u is p ? d p D ` δ q , δ q approximately Lipschitz.Proof. Assume that l “ p l , ¨ ¨ ¨ , l d q . For each 1 ď i ď d , we pick y “ ´ e i l i | l i | , where e i isthe coordinate vector in ϕ i . Then0 “ u p ϕ ` e i q ´ u p ϕ q ď ´| l i | ` D ` δ, so | l i | ď D ` δ . As a result } l } ď ? d p D ` δ q . For any y P r , s d , we have } y } ď ? d and u p ϕ ` y q ´ u p ϕ q ď p? d p D ` δ q ` D } y }q} y } ` δ ă ? d p D ` δ q} y } ` δ. Proof of Proposition 5.4.
Since u is a weak KAM solution, for any ϕ P T d , let γ “p γ st , γ wk q : p´8 , s Ñ T d be a p u, L s ´ c ¨ v, α H p c qq -calibrated curve with γ p q “ ϕ “p ϕ st , ϕ wk q . Then for any T ą u p ϕ q “ u p γ p´ T qq ` ż ´ T p L s ´ c ¨ v ` α H s p c qqp γ, γ q dt. Using (4.6), we get u p ϕ q “ u p γ p´ T qq ` ż ´ T p L st ´ ¯ c ¨ v st qp γ st , γ st q dt ` p α H s p c q ´ c wk ¨ ˜ C ´ c wk q T ` ż ´ T p γ wk ´ B T A ´ γ st ´ ˜ Cc wk q ¨ ˜ C ´ p γ wk ´ B T A ´ γ st ´ ˜ Cc wk q ` U wk p γ p t qq dt. (7.1)We now produce an upper bound using a special test curve. Let γ st0 : r´ T, s Ñ T m be such that ż ´ T p L st ´ ¯ c ¨ v st qp γ st0 , γ st0 q dt “ min ζ ż ´ T p L st ´ ¯ c ¨ v st qp ζ, ζ q dt (7.2)where the minimum is over all ζ p´ T q “ γ st p´ T q and ζ p q “ γ st p q .We define ξ “ p ξ st , ξ wk q : r´ T, s Ñ T d as follows.41. For y P R d , ξ st p t q “ γ st0 p t q ` T ` tT y. The curve ξ st is a linear drift over γ st0 with h “ } y } and f “ y } y } (see Lemma 7.1).2. Define ξ wk p t q “ γ wk p´ T q ` B T A ´ p ξ st p t q ´ γ st0 p´ T qq ` ˜ Cc wk p T ` t q . We note that ξ wk p´ T q “ γ wk p´ T q and ξ wk0 ´ B T A ´ ξ st ´ ˜ Cc wk “ . Using the fact that u is dominated by L s ´ c ¨ v ` α H s p c q , we have u p ϕ st ` y, ξ wk p qq ď u p γ p´ T qq ` ż ´ T p L s ´ c ¨ v ` α H s p c qqp ξ, ξ q dt “ u p γ p´ T qq ` ż ´ T p L st ´ ¯ c ¨ v st qp ξ st , ξ st q dt ` ż ´ T p ξ wk ´ B T A ´ ξ st ´ ˜ Cc wk q ¨ ˜ C ´ p ξ wk ´ B T A ´ ξ st ´ ˜ Cc wk q dt ` p α H s p c q ´ c wk ¨ ˜ C ´ c wk q T ` ż ´ T U wk p ξ q dt and note that the third line in the above formula vanishes, using the definition of ξ wk .Combine with (7.1), we get u p ϕ st ` y, ξ wk p qq ´ u p ϕ st , ϕ wk qď ż ´ T p L st ´ ¯ c ¨ v st qp ξ st , ξ st q dt ´ ż ´ T p L st ´ ¯ c ¨ v st qp γ st , γ st q dt ` } U wk } C . From (7.2) we get u p ϕ st ` y, ξ wk p qq ´ u p ϕ st , ϕ wk qď ż ´ T p L st ´ ¯ c ¨ v st qp ξ st , ξ st q dt ´ ż ´ T p L st ´ ¯ c ¨ v st qp γ st0 , γ st0 q dt ` } U wk } C . Since γ st0 is an extremal of L st ´ ¯ c ¨ v st , the linear drift lemma (Lemma 7.1) applies.Noting that }B v st v st L st } ď } A ´ } , }B ϕ st ϕ st L } ď } U st } C ď R , and B ϕ st v st L “
0. We obtainfrom Lemma 7.1 that ż ´ T p L st ´ ¯ c ¨ v st qp ξ st , ξ st q dt ´ ż ´ T p L st ´ ¯ c ¨ v st qp γ st0 , γ st0 q dt ď l ¨ y ` p} A ´ } ` } U st } C q} y } , l “ B v L st p γ st0 p q , γ st0 p qq . Note that } A ´ } ` } U st } C is a constant depending onlyon B st , Q, R .We now invoke Proposition 5.3 to get | u p ϕ st ` y, ξ wk p qq ´ u p ϕ st ` y, ϕ wk q| ď δ | ξ wk p q ´ ϕ wk | ` δ ď δ, where δ “ M ˚ µ p B wk q ´ p q ´ d ` m q for some M ˚ “ M ˚ p B st , Q, κ, q, R q . Combine all theestimates, we get u p ϕ st ` y, ϕ wk q ´ u p ϕ st , ϕ wk q ď l ¨ y ` p} A ´ } ` } U st } C q} y } ` δ ` } U wk } . We note that in Ω m,dκ,q we have } U wk } C ď ř d ´ mi “ } U wk i } C ď p d ´ m q κ p µ p B wk qq ´ q . Wemay choose M ˚ “ M ˚ p B st , Q, κ, q, R q , such that2 δ ` } U wk } ď M ˚ µ p B wk qq ´p q ´ d ` m q “ : δ . We now apply Lemma 7.2 to get u p¨ , ϕ wk q is p ? d p} A ´ } ` } U st } C ` δ q , δ q approximately Lipschitz. Define M “ ? d p} A ´ } ` } U st } C ` M ˚ q , and the Propositionfollows. For the proof of Proposition 5.3, we need a finer decomposition of the Lagrangian L s which treat all ϕ wk i , 1 ď i ď d ´ m separately. First, we have the following linear algebraidentity. (The proof is direct calculation) Lemma 7.3.
Let S “ „ A BB T C be a nonsingular symmetric matrix in block form.Then „ Id 0 ´ B T A ´ Id „ A BB T C „ Id ´ A ´ B “ „ A
00 ˜ C , where ˜ C “ C ´ B T A ´ B . In particular, ˜ C is positive definite if S is. We write H s p ϕ, I q “ K p I q ´ U p ϕ q “ K p I q ´ U st p ϕ st q ´ U wk p ϕ q and S “ B II K . Wedescribe a coordinate change block diagonalizing B II K . Write S in the following blockform S “ „ X d ´ m y d ´ m y Td ´ m z d ´ m , X d ´ m P M p d ´ qˆp d ´ q , y d ´ m P R d ´ , z d ´ m P R , and for each 1 ď i ď d ´ m ´
1, further decompose each X i ` as X i ` “ „ X i y i y Ti z i , X i P M p m ` i ´ qˆp m ` i ´ q , y i P R m ` i ´ , z i P R . X “ B I st I st K “ A (see (2.5)).Define, for 1 ď i ď d ´ m , E i “ »– Id m ` i ´ ´ X ´ i y i
00 1 00 0 Id d ´ m ´ i fifl , where Id i denote the i ˆ i identity matrix. Then by Lemma 7.3 E Td ´ m SE d ´ m “ „ Id d ´ ´ y Td ´ m X ´ d ´ m „ X d ´ m y d ´ m y Td ´ m z d ´ m „ Id d ´ ´ X ´ d ´ m y d ´ m “ „ X d ´ m
00 ˜ z d ´ m , where ˜ z d ´ m “ z d ´ m ´ y Td ´ m X ´ d ´ m y d ´ m . Moreover, for each 1 ď i ď d ´ m ´ „ Id m ` i ´ ´ y Ti X ´ i X i ` „ Id m ` i ´ ´ X ´ i y i “ „ Id m ` i ´ ´ y Ti X ´ i „ X i y i y Ti z i „ Id m ` i ´ ´ X ´ i y i “ „ X i
00 ˜ z i . (7.3)Let E “ E d ´ m ¨ ¨ ¨ E “ »—————– Id m ´ X ´ y ´ X ´ y ¨ ¨ ¨ ´ X ´ d ´ m y d ´ m fiffiffiffiffiffifl , (7.4)then recursive computation yields E T SE “ E T ¨ ¨ ¨ E Td ´ m SE d ´ m ¨ ¨ ¨ E “ »———– X ˜ z . . . ˜ z d ´ m fiffiffiffifl “ : ˜ S. (7.5)We summarize the characterization of the Lagrangian in the following lemma. For v “ p v st , v wk1 , ¨ ¨ ¨ , v wk d ´ m q P R m ˆ R d , we define t v u “ v st , t v u i “ p v st , v wk1 , ¨ ¨ ¨ , v wk i q , ď i ď d ´ m. (7.6) Lemma 7.4.
For v, c P R d we denote w “ E T v and η “ E ´ c , where E is defined in (7.4) . Explicitly, we have w “ »———– w st w wk1 ... w wk d ´ m fiffiffiffifl “ »———– v st v wk1 ´ y T X ´ t v u ... v wk d ´ m ´ y Td ´ m X ´ d ´ m t v u d ´ m ´ fiffiffiffifl , (7.7)44 nd η st “ c st ` A ´ Bc wk , η “ p η st , η wk q , c “ p c st , c wk q , where A, B are defined in (2.5) . Then we have L s p ϕ, v q ´ c ¨ v “ L st p ϕ st , v st q ´ η st ¨ v st ` d ´ m ÿ i “ ˆ
12 ˜ z ´ i p w wk i ´ ˜ z i η wk i q ´ z i p η wk i q ` U wk i p ϕ q ˙ . (7.8) Remark.
This is a finer version of Lemma 4.2. In particular, the strong component L s ´ η st ¨ v st is identical to the L s ´ ¯ c ¨ v st defined in Lemma 4.2.Proof. Formula (7.7) can be read directly from the definition (7.4) and w “ E T v . Toshow η st “ c st ` A ´ Bc wk , we compute „ A
00 ˜ C „ η st η wk “ ˜ Sη “ ˜ SE ´ c “ E T Sc “ „ Id m ˚ ˚ „ A BB T C „ c st c wk . The first block of the above equation yields Aη st “ Ac st ` Bc wk , hence η st “ c st ` A ´ Bc wk .We now prove (7.8). We have L s p ϕ, v q ´ c ¨ v “ v T S ´ v ´ c T v ` U st ` U wk “ p E T v q ˜ S ´ p E T v q ´ p E ´ c q T p E T v q ` U st ` U wk “ ˆ w st ¨ A ´ w st ´ η st ¨ w st ` U st ˙ ` d ´ m ÿ i “ ˆ
12 ˜ z ´ i p w wk i q ´ η wk i w wk i ` U wk i ˙ . In the above formula, the first group is equal to L st ´ η st ¨ v st , noting w st “ v st . Moreover12 ˜ z ´ i p w wk i q ´ η wk i w wk i “
12 ˜ z ´ i p w wk i ´ ˜ z i η wk i q ´
12 ˜ z i p η wk i q , ď i ď d ´ m, and (7.8) follows.We derive some useful estimates. Lemma 7.5.
There exists M ˚ “ M ˚ p B st , Q, κ, q q ą such that, for L s “ L H s p B wk , p, U st , U st q , p B wk , p, U st , U st q P Ω m,dκ,q , the following hold.1. For each ď i ď d ´ m , we have ř d ´ mj “ i } U wk j } C ď M ˚ | k wk i | ´ q .2. For each ď i ď d ´ m , ˜ z ´ i ď M ˚ | k wk i | i . roof. For item 1, note that for each j ě i , | k wk i | ď κ | k wk j | , hence } U wk j } C ď κ | k wk j | ´ q ď κ ` q | k wk i | ´ q . Item 1 holds for any M ˚ ě p d ´ m q κ ` q .For item 2, inverting (7.3) we get X ´ i ` “ „ Id m ` i ´ ´ X ´ i y i „ X ´ i
00 ˜ z ´ i „ Id m ` i ´ y Ti ` X ´ i ` . Denote f “ p , ¨ ¨ ¨ , , q P T m ` i , then f T X i ` f “ f T „ Id m ` i ´ ´ X ´ i y i „ X ´ i
00 ˜ z ´ i „ Id m ` i ´ y Ti ` X ´ i ` f “ ˜ z ´ i . Moreover, using the definition (see (1.3)) S “ B II K “ “ k st1 ¨ ¨ ¨ k st m k wk1 ¨ ¨ ¨ k wk d ´ m ‰ T Q “ k st1 ¨ ¨ ¨ k st m k wk1 ¨ ¨ ¨ k wk d ´ m ‰ , we have X i ` “ “ k st1 ¨ ¨ ¨ k st m k wk1 ¨ ¨ ¨ k wk i ‰ T Q “ k st1 ¨ ¨ ¨ k st m k wk1 ¨ ¨ ¨ k wk i ‰ “ “ ¯ k st1 ¨ ¨ ¨ ¯ k st m ¯ k wk1 ¨ ¨ ¨ ¯ k wk i ‰ T Q “ ¯ k st1 ¨ ¨ ¨ ¯ k st m ¯ k wk1 ¨ ¨ ¨ ¯ k wk i ‰ “ : ¯ P T Q ¯ P , where ¯ k is the first n components of k . We have assumed Q ě D ´ Id for D ą
1. ByLemma 3.4, there exists a constant c n ą n such that } X ´ i ` } “ p min } v }“ v T X i ` v q ´ “ p min } v }“ v T ¯ P Q ¯ P q ´ ď D } ¯ P ´ } ď Dc n | k st1 | ¨ ¨ ¨ | k st m | | k wk1 | ¨ ¨ ¨ | k wk i | ď Dc n ¯ M m κ i ´ | k wk i | i , where ¯ M “ | k st1 | ` ¨ ¨ ¨ ` | k st m | depend only on B st . In this section we prove Proposition 5.3. We fix p B wk , p, U st , U st q P Ω m,dκ,q Xt} U st } C ď R u ,and write L s “ L H s p B wk , p, U st , U st q .For c P R d , we define L sc,i p ϕ st , ϕ wk1 , ¨ ¨ ¨ , ϕ wk i , v st , v wk1 , ¨ ¨ ¨ , v wk i q “ L sc,i p t ϕ u i , t v u i q“ L st p ϕ st , v st q ´ η st ¨ v st ` i ÿ j “ ˆ
12 ˜ z ´ j p w wk j ´ ˜ z j η wk j q ´ z j p η wk j q ` U wk j p ϕ q ˙ , (7.9)46hen L s p ϕ, v q ´ c ¨ v “ L sc,i p t ϕ u i , t v u i q ` d ´ m ÿ j “ i ` ˆ
12 ˜ z ´ j p w wk j ´ ˜ z j η wk j q ´ z j p η wk j q ` U wk j p ϕ q ˙ . (7.10)Our proof of Proposition 5.3 follows an inductive scheme. Following our notationalconvention, denote e wk i “ e i ` m , which is the coordinate vector of ϕ wk i . Lemma 7.6.
Let u : T d Ñ R be a weak KAM solution of L s ´ c ¨ v . Then for δ d ´ m : “ p ˜ z ´ d ´ m } U wk d ´ m } C q , we have u is δ d ´ m ´ semi-concave and δ d ´ m ´ Lipschitz in ϕ wk d ´ m .Proof. First we have B ϕ wk d ´ m ϕ wk d ´ m L s “ B ϕ wk d ´ m ϕ wk d ´ m U wk d ´ m , B ϕ wk d ´ m v wk d ´ m L s “ , B v wk d ´ m v wk d ´ m L s “ ˜ z ´ d ´ m . The first two equality follows directly from the definition, while the last one uses (7.7)and (7.8).For any ϕ P T d , let γ : p´8 , s Ñ T d be a p u, L s , c q -calibrated curve with γ p q “ ϕ .Then for any T ą u p ϕ q “ u p γ p´ T qq ` ż ´ T p L s ´ c ¨ v ` α H s p c qqp γ, γ q dt. Using the definition of the weak KAM solution, u p ϕ ` he wk i q ď u p γ p´ T qq ` ż ´ T p L s ´ c ¨ v ` α H s p c qqp γ ` thT e wk d ´ m , γ ` hT e wk d ´ m q dt. Subtract the two estimates, and apply Lemma 7.1 to L s ´ c ¨ v ` α H s p c q and γ , weget u p ϕ ` he wk i q ´ u p ϕ q ď pB v wk d ´ m L s p γ p q , γ p qq ´ c d ´ m q h ` ˆ }B v d ´ m v d ´ m L s } T ` }B ϕ wk d ´ m v wk d ´ m L s } ` T }B ϕ wk d ´ m ϕ wk d ´ m L s } ˙ h ď pB v wk d ´ m L s p γ p qq , γ p q ´ c wk d ´ m q h ` ` ˜ z ´ d ´ m { T ` } U wk d ´ m } C T ˘ h , Take T “ p ˜ z d ´ m } U wk d ´ m } C q ´ , and write l “ B v wk d ´ m L s p γ p qq , γ p q ´ c wk d ´ m , we get u p ϕ ` he wk i q ´ u p ϕ q ď lh ` δ d ´ m h . u is Z d periodic, we take h “ l {| l | to get | l | ď δ d ´ m . Therefore for | h | ď | u p ϕ ` he wk i q ´ u p ϕ q| ď p δ d ´ m ` δ d ´ m h q h ď δ d ´ m h. This is the Lipschitz estimate.We now state the inductive step.
Proposition 7.7.
Let u : T d Ñ R be a weak KAM solution of L s ´ c ¨ v . Assumethat for a given ď i ď d ´ m ´ , u is p δ j , δ j q approximately Lipschitz in ϕ wk j for all i ` ď j ď d ´ m . Then for σ i “ ˜ ˜ z ´ i d ´ m ÿ j “ i } U wk j } C ¸ , δ i “ ? d p σ i ` d ´ m ÿ j “ i ` δ j q , we have u is p δ i , δ i q approximately Lipschitz in ϕ wk i .Proof. The proof is very similar to the proof of Proposition 5.4, but uses the finerdecomposition in this section.Since u is a weak KAM solution, then given any ϕ P T d , there exists a calibratedcurve γ : p´8 , s Ñ T d with γ p q “ ϕ . Then for any T ą u p ϕ q “ u p γ p´ T qq ` ż ´ T p L s ´ c ¨ v ` α H s p c qqp γ, γ q dt. Let h P R , χ P R d , and a C curve ξ : r´ T, s Ñ T d satisfies ξ p´ T q “ γ p´ T q , ξ p q “ ϕ ` he wk i ` χ, then u p ϕ ` he wk i ` χ q ď u p γ p´ T qq ` ż ´ T p L s ´ c ¨ v ` α H s p c qqp ξ, ξ q dt ď u p γ p´ T qq ` ż ´ T p L s ´ c ¨ v ` α H s p c qqp γ, γ q dt ` ż ´ T p L s ´ c ¨ v qp ξ, ξ q ´ ż ´ T p L s ´ c ¨ v qp γ, γ q dt “ u p ϕ q ` ż ´ T p L s ´ c ¨ v qp ξ, ξ q ´ ż ´ T p L s ´ c ¨ v qp γ, γ q dt. (7.11)We will first give the precise definition of ξ , then estimate (7.11), before finally obtainthe desired estimate. 48 efinition of ξ . Recall the Lagrangian L sc,i : T m ` i ˆ R m ` i Ñ R defined in (7.9). Let ξ : r´ T, s Ñ T m ` i be an L sc,i minimizing curve satisfying the constraint ζ p´ T q “ t γ u i p´ T q , ζ p q “ t γ u i p q , where t ¨ u i is defined in (7.6). For h P R , we define ξ in the following way.1. The first m ` i components of ξ is ζ with an added linear drift in e wk i , moreprecisely, t ξ u i p t q “ ζ p t q ` thT e wk i . (7.12)2. We define the other components inductively. For i ă j ď d ´ m , suppose t ξ u j ´ p t q “ p ξ st , ξ wk1 , ¨ ¨ ¨ , ξ wk j ´ qp t q has been defined. We define ξ wk j p t q “ γ wk j p t q ` y Tj X ´ j t ξ u j ´ p t q ´ y Tj X ´ j t γ u j ´ p t q . For each i ă j ď d ´ m , we have ξ wk j p´ T q “ γ wk j p´ T q , ξ wk j ´ y Tj X ´ j t 9 ξ u j ´ “ γ wk j ´ y Tj X ´ j t 9 γ u j ´ . (7.13)We define χ “ ξ p q ´ ϕ ´ he wk i , and note that from (7.12), t χ u i “ t ξ u i p q ´ t γ u i p q ´ he wk i “ . Action comparison.
We now compute ż ´ T p L s ´ c ¨ v qp ξ, ξ q dt ´ ż ´ T p L s ´ c ¨ v qp γ, γ q dt “ ż ´ T L sc,i p t ξ u i , t 9 ξ u i q dt ´ ż ´ T L sc,i p t γ u i , t 9 γ u i q dt ` d ´ m ÿ j “ i ` ż ´ T ` U wk j p ξ p t qq ´ U wk j p γ p t qq ˘ dt ` d ´ m ÿ j “ i ` ˜ z ´ j ż ´ T ´ p ξ wk j ´ y Tj X ´ j t 9 ξ u j ´ ´ ˜ z j η wk j q ´ p γ wk j ´ y Tj X ´ j t 9 γ u j ´ ´ ˜ z j η wk j q ¯ ď ż ´ T L sc,i p t ξ u i , t 9 ξ u i q dt ´ ż ´ T L sc,i p t γ u i , t 9 γ u i q dt ` T d ´ m ÿ j “ i ` } U wk j } C . (7.14)In the above formula, the equality is due to (7.10). Moreover, observe that from (7.13),the third line of the above formula vanishes. The inequality follows by replacing U wk j with its upper bound } U wk j } C . 49e now have ż ´ T L sc,i p t ξ u i , t 9 ξ u i q dt ´ ż ´ T L sc,i p t γ u i , t 9 γ u i q dt “ ż ´ T L sc,i p t ξ u i , t 9 ξ u i q dt ´ ż ´ T L sc,i p ζ, ζ q dt ` ż ´ T L sc,i p ζ, ζ q dt ´ ż ´ T L sc,i p t γ u i , t 9 γ u i q dt ď ż ´ T L sc,i p t ξ u i , t 9 ξ u i q dt ´ ż ´ T L sc,i p ζ, ζ q dt, noting that ζ is minimizing for L sc,i .Since ζ is minimizing and hence extremal for L sc,i , from the definition of ξ in (7.12),Lemma 7.1 applies. Hence ż ´ T L sc,i p t ξ u i , t 9 ξ u i q dt ´ ż ´ T L sc,i p ζ, ζ q dt ď l ¨ h ` ˜ T ˜ z ´ i ` T } d ´ m ÿ j “ i U wk j } C ¸ h , where l “ B v i p L sc,i qp ζ p q , ζ p qq . As in the proof of Lemma 7.6, we choose T “ ´ ˜ z i ř d ´ mj “ i } U wk j } C ¯ ´ , we get ż ´ T L sc,i p t ξ u i , t 9 ξ u i q dt ´ ż ´ T L sc,i p ζ, ζ q dt ď l ¨ h ` σ i h , σ i “ ˜ ˜ z ´ i d ´ m ÿ j “ i } U wk j } C ¸ . Combine with (7.14), and use the upper bound ř d ´ mj “ i ` } U wk j } C ď ř d ´ mj “ i } U wk j } C , weget ż ´ T p L s ´ c ¨ v qp ξ, ξ q dt ´ ż ´ T p L s ´ c ¨ v qp γ, γ q dt ď l ¨ h ` σ i h ` σ i . Estimating the weak KAM solution.
Combine the last formula with (7.11), we get u p ϕ ` he wk i ` χ q ´ u p ϕ q ď l ¨ h ` σ i h ` σ i . Since t χ u i “
0, using the inductive assumption, | u p ϕ ` he wk i ` χ q ´ u p ϕ ` he wk i q| ď d ´ m ÿ j “ i ` δ j . Therefore u p ϕ ` he wk i q ´ u p ϕ q ď l ¨ h ` σ i h ` σ i ` d ´ m ÿ j “ i ` δ j . We now use Lemma 7.2 to get for δ i “ ? d p σ i ` d ´ m ÿ j “ i ` δ j q ,u is p δ i , δ i q approximately Lipschitz in ϕ wk i .50 roof of Proposition 5.3. We have shown by induction that for all 1 ď i ď d ´ m , u is p δ i , δ i q approximately Lipschitz in ϕ wk i , where δ i are defined inductively in Lemma 7.6and Proposition 7.7.By Lemma 7.5, for each 1 ď i ď d ´ mσ i “ p ˜ z ´ i } U wk i } C q ď M ˚ | k wk i | ´ q ` i ´ m . Then δ d ´ m “ σ d ´ m ď M ˚ | k wk d ´ m | ´ q ` d ´ m . For each 1 ď i ď d ´ m , we have δ i “ ? d p σ i ` d ´ m ÿ j “ i ` δ i q ď p ? d q i ´ m d ´ m ÿ j “ i σ i ď M ˚ p ? d q i ´ m | k wk i | ´ q ` d ´ m . For any ϕ wk , ψ wk P T d ´ m and ϕ st P T m , | u p ϕ st , ϕ wk q ´ u p ϕ st , ψ wk q| ď d ´ m ÿ i “ δ i | ϕ wk i ´ ψ wk i | ` d ´ m ÿ i “ δ i Since ř d ´ mi “ δ i ď p d ´ m q M ˚ p ? d q i ´ m p µ p B wk qq ´ q ` d ´ m , the proposition follows by re-placing M ˚ by p d ´ m q M ˚ p ? d q i ´ m . A Diffusion path with dominant structure
A.1 Diffusion path for Arnold diffusion
Our main motivation is to prove Arnold diffusion for a “typical” nearly integrablesystem of the form (1.1). The word “typical” here means the cusp residual conditionintroduce by Mather ([Mat03]).
Definition.
For r ě , we say that a property G hold for a cusp residual set of C r nearly integrable systems H ε “ H ` εH , if:• G is an open property in C r topology;• There exists an open and dense set V Ă t} H } C r “ u , and a positive function ε : V Ñ R ` , such that G is C r -dense on U “ t H ` εH : H P V , ă ε ă ε p H qu . We would like to show that the property of topological instability is cusp residual.Instabilities for multidimensional Hamiltonian systems ( n ě
3) are studied in [Moe96;GK14a; CY09; BKZ11; GK14a; KZ14; DLS13; Tre04; Tre12; Zhe10; Mar12a; Mar12b;KS12].The main conjecture of Arnold diffusion in finite regularity may be formulated asfollows.
Conjecture.
There exists r ą such that for each n ě , γ ą , r ď r ă 8 , for acusp residual set of C r nearly integrable system, the system admits an orbit p θ ε , p ε qp t q such that t p ε p t qu t P R is γ ´ dense on the unit ball B n : “ t} p } ď u . n “
2, we refer the reader to [Che13; KZ13] andreference therein. The proof in n “ Step 1 , define the set V , which contains the set of “nondegenerate” H . For H P V , H ` εH possesses certain open structure of instability, such as NHICs and the AMproperty mentioned below. Step 2 , show that for any H ` εH with H P V and ε sufficiently small, one canmake an arbitrarily small perturbation to H ` εH such that there exists diffusionorbits.In Theorem A.1, we prove a weaker version of Step 1 . The heart of the argumentis the construction of a diffusion path, on which all the essential resonances has adominant structure. We expect the same diffusion path can be used to prove the fullconjecture. To avoid excessive length, we will give an outline of the proof with keystatements, and the full details will appear later.A diffusion path P is a subset in R n that the diffusion orbit p ε p t q roughly shadows.We pick a diffusion path that travels along a collection of p n ´ q´ resonances or,equivalently, along a collection of connected 1-dimensional resonant curves. Definition.
A diffusion path P is a compact connected subset of ď t Γ Λ p n ´ q : Λ p n ´ q P L p n ´ q u where L p n ´ q : “ t Λ p n ´ q i u Ni “ is a collection of rank n ´ irreducible resonant lattices(and each Γ Λ p n ´ q is a 1-dimensional resonant curve) . We define the AM property of a mechanical system relative to an integer homologyclass.• Let H “ K ´ U be a mechanical system on T n ˆ R n ,• h be an integer homology class,• S E “ t H “ E u be energy surface.• min U “
0, and the minimum is unique.Denote by π : T n ˆ R n ˆ T Ñ R n the natural projection onto the action component.Recall that homology and cohomology are related by Legendre-Fenichel tranform LF β p h q Ă H p T n , R q (see (5.1)). By a result of Diaz Carneiro [Car95] for each coho-mology c P H p T n , R q the Aubry set A p c q Ă S α p c q . Definition.
Let ρ ą . We say that p H, h, ρ q has the AM property if for any λ suchthat c P LF β p λh q and α H p c q ě ρ the Aubry set A p c q is a finite union of hyperbolicperiodic orbits such that each of these periodic orbits as a closed curve has homology h . A remark on notation: the supscript p n ´ q is not used as an index, but rather an indication for therank of the lattice. emark. Note that in the definition we do not consider the energy ď E ă ρ .The AM property is far from being generic. In Section B we discuss variety of waysthe AM property can fail for an open class of systems. Let Λ p n ´ q Ă Σ p n q be two irreducible lattices of rank n ´ n respectively. Let B p n q e “ r l , . . . , l n s be an ordered basis of Σ p n q and B p n ´ q “ r k , . . . , k n ´ s is be anordered basis of Λ p n ´ q . Then Λ p n ´ q and Σ p n q induce a (unique up to a sign) irreducibleinteger homology class denoted h p B p n ´ q , B p n q e q P Z n » H p T n , Z q (see (A.1) for details).We now state the main theorem of this section. Theorem A.1.
There exists r ą , C ą such that for each n ě , ρ ą , r ď r ă 8 ,for an open and dense set of H from t} H } C r “ u Ă C r p T n ˆ B n ˆ T q , there existsa diffusion path P “ P p H , ρ q with a a finite set E called the punctures or strongresonances, with the following properties.1. P is ρ -dense in B n , i.e. ρ -neighborhood of P contains B n .2. For each -dimensional resonant curve Γ i Ă Γ Λ p n ´ q X P there is a -dimensionalNHWIC r C i whose projection onto the action component dist p π r C i , Γ i q ď C ? (cid:15) .3. (Away from strong resonances) For each c P Γ i with dist p c, Σ n q ě C ? (cid:15) , we have A p c q belongs to r C i .4. (At strong resonance) Each puncture p P E is given by a rank n irreducible lattice Σ p n q , i.e. t p u “ Γ Σ p n q .5. Let p P P X Γ Σ p n q be a puncture. Then p Ă Γ Λ p n ´ q X P ‰ p for some rank n ´ irreducible lattice Λ p n ´ q , with bases B p n ´ q and B p n q e . For the induced homology h “ h p B p n ´ q | B p n q e q and the slow mechanical system H “ H p , B p n q e , defined in (1.2),we have that p H, h, ρ q have AM property. We have the following remarks.• If n “
2, a stronger version of Theorem A.1 hold. Namely, one can prove that foran fixed diffusion path, there exists a cusp residue set of systems H ` εH forwhich the theorem hold. Whether this statement generalizes to higher degrees offreedom is an open question.In our formulation, it is essential that the choise of diffusion path P does depend on the perturbation (cid:15)H .• Item 3 says that 3-dimensional cylinders r C i are minimal in the sense that theycontain the Aubry sets with frequency vector from Γ i away from maximal essentialresonances. 53 It turn out that away from strong resonances for each h P Γ i with dist p h, Σ n q ě C ? (cid:15) and c P LF β p h q we have not only that A p c q belongs to r C i , but also itis a Lipschitz graph over a certain 2-torus T c , i.e. for some submersion π c : T n ˆ B n ˆ T Ñ T we have that π A p c q : A p c q Ñ T c is one-to-one and the inverseis Lischitz. This is similar but more involved than what is presented in [BKZ11].See discussion of n “ r C i for H (cid:15) correspond to 2-dimensional cylinders C foraveraged Hamiltonians.• The cylinder r C i might consists of several connected components. At each maximalessential resonance Γ Λ p n q this cylinder can have two connected components: oneon each local component of Γ Λ p n ´ q z Γ Λ p n q .• The union of hyperbolic periodic orbits gives rise to a NHIC.• In section B we discuss the role of AM property for proving diffusion as well asthe number of ways it can be violated.• Notice that at each strong resonance, due to our definition of AM property, wedo not discuss the case low energy 0 ď E ă ρ . This is why Theorem A.1 does notcomplete Step 1. For n “ n “
3, construction of NHIC for away from critical energyin general and normally hyperbolic invariant manifolds (NHIM) for critical energyfor simple homologies is discussed in [KZ14], sect. 6.3. We expect these methodsextend to arbitrary n ě n “ n ě A.2 Nondegeneracy conditions for Arnold diffusion
We now describe the set V in Theorem A.1, using the conditions [H1] and [H2] to bedefined later. Let ρ ą r ď r ă 8 . We say that H P V if } H } C r “
1, and there exists a diffusion path P that is ρ ´ dense in B n , with the following properties.• For each Λ p n ´ q P L p n ´ q , and each connected component Γ of P X Γ Λ p n ´ q , thereexists λ ą H satisfies condition [H1 λ ] on Γ.• For each λ ą p n ´ q P L n ´ , there exists a finite set of rank n resonant lattices E S p Λ p n ´ q , λ q , with the property Σ p n q Ą Λ p n ´ q for each Σ p n q P E S p Λ p n ´ q , λ q . ThenΓ Σ p n q is a single point contained in Γ Λ p n ´ q . The collection E “ t Γ Σ p n q : Σ p n q P E S p Λ p n ´ q , λ qu is the set of punctures in Theorem A.1.54 Let λ ą λ ] is satisfied for H . For each Σ p n q P E S p Λ p n ´ q , λ q and Λ p n ´ q P L p n ´ q such that Γ Σ p n q P P , we choose basis B p n q e and B p n ´ q . We saythat H satisfies condition [H2] at Γ Σ p n q if for all such Λ p n ´ q Ă Σ p n q , p H sp , B p n q , h p B p n ´ q | B p n q q , ρ q satisfies the AM property.The condition [H1 λ ], and the definition of E S p Λ p n ´ q , λ q and h p B p n ´ q | B p n q q will beexplained below. For the moment we only remark that for a fixed P , the condition that[H1 λ ] holds for some λ ą Σ p n q , the AM property isopen but not always dense. However, it is a dense condition if the lattice Σ p n q satisfies adomination property. The main idea is then, to pick a particular H -dependent diffusionpath P , such that all the essential resonances on this path has this domination property. The condition [H1]
We now describe our first set of non-degeneracy condition. For Λ p n ´ q P L p n ´ q , let usfix a basis B . For λ ą p n ´ q Ă Γ Λ p n ´ q , we say that H satisfies condition [H1 λ ] on Γ p n ´ q if• For all p P Γ p n ´ q , the function Z B p¨ , p q has at most two global maxima.• At each global maxima ϕ ˚ of Z B p¨ , p q , the Hessian B ϕϕ Z B p ϕ ˚ , p q ď ´ λ Id asquadratic forms.• Suppose p is such that there are two global maxima ϕ ˚ p p q and ϕ ˚ p p q . Thenthey extend to local maxima for nearby p P Γ p n ´ q . We assume that the functions Z B p ϕ ˚ p p q , p q and Z B p ϕ ˚ p p q , p q have different derivatives along Γ p n ´ q , with thedifference at least λ .We say that H satisfies [H1] on Γ p n ´ q if it satisfies [H1 λ ] for some λ ą
0. Theseconditions are introduced by Mather ([Mat03]) for n “ λ ] of the condition depends on the choice of basis,while the qualitative version [H1] does not.For H satisfying [H1 λ ], there exists a finite set of rank n lattices containing Λ p n ´ q ,which we will call E S p Λ p n ´ q , λ q . More precisely, assume that the basis for Λ p n ´ q is t k , ¨ ¨ ¨ , k n ´ u and there exists M “ M p Λ p n ´ q , λ q ą E S p Λ p n ´ q , λ q “ t Λ k , ¨¨¨ ,k n ´ ,k : | k | ď M u . For each Λ p n q P E S p Λ p n ´ q , λ q , Γ Λ p n q is a point contained in 1-dimensional curve Γ Λ p n ´ q .The condition [H1] implies the existence of NHIC away from punctures, see [BKZ11].It is not hard to see that item 1-5 of Theorem A.1 are direct consequences of ournon-degeneracy conditions. The difficulty in Theorem A.1 is in showing these conditionsare open and dense. 55 nduce homology and non-degeneracy Fix Λ p n q P E S p Λ p n ´ q , λ q , and let B p n q be an ordered bases of Λ p n q , B p n ´ q is an orderedbasis of Λ p n ´ q and t p u “ Γ Λ p n q . Our second set of non-degeneracy condition concernsthe slow system H sp , B p n q : T n ˆ R n Ñ R , for a particular integer homology class h p B p n ´ q | B p n q q P H p T n , Z q , uniquely defined modulo the sign. We give a more generaldefinition here. Definition.
For ď s ď n , irreducible lattices Λ p s ´ q Ă Λ p s q , with corresponding basis B p s ´ q “ r k , ¨ ¨ ¨ , k s ´ s and B p s q “ r l , ¨ ¨ ¨ , l s s . Since k i P Λ p s q , there exists a uniquecollection a i P Z s z , i “ , . . . , s ´ , such that k i “ “ l ¨ ¨ ¨ l s ‰ a i . Then h p B p s ´ q | B p s q q P Z s is defined by the relations a i ¨ h p B p s ´ q | B p s q q “ , ď i ď s ´ . (A.1)This definition is determined by the resonance relation k i ¨ p ω p p q , q “ , ď i ď s ´ , after converting to the basis r l , ¨ ¨ ¨ , l s s .We require the triplet p H sp , B p n q , h p B p n ´ q | B p n q q , ρ q satisfies the AM property. We have the following consequences of the AM property:• (Robustness) The non-degeneracy condition is open.• (Minimality)
The condition guarantees, among other things, existence of anordered collection of minimal
NHICs with heteroclinic connectionsof neighbors. Each cylinder is minimal in the sense that it is foliated by periodicorbits minimizing action of a certain variational problem.• (Hyperbolicity)
Each cylinder is hyperbolic in the sense that it consists of hyperbolicperiodic orbits.
A.3 Properties of the nondegeneracy condition
Suppose H s p B st , B wk , p, U st , U wk q is a dominant system. Then the AM property extendsnicely from the strong system to the slow system. More precisely, the following propertieshold. Property A0.
For H “ K ´ U , the AM property for p H, h, ρ q is an open conditionin both K and U . 56 roperty A1. (Genericity in 2-degrees of freedom) For a fixed quadratic form K on R and h P Z , there exists an open and dense set of U P C p T q on which p H “ K ´ U, h, ρ q have AM property. Property A2. (Dimension reduction using hyperbolic fixed point) Consider the data p B st , Q , κ, q, ρ q and the space of corresponding dominant system Ω m,m ` κ,q p B st q . Assumethat U st0 P T m admits at most two non-degenerate minima. Note that each correspondsto a hyperbolic fixed point of H st .Then there exists M ą δ ą B st , Q , κ, q, p , U st0 , ρ suchthat the following hold. For each B wk with µ p B wk q ą M, } p ´ p } ă δ, } U st ´ U st0 } C ă δ, and h P Z , for an open and dense set of U wk (in the space Ω m,m ` κ,q p B st q restricted to fixed B wk , p, U st ),the triple ` H s p B st , B wk , p, U st , U wk q , g, ρ ˘ , with g “ p , ¨ ¨ ¨ , , h q , have AM property. Property A3. (Dimension reduction using AM property) Consider the data p B st , Q , κ, q, ρ q and the space of corresponding dominant system Ω m,m ` κ,q p B st q . Assume that p P R n , U st0 P C r p T m q , h P Z m satisfies ` H st p p , U st0 q , h, ρ ˘ have AM property.Then there exists M ą sup k P B st | k | , δ ą B st , Q , κ, q, p , U st0 , h, ρ such that the following hold. For each B wk with µ p B wk q ą M, } p ´ p } ă δ, } U st ´ U st0 } C ă δ, and any nonzero pair of integers z, w , the following hold.For an open and dense set of U wk (in the space Ω m,m ` κ,q p B st q restricted to a fixed setof B st , p, U st ), the triple ` H s p B st , B wk , p, U st , U wk q , g , ρ ˘ , where g “ p zh, w q , has AM property. Remarks :1. The proof of Theorem A.1 uses only Properties A0-A3 instead of the precisedefinition of AM property. Therefore the proof applies if we take Properties A0-A3as ansatz. We expect that the properties required for the full diffusion problemsatisfy the same ansatz and our construction applies to the full diffusion problem.2. The list of properties A0 - A3 provides a setup for proving non-degeneracy usinginduction over degrees of freedom. Assume that U st admits a non-degenerateminimum, then property A2 allows to extend this system by two more degrees offreedom, provided the homology g is only nontrivial in the weak variables. If H st is nondegenerate in a nontrivial homology h , property A3 allows to extend by onedegree of freedom, provided the new homology g is trivial in the weak variable.57. Let us explain the proof briefly. Property A1 is a known result. This propertyis used in Arnold diffusion in 2 degrees of freedom, and we refer to [Mat10],[Mat11], [KZ13], [Che13] for more details.4. Property A2 uses the first type of dimension reduction. The assumption ensuresthat H st admits at most two minimal hyperbolic saddles. An arbitrarily smallperturbation ensures that only one of them is minimal. Using Theorem 2.3, oneobtain that H s admits a minimal four-dimensional NHWIC C . Furthermore,Theorem 2.4 and Proposition 6.1 provide variational characterization for thecylinder. Then the restricted system to C behaves like a system with two degreesof freedom , and an analog of property A1 can be proven. In particular, there willbe an ordered collection of minimal two-dimensional NHIC’s contained in C .5. For property A3, when the triple p H st , h, ρ q satisfies the AM property, the strongsystem admits a family of two dimensional NHICs. Because there is only oneweak component, Theorem 2.3 implies that H s admits a minimal four-dimensional NHWIC. Similar to the previous case, the idea from property A1 can be appliedto prove nondegeneracy.6. One can say that in the case A2 or A3, the slow system H s is “dominated” bythe strong system H st . A.4 Construction of a diffusion path and surgery of resonantmanifolds
To prove Theorem A.1, it remains to construct a diffusion path with our non-degeneracyconditions.
Proposition A.2.
For each γ ą , there exists an open and dense set V Ă t} H } C r “ u , such that for any H P V , there exists a γ ´ dense diffusion path P , such that thenon-degeneracy conditions [H1] and [H2] are satisfied along P . The proof of Proposition A.2 occupies the rest of this section. Since our nondegener-acy conditions are assumed to be open, it suffices to prove density. We fix an arbitraryrelative open set U Ă t} H } C r “ u , we will show there exists H P U such that theconclusions hold. The proof follows an inductive scheme. The strategy is as follows:1. At step s we have a finite collection of integer irreducible lattices L p s q “ t Λ p s q i u ,i.e. each Λ p s q i : “ span Z t k i , ¨ ¨ ¨ , k is u has rank s and is spanned by integer vectors k i , ¨ ¨ ¨ , k is . The union of corresponding codimension s resonant manifolds (Γ Λ p s q )is called P p s q .The lattices Λ p s q has a hierarchical structure in the sense that there is an uniqueelement Λ p s ´ q P L p s ´ q such that Λ p s ´ q Ă Λ p s q . As a result, P p s q Ă P p s ´ q ,we will choose L p s q such that P p s q is p ´ ¨ ´ s q γ ´ dense subset in P p s ´ q and p ´ ´ s q γ ´ dense in B n . 58. A set of essential resonances E p s ` q is a collection of irreducible lattices of rank s `
1. Each element Σ p s ` q P E p s ` q contains at lease one element Λ p s q P L p s q .Roughly speaking, the essential lattices are the collection of lattices Σ p s ` q thatcontain but does not dominate some Λ p s q P L p s q .Essential resonances Σ p s ` q correspond to a codimension s ` Σ p s ` q contained in P p s q .3. A nondegenerate set N s Ă P p s q , which is open, connected and p ´ ¨ ´ s q γ ´ densein B n , such that the following hold:(a) For all Λ p s q P L p s q with basis B p s q “ r k , ¨ ¨ ¨ , k s s , and p P N s X Γ L p s q , theslow system H sp , B p s q is nondegenerate in the sense of Property A2.(b) For all essential lattice Σ p s ` q P E p s ` q with basis B p s ` q e , and p P N s X Γ Σ p s ` q ,and every Λ p s q P L p s q with Λ p s q Ă Σ p s ` q and basis B p s q , the triple p H sp , B p s ` q e , h p B p s q , B p s ` q e qq , ρ q satisfies the AM property, setting up to apply Property A3.4. The next generation of resonant lattices are carefully defined so that we can useProperty A2, A3 to extend the non-degeneracy in item 3 to the next generation.5. The induction finishes at step n ´
1, when we obtain an open, connected and γ ´ dense set N n ´ Ă P p n ´ q in B n which consists of 1-dimensional resonantmanifolds and will be our diffusion path, and all essential resonances have theAM property. A.4.1 An initial step of the induction
Since the union of all 1 ´ resonant manifolds are dense and locally connected, for each γ ą L p q “ t Λ p q i u such that the set P p q “ ď Λ p q P L p q Γ L p q X B n is connected and γ { ´ dense in B n . For each lattice L p q denote its basis by B p q “ r k s ,i.e. L p q “ span R p k q X Z n ` . Denote by K p q “ t B p q i u the union of basis vectors.For any two sets E , E Ă Z n ` , we define E _ E “ span R t E Y E u X Z n ` to be the smallest irreducible lattice containing E and E .We define a first non-degeneracy set Y λ p H , P p q q Ă P p q by the following condition:For any Λ p q P L p q with basis B p q and p P Γ Λ p q X P p q , the averaged potential U p, B p q has at most two λ ´ nondegenerate minima.59 Λ p q Y λ Figure 2: The first nondegeneracy set for n “
3: the shaded part is the λ ´ nondegenerateset. On the bold lines there are two minima for U p, B p q , at the blue dots are there arethree minima. At the green dots the minimum is degenerate. Lemma A.3.
There exists a relative open set U Ă U and λ ą , such that for H P U , the nondegeneracy set Y λ p H , P p q q is open, connected, ´ γ ´ dense in P p q ,and p ´ ¨ ´ γ q´ dense in B n . The set Y λ p H , P p q q is the shaded set on Figure 2. For brevity in what follows weoften omit dependence of Y λ on H and P p q .For each p P Γ Λ p q X Y λ , where Λ p q ’s basis is B p q , the assumption of Property A2is satisfied for B st “ B p q , p and U st “ U p , B p q . Moreover, using compactness, for all m “ , Λ p q P L p q , p P Γ Λ p q X Y λ , U st “ U p , B p q , there exists a uniform M “ M p P p q , Y λ q , such that for all B wk “ r k wk1 , k wk2 s with µ p B wk q ą M the conclusion of Property A2 is satisfied. We assume that M is chosensuch that M p P p q , Y λ q ą max k P K p q | k | . We define the first generation of essential lattices E p q “ E p q p P p q , Y λ q as the setof all rank 2 irreducible lattices Σ p q satisfying the following conditions: there existsΛ p q P L p q such that Σ p q Ă Λ p q , M p Σ p q | Λ p q q ď M p P p q , Y λ q . The requirement M p P p q , Y λ q ą max k P K p q | k | ensures that for any Λ p q , Λ p q P L p q ,the lattice Λ p q _ Λ p q is automatically essential. This corresponds to the intersection ofΓ Λ p q and Γ Λ p q .The essential lattice set contain all lattices that does not “dominate” the lattices in L p q . Let us also denote Γ E p q “ ď Σ p q P E p q Γ Σ p q Λ p q Y λ Γ Λ p q N Γ Σ p q Figure 3: The final nondegeneracy set of step 1: removing all essential resonancesresults in a disconnected set, but removing only the degenerate part does not destroyconnectivity.the union of all resonance manifolds corresponding to the essential lattices.For each essential lattice Σ p q we fix an ordered basis B p q e (the actual choice isirrelevant). We define a second nondegeneracy set Z p H , Σ p q , Y λ q to be the set of p P Γ Σ p q X Y λ such that for each k P Σ p q with B p q “ r k s P K p q , the pair ´ H sp, B p q e , h p B p q | B p q e q , ρ ¯ satisfies the AM property. We then define Z p H , E p q , Y λ q to be the union of all Z p H , Σ p q , Y λ q over essential resonances Σ p q P E p q .Because H sp, B p q e has two degrees of freedom, we can use Property A0 to obtain thefollowing lemma. Lemma A.4.
There exists a relative open set U Ă U and a relative open ˜ Z Ă Γ E p q X Y λ such that the following hold.1. For all H P U , ˜ Z is compactly contained in Z p H , E p q , Y λ q .2. The set N : “ Y λ X ˜ Z is open, connected, and p ´ ´ q γ ´ dense in P p q . We choose ˜ Z compactly contained in Z so that the nondegeneracy on ˜ Z is uniform due to compactness. The idea behind the definition of N is the following: On the set ofessential resonances Γ E p q , domination does not apply, so we should remove it from thenondegeneracy set Y λ . However, in this case the remaining set becomes disconnected because the essential resonances divide the space (see Figure 3, left). Instead we onlyremove only p ’s with the nearly degenerate essential resonances, i.e. Y λ X Γ E p q z ˜ Z (seeFigure 3, right dashed line). 61 .4.2 Step 2 of the induction We completed step 1 with• the collection of rank one lattices L p q , with associated bases B p q ,• a collection of essential rank two lattices E p q ,• a dual collection of codimension one resonant manifolds P p q ,• the nondegenerate set N Ă P p q and is p ´ ´ γ q´ dense in B n .By step 1, for each essential resonance Σ p q Ą B p q e and p P Γ Σ p q X N the pair ´ H sp, B p q e , h p B p q | B p q e q ¯ is nondegenerate. Therefore, Property A3 applies with m “ , B st “ B p q e , U st0 “ U p , B p q e , h “ h p B p q | B p q e q . Moreover, we can choose a uniform constant N “ N p E p q , N q over all Σ p q P E p q , B p q Ă Σ p q and p P Γ Σ p q X N such that the conclusion of Property A3 hold.We are now ready to define the set L p q . We say a rank 2 lattice Λ p q is admissible ifthe following hold.1. There exists Λ p q P L p q such that Λ p q Ă Λ p q .2. Λ p q cannot be generated by the previous generation essential resonances, namelyΛ p q Ć ł t Σ p q P E p q u , (A.2)where Ž is the smallest irreducible lattice that contains all lattices Σ p q P E p q .3. Item 2 ensures that Λ p q Ă Λ p q is unique. Otherwise, suppose we have Λ p q , Λ p q Ă Λ p q with bases B p q , B p q , then Λ p q “ Λ p q _ Λ p q , and M p Λ p q | Λ p q q ď max k P B p q Y B p q t| k |u ď M s p P p s q , N s q , hence Λ p q P E p q , which is a violation of item 2.4. (ghost property) For each Λ p q Ă Λ p q and Λ p q Ă Σ p q , we have M p Σ p q _ Λ p q | Σ p q q ą N s p E p s ` q , N s q . (A.3)In particular, for an adapted basis B st , B wk of Σ p q Ă Σ p q _ Λ p q and p P Γ Σ p q X N the conclusion of Property A3 hold . We also point out in this case Σ p q “ Σ p q _ Λ p q will be an element of the next generation essential resonance. The name “ghost” comes from the fact that we test k against all possible essential lattices Σ p q
62e claim that the lattices that are not admissible can be generated by a finite setof integer vectors. Therefore the resonance manifolds of the admissible lattices form adense set. As a result:
Lemma A.5.
There exists an collection of rank two admissible lattices L p q such that P p q “ ď Λ p q P L p q Γ L p q X N is connected, ´ γ ´ dense in N and p ´ ¨ ´ q γ dense in B n . For each Λ p q P L p q , there is a unique Λ p q P L p q with Λ p q Ă Λ p q . Since Λ p q comeswith a standard basis B p q , we extend it using Proposition 3.1 to obtain a standardbasis B p q of Λ p q . We call the collection of all basis K p q .Similar to step 1, we define the non-degeneracy set Y λ p H , P p q q Ă P p q by thefollowing condition: For any lattice Λ p q P L p q , with basis B p q , and p P Γ Λ p q X P p q , theaveraged potential U p, B p q has at most two λ ´ nondegenerate minima. Lemma A.6.
There exists an open set U Ă U and λ ą , such that for H P U , the nondegeneracy set Y λ p H , P p q q is open, connected, ´ γ ´ dense in P p q , and p ´ ¨ ´ q γ ´ dense in B n . Using compactness, we obtain that for m “ , B st “ B p q P K p q , p P Γ B p q X Y λ , U st “ U p , B p q , there exists M p P p q , Y λ q ą
0, such that the conclusion of Property A3 applies for all µ p B wk q ą M . As before, we require M p P p q , Y λ q ą max k ,k P B p q P K p q p| k | , | k |q . We now define essential lattices. It suffices to define bases of these lattices. Asin step 1, E p q is the set of all rank 3 irreducible lattices Σ p q satisfying the followingconditions: there exists Λ p q P L p q such thatΣ p q Ă Λ p q , M p Σ p q | Λ p q q ď M p P p q , Y λ q . Starting from s “
2, the essential resonances come with a hierarchical structure (seeFigure 5).•
Type p , q : We say Σ p q is of type p , q if there exists Σ p q P E p q such thatΣ p q Ă Σ p q . The collection of p , q essential lattices is denoted E p q .This element Σ p q is necessarily unique, otherwise Σ p q Ą Λ p q can be generatedfrom two elements from E p q , leading to a contradiction with item 2 in the definitionof L p q . Moreover, since L p q cannot be generated by vectors from Σ p q , we haveΣ p q “ Σ p q _ Λ p q . 63y item 3 in the definition of L p q , M p Σ p q _ Λ p q | Σ p q q “ M p Σ p q | Σ p q q ą N p E p q , N q . Recall that Σ p q comes with fixed basis B p q e . We use Proposition 3 to extend thisbasis to an adapted basis B p q e of Σ p q . We take this basis as the fixed basis of Σ p q .For each p P Γ Σ p q , Property A3 applies. We say that H p, B p q e dominates H p, B p q e . • Type p , q : Σ p q is called type p , q if it does not contain any element in E p q .The collection is denoted E p q .In this case, by definition, Σ p q P E p q and we have M p Σ p q | Λ k q ą M p P p q , Y λ q . We use Proposition 3 to extend k to an adapted basis B p q e of Σ p q , taken as thefixed basis for Σ p q . Property A2 applies, and we say that H p, B p q dominates H p, B p q e . We now have the decomposition E p q : “ E p q Y E p q . For each Σ p q Ă E p q with basis B p q e , we define the nondegeneracy set Z p H , Σ p q , Y λ q Ă Γ Σ p q X Y λ to be the subset such that for each Λ p q Ă Σ p q with basis B p q , ´ H sp, B p q e , h p B p q | B p q e q , ρ ¯ satisfies the AM property. We then define Z p H , E p q , Y λ q to be the union of all Z p H , Σ p q , Y λ q over essential resonances Σ p q P E p q .If the essential resonance Σ p q P E p q is type p , q , we use Property A3; if Σ p q is oftype p , q , we use Property A2. This allows us to prove the following non-degeneracylemma. Lemma A.7.
There exists a relative open set U Ă U and a relative open set ˜ Z suchthat the following hold.1. For each H P U , ˜ Z is compactly contained in Z p H , E p q , Y λ q .2. The subset N : “ Y λ X ˜ Z is open, connected, ´ γ ´ dense in P p q and p ´ ¨ ´ q γ ´ dense in B n . Λ p q P p q Figure 4: Hierachy of essential resonances: faint red curves are the previous generationessential resonances, blues line are current generation diffusion path, solid red dots areof type p , q , hollow blue dots are of type p , q . A.4.3 Step s ` of the induction We completed step s with• the collection of rank s lattices L p s q , with associated bases B p s q ,• a collection of essential rank s+1 lattices E p s ` q ,• a dual collection of codimension one resonant manifolds P p s q ,• the nondegenerate set N s Ă P p s q , which is p ´ ´ s γ q´ dense in B n . The diffusion path .• We have the collection of lattices L p q , ¨ ¨ ¨ , L p s q , with L p j q “ t Λ p j q i u are irreduciblerank j lattices. For each Λ p j q P L p j q , there exists a unique Λ p j ´ q Ă Λ p j q and suchthat Λ p j ´ q Ă L p j ´ q .• Each Λ p s q P L p s q has a ordered basis defined in the following way. For each Λ p q wefix a basis B p q “ t k u which is unique up to a sign. From the previous property,Λ p s q comes with the chain of inclusionΛ p q Ă ¨ ¨ ¨ Ă Λ p s q , Λ p j q P L p j q , ď j ď s, and we extend the basis B p q of Λ p q an increasing set of bases B p q Ă ¨ ¨ ¨ Ă B p s q by consecutive application of Proposition 3.1.65 We use K p s q to denote the collection of standard bases. For each B p s q “ r k , ¨ ¨ ¨ , k s s ,we denote | B p s q | “ sup i | k i | .• The diffusion path at step s is P p s q “ ď Λ p s q P L p s q Γ Λ p s q . The set P p s q X B n is connected and p ´ ¨ ´ s q γ ´ dense in B n . Essential resonances .• We have the essential lattices E p q , ¨ ¨ ¨ , E p s ` q , where for each 1 ď j ď s , Σ p j ` q P E p j ` q is a rank j ` p j ` q , there exists at least one,and at most two element Λ p j q P L p j q , such that Λ p j q Ă L p j q .• If essential lattice Σ p s ` q P E p s ` q contains only one element Σ p s q P E p s q , then thereexists 1 ď j ď s , such thatΣ p j ` q Ă ¨ ¨ ¨ Ă Σ p s ` q , Σ p t q P E p t q , j ` ď t ď s ` p j ` q does not contain anyelement of E p j q . We then have the following inclusionΛ p q Ă ¨ ¨ ¨ Ă Λ p j ´ q Ă Σ p j ` q Ă ¨ ¨ ¨ Ă Σ p s ` q . We use Proposition 3.1 to obtain the chain of adapted bases (called ordered basis: B p q Ă ¨ ¨ ¨ Ă B p j ´ q Ă B p j ` q e Ă ¨ ¨ ¨ Ă B p s ` q e , where each B p t q is a basis of Λ p t q P L p t q and each B p t q e is a basis of Σ p t q P E p t q .Recording the increment of rank in the chain, the essential resonance Σ p s ` q iscalled of type p s ` , j q . Denote by E p s ` q j the set of essential resonances with thisproperty. Strong system and nondegeneracy .• For each 1 ď j ď s , there exists the nondegeneracy set N j Ă P p j q , with theproperty that each N j is relative open, connected and p ´ ´ j γ q´ dense in P p j q X B n and p ´ ¨ ´ j q γ ´ dense in B n . The following inclusion hold P p q Ą N Ą ¨ ¨ ¨ Ą P p s q Ą N s . • There exists a sequence of (nonempty) relative open sets t} H } C r “ u Ą U Ą U Ą U Ą ¨ ¨ ¨ Ą U s Ą U s λ s ą H P U s , Λ p s q P L p s q with basis B p s q , and p P N s X Γ Λ p s q , the strong system H p, B p s q is nondegenerate in the sense of PropertyA3 and the averaged potential U p, B p s q has at most two λ -nondegenerate minima.Using compactness, let M s p P p s q , N s q ą sup B p s q P K p s q | B p s q | be a uniform constant such that Property A3 applies.• For each H P U s , Σ p s ` q P E p s ` q with basis B p s ` q e , each Λ p s q Ă Σ p s ` q with basis B p s q , and p P N s X Γ Σ p s ` q , the pair ´ H p, B p s ` q e , h p B p s q , B p s ` q e q ¯ is nondegenerate in the sense of Property A3. Using compactness, let N s p E p s ` q , N s q ą omination Properties • Let Σ p s ` q P E p s ` q j be an essential resonance of type p s ` , j q , then we have thechain Λ p q Ă ¨ ¨ ¨ Ă Λ p j ´ q Ă Σ p j ` q Ă ¨ ¨ ¨ Ă Σ p s ` q . The following domination property holds: M p Σ p j ` q | Λ p j ´ q q ą M j ´ p P p j ´ q , N j ´ q ,M p Σ p t ` q | Σ p t q q ą N t ´ p E p t q , N t ´ q , j ` ď t ď s. • As a corollary of the domination properties, for Σ p s ` q with the type p s ` , j q , let B p q Ă ¨ ¨ ¨ Ă B p j ´ q Ă B p j ` q e Ă ¨ ¨ ¨ Ă B p s ` q e be the chain of basis. Then – For each p P Γ Λ p j ´ q X N j ´ , the system H p, B p j ´ q dominates H p, B p j ` q e in thesense of Property A2. – Fore each j ` ď t ď s , p P Γ Σ p t q X N t ´ , the system H p, B p t q e dominates H p, B p t ` q e in the sense of Property A3.We now define the set L p s ` q . This is essentially an elaboration of step 2. We saythe rank s ` p s ` q is admissible if the following hold.1. There exists Λ p s q P L p s q such that Λ p s q Ă Λ p s ` q .2. Λ p s ` q cannot be generated by any previous generation essential resonances, namelyΛ p s ` q Ć ł t Σ p s ` q P E p s ` q u , (A.4)where Ž is the smallest irreducible lattice that contains all lattices Σ p s ` q P E p s ` q .3. Item 2 ensures that Λ p s q Ă Λ p s ` q is unique. Otherwise, suppose we haveΛ p s q , Λ p s q Ă Λ p s ` q with bases B p s q , B p s q , then Λ p s ` q “ Λ p s q _ Λ p s q , and M p Λ p s ` q | Λ p s q q ď max t| B p s q | , | B p s q |u ď M s p P p s q , N s q , hence Λ p s ` q P E p s ` q , which is a violation of item 2.4. (ghost property) For each Λ p s q Ă Λ p s ` q and Λ p s q Ă Σ p s ` q , we have M p Σ p s ` q _ Λ p s ` q | Σ p s ` q q ą N s p E p s ` q , N s q . (A.5)68 emma A.8. There exists an collection L p s ` q “ t Λ p s ` q u of admissible pairs such that P p s ` q “ ď Λ p s ` q P L p s ` q Γ Λ p s ` q X N s is connected, ´ s ´ γ ´ dense in N s , and p ´ ¨ ´ s ´ q γ dense in B n . The set Y λs ` p H , P p s ` q q Ă P p s ` q is defined by the following condition: For anyΛ p s ` q P L p s ` q with basis B p s ` q , and p P Γ B p s ` q X P p s ` q , there exists 0 ă λ ă λ suchthat the averaged potential U p, B p s ` q has at most two λ ´ nondegenerate minima. Lemma A.9.
There exists an open set U s ` Ă U s and λ s ` ą , such that for H P U s ` , the nondegeneracy set Y λ s ` s ` p H , P p s ` q q is ´ s ´ γ ´ dense in P p s ` q andconnected. Define M s ` p P p s ` q , Y λ s ` s ` q ą sup B p s ` q P K p s ` q | B p s ` q | be the uniform constant over all H P U s ` , p P Y λ s ` s ` , and Λ p s ` q P L p s ` q . The essentiallattice set E p s ` q is defined as the set of all rank s ` p s ` q satisfyingthe following conditions: there exists Λ p s ` q P L p s ` q such thatΣ p s ` q Ą Λ p s ` q , and M p Λ p s ` q | Σ p s ` q q ă M s ` p P p s ` q , Y λ s ` s ` q . We have the following remarks:• Suppose there exists Σ p s ` q Ą Σ p s ` q with Σ p s ` q P E p s ` q , then Σ p s ` q is unique.Otherwise, suppose Σ p s ` q contains both Σ p s ` q , Σ p s ` q , then there exists Λ p s ` q Ă Σ p s ` q “ Σ p s ` q _ Σ p s ` q , this is a violation of (A.4).• In case that Σ p s ` q Ą Σ p s ` q , then for Λ p s ` q P L p s ` q with Σ p s ` q Ą L p s ` q , we get M p Σ p s ` q | Σ p s ` q q “ M p Σ p s ` q | Σ p s ` q _ Λ p s ` q q ą N s p E p s ` q , N s q by (A.5).Finally, for each Σ p s ` q Ă E p s ` q , we define the nondegeneracy set Z s ` p H , Σ p s ` q , Y λ s ` s ` q to be the subset that for each B p s ` q “ p k , ¨ ¨ ¨ , k s ` q Ă Σ p s ` q , the pair ´ H sp, B p s ` q e , h p B p s ` q | B p s ` q e q ¯ is nondegenerate. We then define Z s ` p H , E p s ` q , Y λ s ` s ` q to be the union over essentialresonances Σ p s ` q P E p s ` q .The following lemma is proven using the type of essential resonances, similar to step2. 69 emma A.10. Suppose s ` ă n . There exists an open set U s ` Ă U s ` and a relativeopen set ˜ Z s ` such that the following hold.1. For each H P U s ` , ˜ Z s ` is compactly contained in Z s ` p H , E p s ` q , Y λ s ` s ` q .2. The subset N s ` : “ Y λ s ` s ` X ˜ Z s ` is open, connected, ´ s ´ γ ´ dense in P p q and p ´ ´ s ´ q γ ´ dense in B n .Moreover, if s ` “ n , ď Σ p s ` q P E p s ` q Γ Σ p s ` q X Y λ s ` s ` is a collection of isolated points. Then the same two points hold with ˜ Z s ` “ ď Σ p s ` q P E p s ` q Γ Σ p s ` q X Y λ s ` s ` . This finishes the construction of the lattices and verification of properties for step s ` A.4.4 Concluding the induction
The induction ends when U n ´ , L p n ´ q , P p n ´ q , N n ´ and E p n q are defined. Then P p n ´ q X N n ´ is γ ´ dense diffusion path in B n , and for each p P Γ Λ p n ´ q X N n ´ , H P U n ´ , the potential U p, B p n ´ q has at most two λ n ´ ´ nondegenerate minima.We then have Lemma A.11.
There exists an open and dense set of U n ´ Ă U n ´ such that [H1 λ n ´ ]holds for all H P U n ´ on P p n ´ q X N n ´ . Moreover, from Lemma A.10 we know that condition [H2] holds on all essentialresonances. Therefore, the diffusion path P p n ´ q X N n ´ satisfies all the conditionsrequired. B Diffusion mechanism and AM property
The goal of this section is to give a short review of diffusion mechanisms. Then wefocus on diffusion mechanism using variational methods and discuss difficulties arisingin higher dimensions. After that we explain the role of dominant systems.In [Arn64b] Arnold proposed the following example H p q, p, ϕ, I, t q “ I ` p ` (cid:15) p ´ cos q qp ´ µ p sin ϕ ` sin t qq , where q, ϕ and t are angles and p, I P R . This example is a perturbation of the productof a one-dimensional pendulum and a one-dimensional rotator. There is a rich literature70n Arnold example and we do not intend to give extensive list of references; we mention[AKN06; BB02; DLS06; Tre04], and references therein.The important feature of this example is that it has a 3-dimensional NHIC Λ “t p “ q “ u , which is the direct product of R and 2-dimensional torus T . Later havinga 3-dimensional NHIC means that there is a NHIM diffeomorphic to the direct productof R and 2-dimensional torus T . Whiskered tori and transition chains
In [Arn64b] Arnold noticed that for each ω P R there are an invariant 2-dimensionaltorus T ω “ t p “ q “ , I “ ω u having 3-dimensional stable and instable manifolds W s p T ω q and W u p T ω q resp. Noticethat orbits inside T ω have a well defined rotation number equal to ω .Call an ordered sequence of tori t T ω i u i transition chain if for each i we have W s p T ω i q and W u p T ω i ` q intersect transversally . In [Arn64b] proved that for any a ă b there is a transition chain such that ω “ a and ω N “ b for some N and showed that this implies existence of orbits asymptotic to T b in the future and to T a in the past. Generalized transition chains
In [Mat91a] Mather proposed a diffusion mechanism where invariant tori wherereplaced by Aubry-Mather invariant sets for twist maps.For fiber convex superliear time-periodic Hamiltonians H p θ, p, t q , θ, t P T , p P R foreach rotation number ω there is a “minimal” invariant set A ω consisting “minimal”orbits rotation number ω . The -torus graph property. Let π : p θ, p, t q Ñ p θ, t q is the natural projection. Mather proved that each such aset A ω is a Lipschitz graph over π A ω , i.e. π is one-to-one on A ω and the inverse π ´ : π A ω Ñ A ω is Lipschitz.We say that an invariant set A ω has a -torus graph property , if there is C smoothmap π : T T ˆ T Ñ T having maximal rank in a neighborhood of A ω such that π isone-to-one on A ω and the inverse π ´ : π A ω Ñ A ω is Lipschitz.• (rational case) if ω “ p { q for some integers p, q with q ą A ω contains“minimal” periodic orbits.• (irrational case) if ω R Q the set A ω either a Lipschitz 2-torus, i.e. π A ω “ T orcontains a suspension of a Denjoy set.71 generalized transition chain. Using give a precise meaning of a stable and an unstable set of each invariant set A ω .These sets are not necessarily manifolds, but still denoted W s p A ω q and W u p A ω q resp.One can give a precise meaning of transverse intersection of these sets using the barrierfunction. Call it a generalized transverse intersection. An ordered sequence of “minimal” invariant sets t A ω i u i is called a generalizedtransition chain if• each invariant set A ω i has a 2-torus graph property;• for each i invariant sets W s p A ω i q and W u p A ω i ` q have a generalized transverseintersection .In [Mat91b; Mat93] Mather proposed a generalization of constuction from [Mat91a].Inspired by these ideas, in [Ber08; CY04; CY09] proved that for a generic perturbationin the Arnold example there are generalized transition chains. Moreover, there areorbits shadowing this transition chain. An equivalence of invariant sets in a generalized transition chain.
In [Ber08] replaces generalized transversality condition with forcing relation. Thenhe shows that if A ω forces A ω and vise versa then this is an equivalence relation. Inparticular, if nearby invariant sets A ω i and A ω i ` are equivalent, then there are orbitsheteroclinic orbits for any pair of invariant sets in a generalized transition chain.In [BKZ11] we construct “short” 3-dimensional NHICs. Then we show that each ofsuch cylinders carries a generalized transition chain. Moreover, all invariant sets in sucha chain are equivalent and, therefore, there are orbits connecting any pair of invariantsets in this transition chain.In [KZ13] we construct a “connected” collection of 3-dimensional NHICs and showthat each cylinder carries a generalized transition chain. The -torus graph and AM properties. Partial averaving of nearly integrable system H (cid:15) “ H ` (cid:15)H near a resonant manifoldleads to a mechanical system of d ď n degrees of freedom H p ϕ, I q “ K p I q ´ U p ϕ q , ϕ P T d , I P T d , where K p I q is a positive definite quadratic form and U is a sufficiently smooth function(see (1.2)).In order to find “minimal” invariant set having the 2-torus graph property• we construct 3-dimensional NHICs; 72 we prove that each 3-dimensional NHIC contains a family of “minimal” invariantset and each such a set is localized.• due to localization we prove that the projection along the action component ontothe 2-torus T is one-to-one with a Lipchitz inverse.In order to construct a 3-dimensional NHIC for H (cid:15) near a maximal order resonanceit suffices to construct a 2-dimensional NHIC, diffeomorphic to the standard cylinder,for the averaged mechanical system H .Due to concervation of energy each 2-dimensional NHIC consists of “minimal”hyperbolic periodic orbits. This leads to the following problem: Construct a family of “minimal” invariant sets consisting of hyperbolic periodicorbits!
In the case d “ d ą h on T d and consider infiniteminimizers of homology class h , i.e. the Aubry set A p h q (see section 5.1 for precisedefinition). Then• A p h q does not have to consist of periodic orbits or does not even have to havecountably many invariant components (see e.g. [Mat04]).• A p h q consisting of periodic orbits do not guarantee they have homology h (see[Lev97]).• In the class of Tonelli Hamiltonians minimization within the class of closed loopsin some homology class h might lead to non-hyperbolic minimal periodic orbits(see [Arn98]).The AM property guarantee all these properties. The jump.
In [KZ13] section 12 we show that for each pair of “crossing” cylinders there is ajump from an invariant set from one generalized transition chain with another one.The jump, in particular, means that these invariant sets are equivalent and, therefore,invariant sets from both generalized transition chains are equivalent.One of the main conclusions of this paper is that we construct a diffusion path Γand a “connnected” collection 3-dimensional NHICs and show each of these cylinderscarries a collection of invariant sets haing 2-torus property.Using the technique from [BKZ11; KZ13] it should imply that invariant sets in eachcylinder form a generalized transition chain and are equivalent.Aside of many technical details we beleive that the only important missing partof construction of diffusing orbits along the path Γ is the jump . Construction of avariational problem leading to the jump for 3 -degree of freedom is in section 8 [KZ14].73 Normally hyperbolic invariant manifolds
In this section we state a version of the center manifold theorem and prove Theorem 2.3.While the central manifold theorem is classical, we need an version whose center directionis a non-compact set equipped with a Riemannian metric. This is done in the first twosubsections. In the last subsection, we perform a reduction on our system to apply thecentral manifold theorem.
C.1 Normally hyperbolic invariant manifolds via isolation block
We state an abstract theorem on existence of normally hyperbolic invariant manifoldsfor a smooth map F . based on Conley’s isolation blocks (see McGehee, [McG73]).We introduce a set of notations. We have three components x P R s , y P R u , z P Ω c Ă R c , where Ω c is a (possibly unbounded) convex set. We assume that Ω c admitsa C complete Riemannian metric g . We also consider a Riemannian metric on theproduct space W “ R s ˆ R u ˆ Ω c by taking the tensor product of g and Ω c , and thestandard Euclidean metric on R s , R u .Fix some r ą D s Ă R s and D u Ă R u be closed balls of radius r atthe origin in R s and R u ( r is considered fixed and we omit the dependence). Denote D sc “ D s ˆ Ω c , D uc “ D u ˆ M and D “ D sc ˆ D u . .Consider a C smooth map F : D “ D s ˆ D u ˆ Ω c Ñ R s ˆ R u ˆ Ω c , we state a set of conditions guaranteeing the set W sc p F q “ t Z P D : F k p Z q P D for all k ą u , called the center-stable manifold, is a graph tp X, Y q P D sc ˆ D u : w sc p X q “ Y u for a C function w sc .[C1] π sc F p D sc ˆ D u q Ă D sc .[C2] F maps D sc ˆ B D u into D sc ˆ R u z D u and is a homotopy equivalence.The first two conditions guarantee a topological isolating block : F stretches D sc ˆ B u along the unstable component D u and is a weak contraction along the center-stablecomponent D sc .Now we state the cone conditions . For some µ ą C uµ p Z q “ t v “ p v c , v s , v u q P T Z D : µ } v u } ě } v c } ` } v s } u . Note that p C uµ p Z qq c “ t v “ p v c , v s , v u q P T Z D : µ ´ p} v c } ` } v s } q ě } v u } u “ : C scµ ´ p Z q . K µu p x , y , z q “ tp x , y , z q : µ } y ´ y } ě } x ´ x } ` dist p z , z q u , where the distance is induced by the Riemannian metric g .We assume there is µ ą ν ą Z , Z P D suchthat Z P K µu p Z q we have[C3] F p Z q P K uµ p F p Z qq . [C4] } π u p F p Z q ´ F p Z qq} ě ν } π u p Z ´ Z q} . Proposition C.1. (Lipshitz center-stable manifold theorem) Suppose F satisfies condi-tions [C1-C4], then W sc p F q is given by the graph of a Lipschitz function W sc p F q “ tp x, y, z q P D : w sc p x, z q “ y u . Moreover, for Lebesgue almost every Z P W sc p F q , we have T Z W sc p F q P C scµ ´ p Z q . In order to obtain the center-unstable manifold, consider the involution I : p x, y, z q ÞÑp y, x, z q and assume inv p F q “ I ˝ F ´ ˝ I ´ satisfies the same conditions. Theorem C.2.
Assume that F, inv p F q satisfies the conditions [C1-C4], there exists a C function w c : M Ñ D such that W c p F q : “ W sc p F q X W uc p F q “ tp x, y, z q P D : p x, y q “ w c p z qu . Proof.
Proposition C.1 implies the existence of Lipshitz functions w uc : D uc Ñ D and w sc : D sc Ñ D , with W sc p F q “ t x “ w sc p y, z qu , W uc p F q “ W sc p inv p F qq “ t y “ w uc p x, z qu . Then standard arguments (see Theorem 5.18 in [Shu87]) implies these functions are C .The fact that µ ą T Z W sc p F q P C scµ ´ p Z q , T Z W u p F q P C scµ ´ p Z q implies W sc p F q and W uc p F q intersect transversally, and W sc p F q X W uc p F q is a graphover the center component M . 75 .2 Existence of Lipschitz invariant manifolds We prove Proposition C.1. Let V be the set Γ Ă D satisfying the following conditions:(a) π u Γ “ D u , (b) Z P K uµ p Z q for all Z , Z P Γ, where π u is the projection to theunstable component. These conditions ensures π u : Γ Ñ D u is one-to-one and onto,therefore Γ is a graph over D u . Moreover, condition (b) further implies that the graphis Lipshitz. In particular, each Γ P V is a topological disk. Lemma C.3.
Let Γ P V , then F p Γ q X D P V .Proof. By [C4] for any Z and Z we have that F p Z q belongs to the cone K uF p Z q of F p Z q . Thus, it suffices to show that D u Ă π u p F p Γ q X D q . The proof is by contradiction.Suppose there is Z ˚ P B u such that Z ˚ R π u p F p Γ qq .We have the following commutative diagram B Γ i ã ÝÑ Γ Ó π u ˝ F Ó π u ˝ F R u z D u i ã ÝÑ R u zt Z ˚ u . (C.1)From [C2] and using the fact that B s and Ω c are contractible, π u ˝ F | Γ is a homotopyequivalence. Note that i is a homotopy equivalences, and π u ˝ F | Γ is a homeomorphismonto its image. Let h and g be the homotopy inverses of π u ˝ F |B Γ and i , then h ˝ g ˝ p π u ˝ F q defines a homotopy inverse of i . As a result Γ is homotopic to B Γ, thisis a contradiction.Proposition C.1 follows from the next statement.
Proposition C.4.
The mapping π sc : W sc p F q Ñ D sc is one-to-one and onto, thereforeit is the graph of a function w sc . Moreover w sc is Lipshitz and T Z W sc p F q P p C uµ p Z qq c “ C scµ ´ p Z q , Z P W sc p F q . Proof.
For each X P D sc , we define Γ X “ p π sc q ´ X , clearly Γ X P V . We first showΓ X X W sc p F q is nonempty and consists of a single point. Assume first that Γ X X W sc p F q is empty. Then by definition of W sc p F q , there is n P N such that F n p Γ X q X D “ ∅ . However, by Lemma C.3, Ş ni “ F i p Γ X q X D P V is always nonempty, acontradiction. We now consider two points Z , Z P W sc p F q with π u Z “ π u Z . Notethat F k p Z q , F k p Z q P D for all k ě
0, and Z P K uµ p Z q , by [C4] we have2 ě } π u p F k p Z q ´ F k p Z qq} ě ν k } π u p Z ´ Z q} for all k , which implies Z “ Z .The last argument actually shows Z R K uµ p Z q for all Z , Z P W sc p F q . For any (cid:15) ą
0, for Z “ p X , Y q , Z “ p X , Y q P W sc p F q with dist p X , X q small, we have } Y ´ Y } ď µ ´ dist p X , X q . This implies both the Lipshitz and the cone propertiesin our proposition. 76 .3 NHIC for the dominant system We prove Theorem 2.3 in this section. First, an overview of notations.1. The strong Hamiltonian is H st “ H st p p , B st , U st q defined on T m ˆ R m , and itsassociated Lagrangian vector field is X st (see (2.8)). We call the We denote thetime-1-map of X st by G st0 and we will lift it to the universal cover R m ˆ R m without changing its name.2. The vector field X st is extended trivially to p T m ˆ R m q ˆ p T d ´ m ˆ R d ´ m q (see(2.11)). The time-1-map is denoted G , and we have G “ G st0 ˆ Id. We will alsolift it to the universal cover with the same name.3. The slow Hamiltonian is H s “ H s p B st , B wk , p , U st , U wk q , and consider its La-grangian vector field X sLag . We apply a coordinate change p ϕ st , v st , ϕ wk , v wk q “ Φ p ϕ st , v st , ϕ wk , I wk q as in (2.9), and a rescaling Φ Σ as defined in (2.13). The newvector field is denoted ˜ X s (see (2.10), (2.13)). We denote its time-1-map G , whichis considered a map on the Euclidean space R m ˆ R m ˆ R d ´ m ˆ R d ´ m .By Theorem 2.2, we have: Corollary C.5.
Assume that p B wk , p , U st , U wk q P Ω m,dκ,q p B st q , then for any δ ą , thereexists M ą such that for all p B wk , p , U st , U wk q with µ p B wk q ą M , uniformly on R m ˆ R m ˆ R d ´ m ˆ R d ´ m , we have } Π p ϕ st ,v st q p G ´ G q} ă δ , } DG ´ DG } ă δ . By assumption, the Hamiltonian flow H st admits an NHIC χ st p T l ˆ B l ` a q , where χ st is an embedding. Therefore G st0 admits an NHIC Λ st a “ Φ ´ ˝ p T l ˆ B l ` a q with theexponents α, β . We use local coordinates in a tubular neighborhood to simplify thesetting. Lemma C.6.
There exists a tubular neighborhood N p Λ st a q Ă T m ˆ R m of Λ st a and adiffeomorphism h st : B l ˆ B l ˆ p T l ˆ B l ` a q Ñ N p Λ st a q such that:1. h st p , , z q “ χ st p z q , in particular h st p C st a q : “ h st pt u ˆ t u ˆ p T l ˆ B l ` a qq “ Λ st a .2. For the map F st0 : “ p h st q ˝ G st0 ˝ p h st q ´ :(a) C st a is an NHIC for F st0 with the same exponents α, β .(b) The associated stable/unstable bundles take the form E s “ R l ˆ t u ˆ t u , E u “ t u ˆ R l ˆ t u . In particular, DF st0 is a block diagonal matrix in the blocks corresponding tothe three components. c) Let g denote the Euclidean metric. Then there exists a Riemannian metric g on T l ˆ B l ` a such that the tensor metric g b g b g on B l ˆ B l ˆ p T l ˆ B l ` a q is an adapted metric for the NHIC C st a .Proof. We use the bundles E u , E s , and the parametrization χ st of Λ st a to build acoordinate system for the normal bundle to Λ st a , which is diffeomorphic to the tubularneighborhood. We then pull back the adapted metric of Λ st a using this map to C st a .Denote Ω wk “ R d ´ m ˆ R d ´ m and consider the trivial extension h : B l ˆ B l ˆ pp T l ˆ B l ` a q ˆ Ω wk q Ñ N p Λ st a q ˆ Ω wk by h p x, y, p z st , z wk qq “ p h st p x, y, z st q , z wk q . Define the following maps F “ h ´ ˝ G ˝ h “ p F st0 , Id q , F “ h ´ ˝ G ˝ h. (C.2)Finally, to apply Theorem C.2, we denote Ω st a “ R l ˆ B l ` a which is the universalcover of T l ˆ B l ` a . We lift the maps F , F to the covering space without changing theirnames, namely F, F : B l ˆ B l ˆ p Ω st1 ` a ˆ Ω wk q ö . Ω st1 ` a ˆ Ω wk is our center component and is denoted Ω. While the maps are definedon unbounded regions, we keep in mind that F “ p F st0 , Id q where F st0 is defined on acompact set B l ˆ B l ˆ p T l ˆ B l ` a q .We still need one reduction to apply Theorem C.2. Recall that Ω st a “ R l ˆ B l ` a .Write F “ p F x , F y , F z q , define L p x, y, z q “ p D x F x p , , z q ¨ x, D y F y p , , y q ¨ y, F z p , , z qq (C.3)this is the linearized map at p , , z q (we used F p , , z q “ p , , F z q , and DF is blockdiagonal from Lemma C.6). since F “ p F st0 , Id q and F st0 is defined over a compact set,we obtain as r Ñ } L ´ F } “ o p r q , } DL ´ DF } “ o p q on B lr ˆ B lr ˆ p Ω st a ˆ Ω wk q . (C.4)Moreover, since F preserves t u ˆ t u ˆ B Ω, we get L p B l ˆ B l ˆ B Ω q Ă R l ˆ R l ˆ B Ω . (C.5)Namely, the linearized map L preserves the boundary of the center component. Finally,we modify the map F so that it also fixes the center boundary. Let ρ be a standardmollifier satisfying ρ p x, y, p z st , z wk qq “ ρ p z st q “ z st P Ω st0 ρ p x, y, p z st , z wk qq “ ρ p z st q “ z st P Ω st a z Ω st a { . Let ˜ F “ F p ´ ρ q ` Lρ, (C.6)we have: 78 emma C.7.
For any µ ą , (cid:15) ą and r ą , there exists δ ą and ă r ă r such that if G and G satisfies } Π p ϕ st ,v st q p G ´ G q} ă δ , } DG ´ DG } ă δ , the map ˜ F defined by (C.2) , (C.3) and (C.6) satisfies conditions [C1]-[C4] with theparameters µ and ν “ α ´ ´ (cid:15) on B lr ˆ B lr ˆ Ω . The same hold for the map inv p ˜ F q .Proof. First of all, from Lemma C.6, DF p , , z q “ diag t D x F x , D y F y , D z F z u with } D x F x } , }p D y F y q ´ } ´ ď α, } D z F z } , }p D z F z q ´ } ě β. (C.7)Recall that F “ p F st0 , Id q where F st0 is defined over a compact set. Therefore forsufficiently small r ą
0, we have } Π x F p x, y, z q} ď p α ` (cid:15) q} x } , } Π y DF p x, y, z q} ě p α ` (cid:15) q ´ } y } , hence Π x F p B lr ˆ B lr ˆ Ω q Ă B l p α ` (cid:15) q r , } Π y F p B lr ˆ B B lr ˆ Ω q} ě p α ` (cid:15) q ´ r. Since } ˜ F ´ F } ď }p ´ ρ qp F ´ F q} ` } ρ p L ´ F q} , } Π p x,y q p F ´ F q} ď C } Π p ϕ st ,v st q p G ´ G q} ď Cδ , and } L ´ F } “ o p r q , by choosing δ , r small enough, we getΠ x ˜ F p B lr ˆ B lr ˆ Ω q Ă B lr , } Π y ˜ F p B lr ˆ B B lr ˆ Ω q} ą r. The first half of the above formula combined with (C.5) gives [C1], and the second halfgives [C2].We now prove the cone conditions [C3] and [C4]. We first show the map ˜ F is wellapproximated by the linearized map DF p , , z q . Given any (cid:15) ą
0, we use Corollary C.5to choose δ so small such that } D p F ´ F q} ` } Π z st p F ´ F q}} dρ } ď C } D p G ´ G q} ` } Π ϕ st ,v st p G ´ G q}} dρ } ă (cid:15) { r such that for 0 ă r ă r , } D p F ´ L q} ` } F ´ L }} dρ } ă (cid:15) { B lr ˆ B lr ˆ p Ω st a ˆ Ω wk q . Then from ˜ F “ F ` p L ´ F q ρ , and the fact that ρ dependsonly on z st gives } D ˜ F p x, y, z q ´ DL p x, y, z q} ď } DF ´ DL } ` } Π z st p F ´ L q}} dρ }ď } D p F ´ F q} ` } D p F ´ L q} ` p} Π z st p F ´ F q} ` } Π z st p F ´ L q}q} dρ } ă (cid:15). Consider p x , y , z q , p x , y , z q P B lr ˆ B lr ˆ Ω, denote p ∆ x, ∆ y, ∆ z q “ p x , y , z q ´p x , y , z q and d “ } ∆ x } ` } ∆ y } ` dist p z , z q . For d small enough } ˜ F p x , y , z q ´ ˜ F p x , y , z q ´ DF p , , z qp ∆ x, ∆ y, ∆ z q}“ } L p x , y , z q ´ L p x , y , z q ` p ˜ F ´ L qp x , y , z q ´ p ˜ F ´ L qp x , y , z q´ DF p , , z qp ∆ x, ∆ y, ∆ z q}ď } D p ˜ F ´ L qp x , y , z q} d ď (cid:15)d.
79o prove [C3], we first show the linear map preserves the unstable cone. Forany µ ą p v x , v y , v z q P T p x ,y ,z q B lr ˆ B lr ˆ Ω with µ } v y } ě } v z } ` } v x } , let p v x , v y , v z q “ DF p , , z qp v x , v y , v z q , we have µ } v y } ě µα ´ } v y } ě α ´ p} v x } ` } v z } q ě α ´ p} v x } ` β } v z } q ě βα p} v x } ` } v z } q . In other words, for any µ ą
1, we have DF p , , z q C uµ Ă C uαµ { β .Coming to the non-linear map ˜ F , for p x , y , z q , p x , y , z q P B lr ˆ B lr ˆ Ω, let p x i , y i , z i q “ ˜ F p x i , y i , z i q , and p ∆ x, ∆ y, ∆ z q , p ∆ x , ∆ y , ∆ z q be the correspondingdifference. If p x , y , z q P K uµ p x , y , z q , then µ } ∆ y } ě } ∆ x } ` dist p z , z q . Inparticular dist p z , z q ď µ } ∆ y } ď µr . When r is small enough }p ∆ x , ∆ y , ∆ z q ´ DF p , , z qp ∆ x, ∆ y, ∆ z q} ď (cid:15)d . Furthermore assume r is so small that p ´ (cid:15) q dist p z, z q ď} ∆ z } p , ,z q ď p ` (cid:15) q dist p z, z q , where } ∆ z } p , ,z q is measured using the local Riemannianmetric. We drop the subscript from now on. Using the linear calculation, there exists auniform constant C ą µ } ∆ y } ě βα p} ∆ x } ` } ∆ z } q ´ C(cid:15) d ě βα p} ∆ x } ` } ∆ z } q ´ C p ` µ q (cid:15) } ∆ y } ě p ´ (cid:15) q βα p} ∆ x } ` dist p z , z q q ´ C p ` µ q (cid:15) } ∆ y } . noting that } D ˜ F ´ } , } D ˜ F } are uniformly bounded. When (cid:15) is small enough we get µ } ∆ y } ě } ∆ x } ` dist p z , z q . [C3] is proven.[C4] follows directly from }p ∆ x , ∆ y , ∆ z q ´ DF p , , z qp ∆ x, ∆ y, ∆ z q} ď (cid:15)d and(C.7). The proof for inv p ˜ F q is identical and is omitted. Proof of Theorem 2.3.
For any δ ą
0, we choose 0 ă r ă δ { C and µ ą
1, where C is aconstant specified later. Apply Lemma C.7, there exists M ą µ p B wk q ą M , the map ˜ F associated to H s p B st , B wk , p , U st , U wk q satisfies [C1]-[C4] on B lr ˆ B lr ˆ p Ω st a ˆ Ω wk q . As a result, we obtain a function w c : Ω st a ˆ Ω wk Ñ B m ´ lr ˆ B m ´ lr such that W c “ Graph p w c q “ tp x, y, p z st , z wk qq : p x, y q “ w c p z st , z wk qu is invariant under ˜ F , and is the maximally invariant set on B m ´ lr ˆ B m ´ lr ˆ Ω st a ˆ Ω wk .Since ˜ F “ F on whenever z st P Ω st0 , any F invariant set with z st P Ω st0 is also ˜ F invariantand hence is contained in W c . We now consider the map ζ : p z st , z wk q ÞÑ h p w c p z st , z wk q , z st , z wk q , then Graph p ζ q is an F ´ invariant set. Finally we invert the coordinate changes (2.9)and (2.13) to obtain the desired embedding η s “ Φ ˝ Φ Σ ˝ p ζ, Id q .Moreover, we have } w c } C ď r , using the fact that h st p , , z q “ χ st p z q , and that Φdoes not change the strong component, there is C ą } ζ ´ Φ ´ ˝ χ st } C ď C r .80inally, since the rescaling Φ Σ do not change the strong component, there exists C ą } Π p ϕ st ,I st q η s ´ χ st } “ } Φ p ζ ´ Φ ´ ˝ χ st q} ď Cr ă δ. We choose the open set V “ Φ ˝ h st p B m ´ lr ˆ B m ´ lr ˆ Ω st0 q , then any invariant set in V ˆ Ω wk must be contained in η s p Ω st0 ˆ Ω wk q . This concludes the proof. Acknowledgments
The first author acknowledges NSF for partial support grant DMS-5237860. The authorswould like to thank John Mather, Marcel Guardia, and Abed Bounemoura for usefulconversations.
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