Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices
aa r X i v : . [ m a t h . D S ] J un CONSTRUCTION OF INVARIANT WHISKERED TORI BY APARAMETERIZATION METHOD. PART II: QUASI-PERIODICAND ALMOST PERIODIC BREATHERS IN COUPLED MAPLATTICES.
ERNEST FONTICH, RAFAEL DE LA LLAVE, AND YANNICK SIRE
Abstract.
We construct quasi-periodic and almost periodic solutions for cou-pled Hamiltonian systems on an infinite lattice which is translation invariant.The couplings can be long range, provided that they decay moderately fastwith respect to the distance.For the solutions we construct, most of the sites are moving in a neigh-borhood of a hyperbolic fixed point, but there are oscillating sites clusteredaround a sequence of nodes. The amplitude of these oscillations does not needto tend to zero. In particular, the almost periodic solutions do not decay atinfinity.The main result is an a-posteriori theorem. We formulate an invarianceequation. Solutions of this equation are embeddings of an invariant toruson which the motion is conjugate to a rotation. We show that, if there isan approximate solution of the invariance equation that satisfies some non-degeneracy conditions, there is a true solution close by.This does not require that the system is close to integrable, hence it canbe used to validate numerical calculations or formal expansions.The proof of this a-posteriori theorem is based on a Nash-Moser iteration,which does not use transformation theory. Simpler versions of the scheme weredeveloped in E. Fontich, R. de la Llave,Y. Sire
J. Differential. Equations. ,3136 (2009).One technical tool, important for our purposes, is the use of weighted spacesthat capture the idea that the maps under consideration are local interactions.Using these weighted spaces, the estimates of iterative steps are similar to thosein finite dimensional spaces. In particular, the estimates are independent of thenumber of nodes that get excited. Using these techniques, given two breathers,we can place them apart and obtain an approximate solution, which leads toa true solution nearby. By repeating the process infinitely often, we can getsolutions with infinitely many frequencies which do not tend to zero at infinity.
Contents
1. Introduction 22. Basic setup and preliminaries 62.1. Phase spaces 72.2. Some functional analysis in ℓ ∞ ( Z N ) and the spaces of decay functions 82.3. Symplectic geometry on lattices 112.4. Diophantine properties 123. Formulation of the results 124. The Newton step 21 Supported by NSF grants. k -linear maps over ℓ ∞ ( Z N ) 71A.2. Spaces of differentiable and analytic functions on lattices 73A.3. Spaces of embeddings from C l to M with decay properties 73A.4. Spaces of localized vectors 74A.5. Regularity of the composition operators 75Appendix B. Appendix: Symplectic geometry on lattices 77B.1. Forms on lattices 77B.2. Some symplectic geometry on lattices 80Appendix C. Appendix : Construction of deformations of symplectic mapswhich are not exact symplectic 81References 831. Introduction
The goal of this paper is to prove theorems on persistence of invariant tori insome lattice systems. These models describe copies of identical systems placed onthe nodes of a lattice and interacting with all the other systems in the lattice. Theinteraction can be of infinite range, but it has to decay sufficiently fast with thedistance. We will assume that the dynamics is Hamiltonian and, for simplicity,we will also assume that the dynamics is analytic. We will consider “whiskeredtori” . These are invariant tori such that the motion on them is a rotation andwhich are as hyperbolic as possible, compatible with the fact that the motion is an
UASI-PERIODIC BREATHERS 3 irrational rotation (it is well known that the directions symplectically conjugate tothe tangent of the tori have to be neutral). See Definition 3.1.The main technical tool we will develop is a theorem of persistence of finitedimensional whiskered tori, namely Theorem 3.6 below, which has sufficiently goodproperties to allow us to use it recursively to construct tori with infinite frequencies.The tori we consider in Theorem 3.6 are finite dimensional whiskered tori withsome local character. The motion on the torus is a rigid rotation with a Diophantinefrequency. The preservation of the symplectic structure and the rotational motionon the tori force that there are some neutral directions in the normal directions. Wewill assume that, except for these directions, the normal directions are hyperbolic(they expand at exponential rates either in the future or in the past). In particular,the hyperbolic spaces are infinite dimensional.The main technique to prove Theorem 3.6 is to derive an equation that impliesinvariance of the torus and that the motion on it is a rotation and to develop atheory for solutions of the equation.Given a map F on a phase space M and a frequency ω ∈ R l , it is easy to seethat K : T l → M is a parameterization of a torus with a rotation ω , if and only if(1) F ◦ K = K ◦ T ω , where T ω denotes the rotation on the torus by ω . Similarly, for a vector field X ,we seek parameterizations K satisfying(2) X ◦ K = ∂ ω K, where ∂ ω is the derivative along the direction ω .Our main result will be Theorem 3.6, which shows that if we have an approximatesolution of the invariance equation which is also not too degenerate, there is atrue solution which is close to the approximate one. Theorems of this form, thatvalidate an approximate solution, will be called a posteriori , following the languagein numerical analysis.We emphasize that Theorem 3.6 does not assume that the system is close to inte-grable, so that the approximate solution could be produced in any way. Of course,when the system is close to integrable, we can take as approximate solutions thesolutions of the integrable system, so that we recover the standard formulationsof KAM theorems for quasi-integrable systems. The approximate solutions can beproduced by a variety of methods, including Lindstedt series or numerical computa-tions. In finite dimensions, some whiskered tori are generated by resonant averaging[dlLW04, Tre94] or by homoclinic tangencies [Dua08]. In such cases, Theorem 3.6leads to justifications of the expansions or the numerical computations. We alsonote that Theorem 3.6 does not assume that the system is translation invariant (itassumes only the existence of some uniform bounds).The a posteriori approach to KAM theorem was emphasized in [Mos66b, Mos66a,Zeh75, Zeh76a, Zeh76b]. There, it was pointed out that this a posteriori approachautomatically allows to deduce results for finitely differentiable systems as well asto prove smooth dependence on parameters or analyticity of perturbative series.We refer the reader to [dlL01] for a comparison of different KAM methods.In this paper, we use the a posteriori format to construct more complicatedquasi-periodic solutions by juxtaposing two simpler solutions separated by a suffi-ciently long distance. The a posteriori format of Theorem 3.6, allows us to controlthe limit of the solutions, which will be an almost periodic solution. The ability E. FONTICH, R. DE LA LLAVE, AND Y. SIRE to superimpose solutions far apart is greatly facilitated by assuming translationinvariance, which will be an assumption in the second part of this work. One couldassume significantly less (e.g. some amount of uniformity). Nevertheless, this seemsa natural assumption.One important technical tool in this paper is the use of spaces of decay functionsfollowing [JdlL00, FdlLM11a]. These are spaces of functions whose norms quantifythe effect that the motion of one particle does not affect much the motion of particlesfar apart. Besides that, they also enjoy certain Banach algebra properties so thatthe hyperbolic directions can be dealt with in the same manner than in the finitedimensional spaces.Using spaces of decay functions, we can make quantitative the observation that,since the oscillations at one site almost do not affect those further apart, super-imposing oscillations centered around sites far apart produces a very approximatesolution. We will call the localized oscillating solutions “breathers” . The error inthe invariance equation (measured in the sense of an appropriate space of decayfunctions) is arbitrarily small if the centers are placed far enough. A rather simplecalculation shows that the non-degeneracy conditions deteriorate also by an arbi-trarily small amount. In summary, if the frequencies of the oscillations are jointlyDiophantine (even if the constant is bad), we can satisfy all the requirements of thetheorem by displacing the breathers far apart. If one makes appropriate choices –placing the subsequent centers of oscillation far enough apart – we will show thatthe process can be repeated infinitely often and that it converges in a sense whichis strong enough to justify that the limit is a solution of the system. This solutioncontains infinitely many frequencies.The process of coupling the breathers does not require any smallness conditionsin the coupling (it suffices to place the breathers far enough apart). On the otherhand, establishing the existence of breathers by perturbing from those of the un-coupled system, does require some smallness conditions. We also require somemild smallness conditions on the perturbations to ensure that the system remainsnon-degenerate.In the solutions that we construct most of the sites are near a hyperbolic equi-librium. These solutions, therefore, have an average energy close to that of theequilibrium solutions and are at the border of chaos (in particular, they are dy-namically unstable). There are indications that these solutions play an importantrole in instability.The results of the paper were summarized in [FdlLS09a], which perhaps can beused as a reading guide to the present paper.We also note that, after this paper was finished, the work [BdlL14], used theresults of this paper to construct the whiskers of the whiskered tori constructed inthis paper in a very similar functional formulation, so that the whiskers also havedecay properties.To provide some motivation, we now mention several models found in the litera-ture for which our method applies. These models can be described by the followingformal Hamiltonian(3) H ( p, q ) = X i ∈ Z N (cid:16) | p i | + W ( q i ) (cid:17) + X k ∈ Z N X i ∈ Z N V k ( q i − q i + k ) UASI-PERIODIC BREATHERS 5 under some assumptions on the potentials W and V k . Here the formal Hamiltonianstructure is Ω ∞ = X i ∈ Z N dp i ∧ dq i . Note that, even if the sum defining the Hamiltonian and the symplectic form areformal and not meant to converge, Hamilton’s equations are a well behaved systemof differential equations (if the V k decay fast enough, e.g. if they are finite range).In fact the equations of motion are˙ q i = p i ˙ p i = −∇ W ( q i ) − X k ∈ Z N n ∇ V k ( q i − q i + k ) − ∇ V k ( q i − k − q i ) o . The model (3) involves a local potential W for each particle and interaction po-tentials among pairs of particles. Of course, the interaction potentials are assumedto decay with | k | fast enough. The method of proof also accommodates many bodyinteractions. One important feature of the method is that, in some appropriateweighted spaces, the estimates we obtain are independent of the number and theposition of the centers of oscillation.If we take the lattice to be with one degree of freedom, the potential V to bejust nearest neighbor (i.e. V k = 0 for | k | >
1) and set V ( s ) = γ s , we obtain theso called 1-D Klein-Gordon system described by the formal Hamiltonian H ( q, p ) = + ∞ X n = −∞ (cid:16) p n + W ( q n ) + γ q n +1 − q n ) (cid:17) , and whose equations of motion are(4) ¨ q n + W ′ ( q n ) = γ ( q n +1 + q n − − q n ) , n ∈ Z . We note that the method we present applies to higher dimensional lattices andhigher dimensional systems. We also do not need to assume that the symplecticform is the standard one. This is convenient when the symplectic form is degenerate.Changes in the symplectic form correspond to magnetic fields [Thi97]. Note thatthe systems with magnetic fields are not reversible.For a review of the physical relevance of these models we refer the reader to[FW98]. Concerning the existence proof of periodic breathers, we refer to [MA94,AGT96, AG96, AKK01]. In the latter papers, the technique is based on a variationalargument whereas in [MA94], the authors use an implicit function theorem. Forquasi-periodic breathers in finite – but arbitrarily large systems, we mention [BV02,GY07, GVY08, CY07b]. The paper [Yua02] proves the existence of quasi-periodicbreathers in the Fermi-Pasta-Ulam lattice. In all the cases above, the breathers arenormally elliptic or dissipative. Quasi-periodic and almost periodic breathers forlattices of reversible systems with dissipation are considered in [CY07a].
Remark 1.1.
There is a variety of results showing that for hyperbolic PDE’s thereare no quasi-periodic solutions of finite energy [Pyk96, SW99, KK08, KK10]. Sincesome of the models we consider are obtained as discretizations of nonlinear waveequations, it is interesting to understand why the results above do not apply to thediscretized model, even if they apply to the PDE.
E. FONTICH, R. DE LA LLAVE, AND Y. SIRE
The reason is that the mechanism behind the proofs in the above papers is thatquasi-periodic solutions of non-linear PDE’s have to radiate and send energy toinfinity.In the models we consider, there is no radiation because most of the media isnear the hyperbolic regime.We can understand the lack of radiation in the model but representative problem(5) ¨ q n + Aq n = γ ( q n +1 + q n − − q n ) , n ∈ Z , γ > . where A <
0. The equation (5) is a linearization of (4) near the point q = 0, whichis a maximum of the potential (or a mimimum of W in the notation of (4)).We see that if we substitute solutions of the form q n = exp( i ( ωt + kn )) in (5),we are lead to the dispersion relation − ω + A = γ (2 cos( k ) − . If | γ | is small enough, this dispersion relation does not have any real solutionsfor ω and the only square roots are imaginary.In this model, near the hyperbolic fixed points the equations do not propagatewaves, so that there is no radiation and the arguments excluding quasi-periodicsolutions in the above papers do not apply.However, for PDE’s, the dispersion relation would be − ω + A = − γk . Theunboudedness of the k factor makes it possible to have propagating waves nomatter how small | γ | is.Note also that in the model in [FSW86], there is no propagation either becauseof the random nature of the media.2. Basic setup and preliminaries
The main goal of this paper is to extend the method introduced in [FdlLS09a] forthe study of whiskered tori to some systems on infinite dimensional manifolds. Thesystems we will consider consist of infinitely many finite dimensional Hamiltoniansystems, each of them corresponding to a site on a lattice, subject to some coupling.We will assume that the coupling decays fast enough with respect to the distanceamong the sites. These are standard models in many applied fields and there isa large mathematical theory, which we cannot survey systematically (but we willmake some indication of the results we use or the one closer to our goals).An important tool for us will be appropriate function spaces for these interac-tions. There are many other methods to establish the existence of whiskered tori[Gra74, Zeh76a, You99]. The present method has the advantage that it dependsmuch less on the subtle geometric properties, so that it applies easily to infinitedimensional contexts.Since we are interested in translation invariant problems and want to producesolutions that do not go to zero at infinity, it is natural to model the functionalanalysis in ℓ ∞ , which has some subtle points that require attention.The goal of this section is to set up the functional analysis spaces modelled after ℓ ∞ and capturing the idea that changes in one site have very small effect in sites thatare far away. We anticipate that we need two types of spaces. One family of spacesfor the mappings from the infinite dimensional space to itself and another kind ofspaces for the mappings from a finite dimensional torus to the infinite dimensionalphase space. This corresponds to the F , K in (1) or the X , K in (2). The spaceswe choose are patterned after the choices in [JdlL00, FdlLM11a]. Other Banach UASI-PERIODIC BREATHERS 7 spaces of functions in lattice systems are in [Rug02, BK95]. Indeed, the choice oftopologies in these infinite dimensional systems is rather subtle and arguments inergodic theory which rely more on measure theory than on geometry find usefultoplogies in which the phase space is compact.2.1.
Phase spaces.
In this paper we will assume that the phase space at each nodeis given by an Euclidean exact symplectic manifold ( M = T l × R d − l , Ω = dα ), where T = R / Z . We will not assume that the symplectic form is given in the standardform of action-angle variables. For some calculations we consider ˜ M = R l × R d − l ,the universal covering of M or the complex extensions of the above. This is naturalsince the KAM method requires to consider Fourier series.It is possible to adapt our method to the case of a non-Euclidean manifold M using connectors and exponential mappings. This just requires some typographicaleffort. See the discussion in [FdlLS09a].Then, the phase space of the lattice system will be a subset of(6) M Z N = Y j ∈ Z N M. Since M is unbounded we will take as phase space(7) M = ℓ ∞ ( Z N , M ) = n x ∈ M Z N | sup i ∈ Z N | x i | < ∞ o which is a strict subset of M Z N . We will endow M with the distance d ( x, y ) = sup i ∈ Z N d ( x i , y i ) , where d ( x i , y i ) is the distance on the finite dimensional manifold M .When M is R d , M is a Banach space with the norm k x k ∞ = sup i ∈ Z N | x i | . When M = T l × R d − l , M is a Banach manifold modelled on ℓ ∞ ( Z N ).Notice that because M is an Euclidean space, the tangent space of M is trivialand can be identified with ℓ ∞ . Given x ∈ M and ξ ∈ ℓ ∞ , we can define x + ξ byjust adding the components. We see that if k ξ k ∞ < /
2, the mapping ξ → x + ξ isinjective, so that this defines a chart in M . Remark 2.1.
The fact that we assume that the manifold has the structure M = T l × R d − l is important here since it implies that H ( M ) ∼ H ( T l ) is non-trivial.This allows us to perform the construction in Appendix C. If the manifold wassuch that its first de-Rham cohomology group were trivial, all the symplectic mapswould be exact symplectic and the construction would not work without changes.To deal with manifolds such that H ( M ) is trivial ( M = R d for instance), onecan use the method developed by the authors in the finite dimensional case (see[FdlLS09a, FdlLS09b]). This consists in perturbing the invariance equation for thetori by a translation term and prove, at the end of the convergence scheme, thatthe geometry implies that this term is zero. The method of [FdlLS09a] allows todeal with secondary tori (i.e. tori which are contractible to tori of lower dimension)directly. The present method would require to make some preliminary changes ofvariables. E. FONTICH, R. DE LA LLAVE, AND Y. SIRE
The choice of ℓ ∞ is dictated by the fact that we want to deal with solutions thatneither grow nor decrease at ∞ . This, however, will lead to some complications,the functional analysis in ℓ ∞ being rather delicate. On the other hand, obtainingestimates in ℓ ∞ for several of our objects will be relatively easy.Since we are going to deal with analytic functions, one has to define what isthe complex extension of the manifold M . By assumption, the manifold M isan Euclidean manifold, hence, it admits a complex extension M C . We define thecomplex extension of M as a subspace of the product of the complex extensions of M , i.e. M C = n z ∈ Y j ∈ Z N M C (cid:12)(cid:12)(cid:12) sup i ∈ Z N | z i | < ∞ o . In the following, we will be considering mostly M C but to simplify the notation wewill not write the superscript C , if it does not lead to confusion.2.2. Some functional analysis in ℓ ∞ ( Z N ) and the spaces of decay functions. As emphasized in [FdlLM11a], ℓ ∞ ( Z N ) has a very complicated dual space whichcannot be identified with a space of sequences since there is no Riesz-representationtheorem. As a consequence, we have that the matrix elements of an operator do notcharacterize the operator and, relatedly, the differential of a map is not representedby its partial derivatives. The physical meaning is that one has to take into account“boundary conditions at infinity”.For example, consider the functional T defined on the closed subspace of ℓ ∞ ( Z )consisting of convergent sequences by the formula T ( u ) = lim n → + ∞ u n . By the Hahn-Banach theorem, T extends to ℓ ∞ ( Z ). The extended functional T isnon-trivial but we have ∂ u i T ( u ) = 0since the limit does not depend on u i . Of course, the functional T is linear but itis not represented by a matrix. Similar phenomena have been known in statisticalmechanics for a while under the name observables at infinity .This phenomenon can be eliminated by restricting our attention to functionswhose derivative is a linear functional which is given by the matrix of partial deriva-tives. We will develop some technology that allows to verify this assumption rathercomfortably in the cases of interest. A much more thorough treatment can be foundin [FdlLM11a].2.2.1. Weighted norms to formulate decay properties.
To formulate quantitativelythe approximate locality of the maps we will consider Banach spaces whose normmakes precise that changing one coordinate affects little the outcome of other co-ordinates far away.We will make use of the so-called decay functions introduced in [JdlL00].
Definition 2.2.
We say that a function
Γ : Z N → R + , is a decay function whenit satisfies (1) X j ∈ Z N Γ( j ) ≤ , (2) X j ∈ Z N Γ( i − j )Γ( j − k ) ≤ Γ( i − k ) , i, k ∈ Z N . UASI-PERIODIC BREATHERS 9
The algebraic property (2) in definition 2.2 is important since it is the one thatallows us to construct Banach algebras.The following elementary proposition is proved in detail in [JdlL00] and providesan example of a decay function.
Proposition 2.3.
Given α > N , θ ≥ , there exists a > , depending on α, θ , N such that the function defined by Γ( i ) = (cid:26) a | i | − α e − θ | i | if i = 0 , a if i = 0 is a decay function on Z N . We note, as it is easily verified in [JdlL00], that Γ( i ) = C exp( − β | i | ) is not adecay function for any β, C > Z N , such as the Bethe lattice which alsoadmit decay functions (see [JdlL00]), many of the results of the present paper canbe adapted with little change. Definition 2.4.
Given two decay functions Γ , Γ ′ we say that Γ dominates Γ ′ andwrite Γ ′ ≪ Γ when lim k →∞ Γ ′ ( k ) / Γ( k ) = 0 . We say that a family of decay functions Γ β , β ∈ [0 , , is an ordered family when ˜ β < β implies Γ ˜ β ≪ Γ β . Of course the examples in Proposition 2.3 constitute an ordered family. For someof the arguments later, in the proof of Theorem 3.11, when we are increasing thescales increasing the number of breathers, it will be useful to have a full scale sothat the longer scales have a weaker decay. This is the reason why Theorem 3.11is only stated for these functions.Of course, the examples in Proposition 2.3 enjoy several other nice properties, forexample that Γ( i ) is a decreasing function of | i | . We refer the reader to AppendixA where a deeper study of spaces of decay functions is performed. In the following,we just give the definitions needed to state our main results. Remark 2.5.
Prof. L. Sadun pointed out that there is a very natural physicalinterpretation of the definition of decay functions. We note that a site i can affectanother site j either directly or by affecting another site k which in turn affects thesite j . Of course, more complicated effects involving longer chains of intermediatesites are also possible. If the direct interaction between two sites is bounded bya decay function, it follows that the effect mediated through intermediate sites isbounded by the same function. This makes it possible to comfortably carry outperturbation calculations.2.2.2. Banach spaces of functions with good localization properties.
We now intro-duce the functional spaces needed for our purposes. We introduce: • The Banach space of decay linear operators(8) L Γ ( ℓ ∞ ( Z N )) = A ∈ L ( ℓ ∞ ( Z N )) | ∃ { A ij } i,j ∈ Z N , A i,j ∈ L ( M )( Au ) i = P j ∈ Z N A ij u j , i ∈ Z N , sup i,j ∈ Z N Γ( i − j ) − | A ij | < ∞ , where L ( ℓ ∞ ( Z N )) denotes the space of continuous linear maps from ℓ ∞ ( Z N )into itself. We endow L Γ ( ℓ ∞ ( Z N )) with the norm(9) k A k Γ = sup i,j ∈ Z N Γ( i − j ) − | A ij |• The space of C functions on an open set B ⊂ M C ( B ) = (cid:26) F : B → M | F ∈ C ( B ) , DF ( x ) ∈ L Γ ( ℓ ∞ ( Z N )) , ∀ x ∈ B sup x ∈B k F ( x ) k < ∞ , sup x ∈B k DF ( x ) k Γ < ∞ (cid:27) with the norm k F k C = max (cid:0) sup x ∈B k F ( x ) k , sup x ∈B k DF ( x ) k Γ (cid:1) . For r ∈ N , we define C r Γ ( B ) = (cid:26) F : B → M | F ∈ C r ( B ) , DD j − F ∈ C ( B ) , ≤ j ≤ r = 1 (cid:27) . Of course, we can give an equivalent recursive definition of the C r as theset of fucntions whose derivative is given by a matrix valued function whichis in C r − .We define a notion of analyticity for maps on lattices. Definition 2.6.
Let B be an open set of M . We say that F : B → M is analytic if it is in C ( B ) with the derivatives understood in the complexsense. • The space of analytic embeddings on a strip D ρ = (cid:8) z ∈ C l / Z l | | Im z i | < ρ, i = 1 , . . . , l (cid:9) . Let R ≥ c ∈ ( Z N ) R , i.e. c = ( c , . . . , c R ) . We introduce the following quantity k f k ρ,c, Γ = sup i ∈ Z N min j =1 ,...,R Γ − ( i − c j ) k f i k ρ , where k f i k ρ = sup θ ∈ D ρ | f i ( θ ) | . We denote(10) A ρ,c, Γ = (cid:26) f : D ρ → M | f ∈ C ( D ρ ) , f analytic in D ρ , k f k ρ,c, Γ < ∞ (cid:27) . This space, with the norm k · k ρ,c, Γ , is a Banach space. If we considera map A from D ρ into the set of linear maps L Γ ( ℓ ∞ ( Z N )), the associatednorm is k A k ρ, Γ = sup i,j ∈ Z N sup θ ∈ D ρ Γ − ( i − j ) | A ij ( θ ) | = sup θ ∈ D ρ k A ( θ ) k ΓUASI-PERIODIC BREATHERS 11
Symplectic geometry on lattices.
In this section, we introduce the littlegeometry we need on the manifold M to be able to perform the iteration. We referthe reader to Appendix B where a more systematic description and properties ofthe objects is performed. We basically need symplectic geometry for the KAM stepon the center manilfolds – which will be finite dimensional — and we will needthe exactness properties for the vanishing lemma 6.1 in Section 6. These uses canbe accomplished by just saying that the pullback of the symplectic form by decayembeddings from a finite dimensional torus make sense. It is also very useful thatthe proof presented does not require transformation theory and, hence, we do notneed to discuss a systematic theory of symplectic mappings.Consider our finite dimensional exact symplectic manifold ( M, Ω = dα ) and theassociated lattice M = ℓ ∞ ( Z N , M ) . Define α ∞ and Ω ∞ to be the formal sums (later, we will give them some precisemeaning) α ∞ = X j ∈ Z N π ∗ j α, Ω ∞ = X j ∈ Z N π ∗ j Ω , where π j are the standard projections from M to M at the node j ∈ Z N . Let J bethe symplectic matrix associated to the symplectic two-form Ω on M . We denote J ∞ the operator defined on T M by J ∞ ( z ) = diag (cid:0) . . . , J ( π i z ) , . . . (cid:1) , z ∈ M . We introduce the following definitions.
Definition 2.7.
We say that a C function F : M → M is symplectic if thefollowing identity holds for any z ∈ M DF ⊤ ( z ) J ∞ ( F ( z )) DF ( z ) = J ∞ ( z ) , where the product of two operators A and B in C is given component-wise by ( AB ) i,j = P k ∈ Z N A ik B kj , i, j ∈ Z N . Note that, due to the decay, properties, the products involved in the definitionof a symplectic matrix are absolutely convergent sums.Similarly, we have the following definition. Let ˆ A be the linear operator associ-ated to the Liouville form α on M . We denote ˆ A ∞ the operator defined on T M by ˆ A ∞ ( z ) = diag (cid:0) . . . , ˆ A ( π i z ) , . . . (cid:1) , z ∈ M . Definition 2.8.
We say that a C function F : M → M is exact symplectic on M if there exists a one-form ˜ α defined on T M with matrix ˜ A such that • For every j ∈ Z N , there exists a smooth function W j on M such that ˜ α j = dW j , where d is the exterior differentiation on M . • The following formula holds component-wise on the lattice DF ( z ) ⊤ ˆ A ∞ ( F ( z )) = ˆ A ∞ ( z ) + ˜ A ( z ) . The previous definitions are completely equivalent to the standard definitions ofsymplectic and exact symplectic maps in the finite dimensional case, but they areamong the mildest ones that we can imagine in infinite dimensions.We anticipate that the symplectic structure, will only enter in this paper in twoplaces: 1) The automatic reducibility in the center directions, 2) The vanishinglemma to show that for exact symplectic mappings several averages vanish. Theseapplications are very finite dimensional.The following lemma will be usefull for us (see Appendix 11).
Lemma 2.9.
Consider a function ψ defined on T l (or a subset of it) with valuesin M and belonging to A ρ,c, Γ for some ρ > . Then the bilinear form ψ ∗ Ω ∞ = X j ∈ Z N ψ ∗ Ω( π j ) is a two-form on the torus T l . Diophantine properties.
KAM relies on approximation properties of thefrequencies by rational numbers. In this section, we recall some well known notions.For diffeomorphisms, the relevant notion of Diophantine properties is given by thefollowing
Definition 2.10.
Given κ > and ν ≥ l , we define D ( κ, ν ) as the set of frequencyvectors ω ∈ R l satisfying the Diophantine condition: | ω · k − n | − ≤ κ | k | ν , for all k ∈ Z l − { } and n ∈ Z with | k | = | k | + · · · + | k l | , where k i are the coordinates of k . For vector fields, one uses the following
Definition 2.11.
Given κ > and ν ≥ l − , we define D h ( κ, ν ) as the set offrequency vectors ω ∈ R l satisfying the Diophantine condition: | ω · k | − ≤ κ | k | ν , for all k ∈ Z l − { } , where | k | = | k | + · · · + | k l | . Given f ∈ L ( T l ), we denoteavg ( f ) = Z T l f ( θ ) dθ. We also denote by T ω the rotation on T l by ω : T ω ( θ ) = θ + ω. In Section 9.2 we will discuss extensions of these definitions to infinite dimen-sional vectors which are well adapted to our applications.3.
Formulation of the results
We will first obtain a translated tori result, i.e. a KAM theorem for parameteri-zed families of maps F λ which are symplectic for all λ and such that F is exact symplectic. This will allow us to avoid the considerations of vanishing of averagesat each stage of the iteration. Then, we will prove a simple vanishing lemma(see Section 6 ) that shows that the added extra parameter vanishes. This yieldsto the desired invariant tori theorem. Going through translated curve theorems UASI-PERIODIC BREATHERS 13 has become quite standard in KAM theory (see [Mos67, R¨us76a]) especially since[Sev99] pointed out that it deals with very degenerate situations. In our case, it isparticularly advantageous since the parameters we need are finite dimensional andit avoids many infinite dimensional considerations.The problem is the following: given an exact symplectic map F and a vector offrequencies ω ∈ D ( κ, ν ) we wish to construct an invariant torus for F such that thedynamics of F restricted on it is conjugated to the translation T ω . To this end, wesearch for an embedding K : D ρ ⊃ T l → M in A ρ,c, Γ such that for all θ ∈ D ρ , K satisfies the functional equation (1).Notice that if (1) is satisfied, the image under F of a point in the range of K will also be in the range of K . If the range of DK ( θ ) is l -dimensional for all θ , then K ( T l ) is an l -dimensional invariant torus. (Similarly, the geometric interpretationof (2) is that the vector field X at a point in the range of K is tangent to the rangeof K .)The assumptions are that we are given a mapping K that satisfies (1) up to avery small error and that fullfills some non-degeneracy assumptions. We prove thatthe embedding K exists and also that the solution is unique up to composition onthe right with translations.Actually we are going to prove a more general result which works for parame-terized families of symplectic maps F λ , such that F is exact symplectic, but onlyprovides translated (and not invariant) tori. That is, given ω ∈ D ( κ, ν ) and anapproximate solution K of F λ ◦ K − K ◦ T ω = 0 satisfying a set of non-degeneracyconditions, we search for an embedding K : D ρ ⊃ T l → M in A ρ,c, Γ such that(11) F λ ◦ K = K ◦ T ω for some λ close to λ . The geometric interpretation of the invariance equations isillustrated in Figure 1.We go through a Newton scheme to prove the existence of such a pair ( λ, K ).To this end, we introduce the operator F ω F ω ( λ, K ) = F λ ◦ K − K ◦ T ω . In the paper [FdlLS09a], the authors constructed invariant tori using a posteriori
KAM theorems in finite dimensional systems. The general principles of this methodremain valid in some infinite dimensional systems such as lattices. We first introducesome notations and several non-degeneracy conditions.
Definition 3.1.
Consider ρ > , ω ∈ R l , c = ( c , . . . , c R ) ∈ ( Z N ) R , R ≥ , adecay function Γ , λ ∈ R l and F λ : M → M be a C map.We say that K : D ρ → M ∈ A ρ,c, Γ is a whiskered embedding for F λ when wehave:The tangent space T K ( θ ) M has an invariant analytic splitting for all θ ∈ D ρ (12) T K ( θ ) M = E sK ( θ ) ⊕ E cK ( θ ) ⊕ E uK ( θ ) , where E sK ( θ ) , E cK ( θ ) and E uK ( θ ) are the stable, center and unstable invariant spacesrespectively, which satisfy: • The projections Π sK ( θ ) , Π cK ( θ ) and Π uK ( θ ) associated to this splitting areanalytic with respect to θ considered as operators in L Γ ( ℓ ∞ ( Z N )) . • The splitting (12) is characterized by asymptotic growth conditions (co-cycles over T ω ): there exist < µ , µ < , µ > such that µ µ < , K F T d K( T d ) K K T d K( T d ) Figure 1.
Illustration of the invariance equations (1), (2). µ µ < and C h > such that for all n ≥ , θ ∈ D ρ and λ ∈ R l k DF λ ◦ K ◦ T n − ω × · · · × DF λ ◦ Kv k ρ,c, Γ ≤ C h µ n k v k ρ,c, Γ ⇐⇒ v ∈ E sK ( θ ) (13) and k DF − λ ◦ K ◦ T − ( n − ω × · · · × DF − λ ◦ Kv k ρ,c, Γ ≤ C h µ n k v k ρ,c, Γ ⇐⇒ v ∈ E uK ( θ ) . (14) • The center subspace E cK ( θ ) is finite dimensional, has dimension l and it ischaracterized by: k DF λ ◦ K ◦ T n − ω ( θ ) × · · · × DF λ ◦ K ( θ ) v k ρ,c, Γ ≤ C h µ n k v k ρ,c, Γ k DF − λ ◦ K ◦ T − ( n − ω ( θ ) × · · · × DF − λ ◦ K ( θ ) v k ρ,c, Γ ≤ C h µ n k v k ρ,c, Γ ⇐⇒ v ∈ E cK ( θ ) . (15)It is important for applications that the spectral condition in Definition 3.1 isimplied by a condition that can be verified by a finite calculation (see Definition 3.2below). Approximate invariance of the splitting is sufficient (see Proposition 4.2below) to ensure there is a truly invariant splitting. So, the final version of ourresults will have as a hypothesis the existence of approximately invariant tori withapproximately invariant splitting (Definition 3.2). The final version of the results UASI-PERIODIC BREATHERS 15 will have as a conclusion the existence of exactly invariant tori with exactly invariantsplittings (Definition 3.1).
Definition 3.2.
Consider ρ > , ω ∈ R l , c = ( c , . . . , c R ) ∈ ( Z N ) R , R ≥ , adecay function Γ , λ ∈ R l and F λ : M → M be a C map.We say that ˜ K : D ρ → M ∈ A ρ,c, Γ satisfies the η -hyperbolic condition (or hasan η − invariant splitting), if there exists an analytic splitting of T ˜ K ( T l ) M , (16) T ˜ K ( θ ) M = E s ˜ K ( θ ) ⊕ E c ˜ K ( θ ) ⊕ E u ˜ K ( θ ) such that, denoting Π s,c,u ˜ K ( θ ) be the corresponding projections, we have (1) The splitting is approximately invariant under the co-cycle DF ◦ ˜ K over T ω in the sense that dist (cid:16) DF λ ( ˜ K ( θ )) E s,c,u ˜ K ( θ ) , E s,c,u ˜ K ( θ + ω ) (cid:17) < η. (2) There exists N ∈ N , < ˜ µ , ˜ µ < and ˜ µ > such that ˜ µ , ˜ µ < , ˜ µ ˜ µ < and k DF λ ◦ ˜ K ◦ T N − ω ( θ ) × · · · × DF λ ◦ ˜ K ( θ ) v k ρ,c, Γ ≤ ˜ µ N k v k ρ,c, Γ , ∀ v ∈ E s ˜ K ( θ ) , (17) k DF − λ ◦ ˜ K ◦ T − ( N − ω ( θ ) × · · · × DF − λ ◦ ˜ K ( θ ) v k ρ,c, Γ ≤ ˜ µ N k v k ρ,c, Γ , ∀ v ∈ E u ˜ K ( θ ) (18) and k DF λ ◦ ˜ K ◦ T N − ω ( θ ) × · · · × DF λ ◦ ˜ K ( θ ) v k ρ,c, Γ ≤ ˜ µ N k v k ρ,c, Γ k DF − λ ◦ ˜ K ◦ T − ( N − ω ( θ ) × · · · × DF − λ ◦ ˜ K ( θ ) v k ρ,c, Γ ≤ ˜ µ N k v k ρ,c, Γ ∀ v ∈ E c ˜ K ( θ ) . (19) Remark 3.3.
Note that in Definition 3.2 we are using that the phase space isEuclidean. On a general manifold, the products used in (17), (18), (19) cannotbe defined because, in general, DF ( x ) : T x M → T F ( x ) M . Hence, in a generalmanifold, if F ◦ K ( θ ) = K ( θ + ω ), we cannot define DF ◦ K ( θ + ω ) DF ◦ K ( θ ). In[FdlLS09a] one can find a definition of approximately invariant cocycles for generalmanifolds. In this paper, we will not consider such generality.We will define Ω cK ( θ ) = Ω | E cK ( θ ) ∀ θ ∈ T ℓ and we introduce the symplectic linear map J c ( K ( θ )) : E cK ( θ ) → E cK ( θ ) byΩ cK ( θ ) ( u, v ) = h u, J cK ( θ ) v i ∀ u, v ∈ E cK ( θ ) . Obviously, we have J c ( K ( θ )) ⊤ = − J c ( K ( θ )). We also have (See Lemma 4.8) thatΩ c is non-degenerate, hence J c is invertible. Definition 3.4.
Given ρ > , ω ∈ R l , c = ( c , . . . , c R ) ∈ ( Z N ) R , R ≥ , a decayfunction Γ , λ ∈ R l and an embedding K : D ρ → M ∈ A ρ,c, Γ , a pair ( λ, K ) issaid to be non-degenerate (and we denote ( λ, K ) ∈ N D loc ( ρ, Γ) ) if it satisfies thefollowing conditions • Non degeneracy of the embedding:
We have that the l × l matrix DK ⊤ ( θ ) DK ( θ ) is invertible for all θ in D ρ . We denote N ( θ ) = (cid:0) DK ⊤ ( θ ) DK ( θ ) (cid:1) − andwe assume that k N k ρ, Γ < ∞• Twist condition: let P ( θ ) = DK ( θ ) N ( θ ) .The average on T l of the l × l − matrix (20) A λ ( θ ) = P ( θ + ω ) ⊤ (cid:16) [ DF λ ( K )( J c ◦ K ) − P ]( θ ) − [( J c ◦ K ) − P ]( θ + ω ) (cid:17) is non-singular. • Parameter cohomological non-degeneracy:
The average on T l of the l × l − matrix (21) Q λ ( θ ) = (cid:16) ( DK ⊤ ( ω + θ ) J c ( K ( ω + θ )) ∂F λ ( K ( θ )) ∂λ (cid:17) is non-singular. It is clear that the meaning of k N k ρ, Γ is a measure of the quality of the embed-ding. It grows if the embedding comes close to having a singularity. During theproof it will become clear that the meaning of A λ is the change of the rotation whenwe move in the direction transversal to the torus. As we will see in calculations,the meaning of the invertibility of the average of Q λ is that, by changing λ , we canadjust the obstructions to the cohomology equations.For applications, it is important to note that the non-degeneracy hypothesisonly depend on the approximate solution considered and that they are readily com-putable algebraic expressions. They are quite analogous to the condition numbersin numerical analysis.First we state our main theorem, which provides the existence of a solution( λ, K ) to the functional equation (11). This is the translated tori KAM theorem. Theorem 3.5.
Let F λ : M → M be a family of symplectic maps parameterizedby λ ∈ R l , ω ∈ D ( κ, ν ) for some κ > , ν ≥ l , ρ > , Γ a decay function and c = ( c , . . . , c R ) ∈ ( Z N ) R . Assume we have λ ∈ R l and K : D ρ ⊃ T l → M satisfying the following hypotheses • For all λ ∈ R l , the maps F λ belong to C and satisfy sup i ∈ Z N Γ − ( i )( F λ (0)) i < ∞ . • The map F λ is real analytic and it can be extended holomorphically to somecomplex neighborhood of the image under K of D ρ : B r = { z ∈ M| ∃ θ s.t. | Im θ | < ρ , | z − K ( θ ) | < r } , for some r > and such that k DF λ k C ( B r ) is finite. • ( λ , K ) ∈ N D loc ( ρ , Γ) i.e , the embedding K is non-degenerate in thesense of Definition 3.4. • The embedding K is η -hyperbolic in the sense of Definition 3.2 with η sufficiently small (depending on k Π s,c,u k ρ ,c, Γ , µ , , , N , k F k C ( B r ) ).Define the error E by E = F λ ◦ K − K ◦ T ω . Denote also ˜ ε = max( k E k ρ ,c, Γ , η ) . UASI-PERIODIC BREATHERS 17
There exists a constant
C > depending on l , κ , ν , ρ , k DF λ k C ( B r ) , k DK k ρ ,c, Γ , k N k ρ , k ∂F λ ( K ) ∂λ k ρ ,c, Γ , k A λ k ρ , | avg ( A λ ) | − , | avg ( Q λ ) | − (where A λ , Q λ and N are as in Definition 3.4, replacing K with K ) and on k Π c,s,uK ( θ ) k ρ , Γ suchthat, if for some δ , < δ < min(1 , ρ / , we have the following conditions satisfied Cκ δ − ν ˜ ε < and Cκ δ − ν ˜ ε < r then, there exist an embedding K ∞ ∈ N D loc ( ρ ∞ = ρ − δ, Γ) and a vector λ ∞ ∈ R l such that (22) F λ ∞ ◦ K ∞ = K ∞ ◦ T ω . Furthermore, we have the following estimates k K ∞ − K k ρ ∞ ,c, Γ ≤ Cκ δ − ν ˜ ε, | λ − λ ∞ | < Cκ δ − ν ˜ ε. (23) Additionnally, we have that the invariant embedding K ∞ admits invariant split-tings, satisfying Definition (3.4) .Denoting the non-degeneracy constants corresponding to K ∞ by index ∞ , wehave: k Π s,c,u ∞ ◦ K ∞ − Π s,c,u ◦ K k ρ ∞ , Γ ≤ Cκ δ − ν ˜ ε, | µ ∞ , , − µ s,c,u | ≤ Cκ δ − ν ˜ ε (24) and |k N k ρ − k N ∞ k ρ ∞ | ≤ Cκ δ − ν ˜ ε, |k A λ k ρ − k A ∞ λ ∞ k ρ ∞ | ≤ Cκ δ − ν ˜ ε, |k Q λ k ρ , Γ − k Q ∞ λ ∞ k ρ ∞ , Γ | ≤ Cκ δ − ν ˜ ε. (25)The previous theorem will allow us to construct increasingly complicated solu-tions, the solutions of one stage being an approximate solution for the next stage.We will however be able to maintain enough control of the non-degeneracy condi-tions.Of course, (25) is an easy consequence of (23) since the objects that enter in thedegeneracy estimates are algebraic expressions of K .We now come to the result on the existence of invariant tori. They correspondto localized quasi-periodic orbits on the manifold M . These orbits are known as“breathers”. Theorem 3.6.
Let F λ : M → M be a family of symplectic maps parameterizedby λ ∈ R l , ω ∈ D ( κ, ν ) for some κ > , ν ≥ l , ρ > , Γ a decay function and c = ( c , . . . , c R ) ∈ ( Z N ) R . Assume we have λ ∈ R l and K : D ρ ⊃ T l → M satisfying the following hypotheses • The map F λ is exact symplectic and F λ (0) = 0 . • For all λ ∈ R l , the maps F λ belong to C and satisfy sup i ∈ Z N Γ − ( i )( F λ (0)) i < ∞ . • The map F λ is real analytic and it can be extended holomorphically to somecomplex neighborhood of the image under K of D ρ : B r = { z ∈ M| ∃ θ s.t. | Im θ | < ρ , | z − K ( θ ) | < r } , for some r > and such that k DF λ k C ( B r ) is finite. • (0 , K ) ∈ N D loc ( ρ , Γ) i.e , the embedding K is non-degenerate in thesense of Definition3.4. • The embedding K is η -hyperbolic in the sense of Definition 3.2 with η sufficiently small (depending on k Π s,c,u k ρ , Γ , µ , , , N , k F k C ( B r ) ).Define the error E by E = F λ ◦ K − K ◦ T ω . Denote also ˜ ε = max( k E k ρ ,c, Γ , η ) . There exists a constant
C > depending on l , κ , ν , ρ , k DF λ k C ( B r ) , k DK k ρ ,c, Γ , k N k ρ , k ∂F λ ( K ) ∂λ k ρ ,c, Γ , k A k ρ , | avg ( A ) | − , | avg ( Q ) | − (where A , Q and N are as in Definition 3.4, replacing K with K ) and on k Π c,s,uK ( θ ) k ρ ,c, Γ such that, iffor some δ , < δ < min(1 , ρ / , we have the following conditions satisfied Cκ δ − ν ˜ ε < and Cκ δ − ν ˜ ε < r Then, we have in (22) λ ∞ = 0 , i.e. the torus K ∞ is actually an invariant torus for F λ and we have F λ ◦ K ∞ = K ∞ ◦ T ω . Remark 3.7.
It is important to mention that we do not assume that the symplecticforms are the standard ones. This allows to consider the existence of externalmagnetic fields and magnetic interactions among the sites since the effect of amagnetic field is just a change of the symplectic form [Thi97]. Alternatively, if ( p, q )are the conjugated coordinates, one can change p → p − A where A is the vectorpotential. Note that the introduction of a magnetic field destroys the reversibilityunder the usual involution S ( p, q ) = ( − p, q ). Remark 3.8.
The whiskered tori that satisfy the spectral hypothesis have invariantmanifolds that make them important in problems of stability. However, the proofof the stable manifold is not completely straightforward since the space ℓ ∞ does nothave smooth cut-off functions. It is possible to show that these invariant manifoldshave also some decay properties. This has been established in [FdlLM11b].We have also the following result which provides local uniqueness. Theorem 3.9.
Let ω ∈ D ( κ, ν ) for some κ > , ν > l and K ∈ N D ( ρ ) and K ∈ N D ( ρ ) be two solutions of equation (1) such that K ( D ρ ) ⊂ B r , K ( D ρ ) ⊂ B r .There exists a constant C > depending on l , κ , ν , ρ , ρ − , k F k C , k K k ρ,c, Γ , k N k ρ, Γ , k A k ρ, Γ , | avg ( A ) | − such that if k K ◦ T τ − K k ρ,c, Γ satisfies for some τ ∈ R l Cκ δ − ν k K ◦ T τ − K k ρ ,c, Γ ≤ , where δ = ρ/ , then there exists a phase ˜ τ ∈ R l such that K ◦ T ˜ τ = K in D ρ .Moreover, | τ − ˜ τ | ≤ Cκ ρ − ν k K ◦ T τ − K k ρ ,c Γ . UASI-PERIODIC BREATHERS 19
Remark 3.10.
It is important to remark that ALL constants in the previoustheorems are independent of c . This fact is crucial for the next result, whichprovides an existence theorem for almost-periodic functions.The idea of the construction follows the one in the finite dimensional case (see[FdlLS09a]). Notice here that the decay properties are on the hyperbolic subspace,the center one being finite dimensional. Some small differences between the schemeof the present paper and [FdlLS09a] are detailed in Remark 4.20.We also have the analogous result to Theorem 3.6 for vector-fields, Theorem 8.4.We will postpone the statement of Theorem 8.4 till Section 8 where we also presenta proof.As an application of Theorem 8.4, we will present a result on existence of solu-tions with infinitely many frequencies (also called almost periodic solutions). Wenote that, as indicated before, we will establish the theorem in two stages. In a firststage, we will continue the breathers from the uncoupled system to the whole sys-tem. In the second stage, we will couple infinitely many of these breathers so thatwe obtain solutions with infinitely many frequencies. We note that the smallnessconditions and the elimination of a positive measure set of frequencies only occursin the first stage. In the second stage, we only need to eliminate a zero measureset of frequencies (in many different measures) and we do not need any smallnesscondition. The reason is that in the second stage, we adjust all the smallness con-ditions by placing the individual breathers far enough. Of course, if we wanted tolet the breathers not to be so far appart, it could be done with other assumptions.The models we consider (26) have been considered in the Physics and Math-ematics literature. They are models of many microscopic processes. See [BK04,BK98, DRAW02, FBGGn05, CF05, Gal08, BEMW07] and references there amongmany others.A large variety of solutions for equations of this type have been constructed:space-localized periodic in time solutions, known as breathers (see [MA94, Jam01]),solitary waves ([Ioo00, IK00, FW94], [FP99, FP02, FP04a, FP04b]), pulsating trav-eling waves (see [JS05, Sir05]). The relevance of these solutions in biological phe-nomena has also been discussed (see [DPW92, PS04, Pey04]).There are already several other papers that have produced solutions with infin-itely many frequencies. The paper [FSW86] produced such solutions by introducingsome random terms and making the excitation of each oscillator goes to zero, sothat its effect on the others was small. Fr¨olich, Spencer and Wayne also assumethat the coupling is high order in terms of the amplitude. The paper [CP95, Per03]considered oscillators but made the natural frequencies increase very fast so thatthere were no resonances in each of them. The paper [P¨os90] proved a very ab-stract theorem that applies to perturbations of integrable systems and managedto recover several results as applications of this theorem. The paper [GY07] alsoconsiders coupled systems in one dimension, but produces tori with finitely manyfrequencies.The solutions we construct are based on a different principle. We use that thesolutions which are far apart interact very weakly even if they are large. Therefore,by placing solutions far apart, we will be able to make them interact weakly andwe can satisfy the smallness conditions assumed by the general theorem. Noticethat we are assuming that most of the sites are close to a hyperbolic orbit. Hence,the system will be very hyperbolic. This will allow us to deal with most of the normal directions using the methods of hyperbolic splittings and we will not needto consider the resonances that appear in the normally elliptic modes, which requiremore delicate estimates. We emphasize that we do not assume that the system isclose to integrable. Theorem 3.11.
Consider a lattice M = M Z N with the symplectic form given by Ω ∞ = P n ∈ Z N dq n ∧ dp n . Consider the following Hamiltonian with respect to Ω ∞ given by (26) H ( q, p ) = X n ∈ Z N (cid:16) | p n | + W ( q n ) (cid:17) + ε X j ∈ Z N X n ∈ Z N V j ( q n − q n + j ) . Let Γ , Γ ′ be decay functions as in Proposition 2.3.Denote by X the vector field associated to the Hamiltonian (26) .Assume: H1 The system ¨ q + W ′ ( q ) = 0 admits a hyperbolic fixed point, which we will setwithout loss of generality at q = 0 . H2 There exists a set Ξ ⊂ R l of positive Lebesgue measure such that for all ω ∈ Ξ , there exists a KAM torus invariant under the flow of ¨ q + W ′ ( q ) = 0 andnon-degenerate in the sense of the standard KAM theory (twist condition). H3 The potentials V j and W are real analytic. Moreover, we assume that thereexists a constant C V such that k V k k C ,ρ ≤ C V Γ( k ) . and also that ∇ V k (0) = 0 for every k .Fix ρ ′ < ρ . Then, (1) A)For all ε ∗ sufficiently small, we can find a set Ξ ( ε ∗ ) ⊂ Ξ , such that if ω ∈ Ξ ( ε ∗ ) and | ε | < ε ∗ , then the system (26) has a localized breather offrequency ω . There exists K : T l → M , K ∈ A ρ ′ ,c, Γ such that X ◦ K = ∂ ω K The embedding satisfies Definition 3.1 and we can choose the hyperbolicityand non-degeneracy constants uniformly.Furthermore we have meas(Ξ \ Ξ ( ε ∗ )) → as ε ∗ → . (2) B) Consider now Ξ ∞ = Ξ ( ε ∗ ) N endowed with the probability measure (cid:16) meas( · )meas(Ξ ( ε ∗ ) (cid:17) N . Then, there exists a set Ξ ∗∞ ⊂ Ξ ∞ , meas(Ξ ∗∞ ) = 1 , suchthat if ω ∈ Ξ ∗∞ , there exist a sequence of centers c and a K analytic insome strip of ( T l ) N so that J ∞ ∇ H ◦ K = ∂ ω K. Note that we have stated Theorem 3.11 only for decay functions of the formgiven in Proposition 2.3. It is clear that the proof only uses a few properties of thefunction (e.g. monotonicity in the modulus of the argument). We have refrainedfrom reformulating the theorem in more abstract terms.We note that the only smallness conditions in ε enter just in the first stageof creating individual breathers around each site and in the preservation of the UASI-PERIODIC BREATHERS 21 hyperbolic structure and other non-degeneracy conditions. The second stage, onthe other hand does not require any other smallness conditions.In the construction of infinite dimensional breathers out of single breathers wejust need to exclude a few sequences of frequencies which are very resonant (theyhave measure zero in the probability measure indicated above). See Section 9.3.The smallness assumptions that we need to couple the sequences can be adjustedjust by placing the different breathers far apart and we do not need any furthersmallness conditions in ε .Note also that the only hypothesis on the one site system is the existence ofpositive measure of KAM tori (and the existence of a hyperbolic fixed point). Thisis implied if the system is close to a non-degenerate integrable system. Nevertheless,there are other arguments to show existence of KAM tori in systems very far fromintegrable [Dua94, Dua08]. Any of these systems could be taken as the basis forTheorem 3.11. 4. The Newton step
The sketch of the proof of Theorem 3.5 is roughly the same as for finite di-mensional systems, with some minor changes detailed in Remark 4.20. Of course,even if the strategy is similar to that in finite dimensions, all the details need tobe different since the situation is very different and we need to pay attention tothe decay properties. With a view to applications to almost periodic solutions ofTheorem 3.11, we also need to pay attention to the change in the non-degeneracyconditions and in the hyperbolicity properties and establish that many of the small-ness assumptions are independent of the number and the geometry of the centersof oscillation.The proof of Theorem 3.5 is based on a Newton iteration of Nash-Moser type.The estimates of the Newton step – including uniqueness – are summarized inSection 4.1 (See, Lemma 4.1). The fact that the inductive step can be iterated ismore or less standard in KAM theory and it is done in Section 5.The proof of the estimates of the Newton step are obtained in different stages(1) We show that the approximate invariant hyperbolic splitting can be trans-formed in an invariant splitting. See Section 4.2.(2) The equations for the Newton step can be divided into equations along thehyperbolic spaces (studied in Section 4.4) and the center space (studied inSection 4.3).As usual in the study of cohomology equations, the equations in the cen-ter direction are much more subtle. In particular, the Diophantine prop-erties and the geometric properties are only used in the equations in thecenter space.(3) Once we have the estimates for the approximate solutions of the linearizedequation, we show that, using the linearized equation, they improve thesolutions of the translated equation. Furthermore, we estimate the changesin the hyperbolicity constants and the non-degeneracy estimates.(4) The passage from Theorem 3.5 to Theorem 3.6 is a geometric argument (avanishing lemma) undertaken in Section 6.4.1.
Estimates for the inductive step.
In this section we describe the inductivestep of the proof of Theorem 3.5.
By Taylor’s theorem we can write F ω ( λ + Λ , K + ∆) = F ω ( λ, K ) + D λ,K F ω ( λ, K )(Λ , ∆) + O ( | (Λ , ∆) | ) . Assuming that ( λ, K ) is a pair that satisfies F ω ( λ, K ) = 0 approximately with anerror E ( θ ) = F ω ( λ, K )( θ ) we look for (Λ , ∆) such that F ω ( λ + Λ , K + ∆) is assmall as possible. Then we are lead to consider the following Newton equation(27) D λ,K F ω ( λ, K )(Λ , ∆) = − E, where D λ,K F ω ( λ, K )(Λ , ∆)( θ ) = ∂F λ ( K ( θ )) ∂λ Λ + DF λ ( K ( θ ))∆( θ ) − ∆( θ + ω ) . To solve (27) we project the equation on both the center and the hyperbolic sub-spaces, taking advantage of the invariant splitting. Then we try to solve the pro-jected equations. The one on the center subspace is reduced to two small divisorsequations, essentially one on the tangent of the torus and the other on its con-jugated directions. Taking advantage of the extra variable λ , we can solve theseequations up to a quadratic error. Using the conditions on the co-cycles over T ω ,we solve the projection on the stable and unstable subspaces.The next result gives an approximate solution of (27) with precise estimates. Lemma 4.1.
Under the hypotheses of Theorem 3.5 the equation D λ,K F ω ( λ, K )(Λ , ∆) = − E has an approximate solution (Λ , ∆) in the following sense: let ˜ E = D λ,κ F ω ( λ, K ) (Λ , ∆) + E .
For < δ < ρ we have the following estimates k ∆ k ρ − δ,c, Γ ≤ Cκ δ − ν k E k ρ,c, Γ , | Λ | ≤ C k E k ρ,c, Γ , k ˜ E k ρ − δ,c, Γ ≤ Cκ δ − (2 ν +1) k E k ρ,c, Γ kF ω ( λ, K ) k ρ,c, Γ . Moreover, if ∆ and ˜∆ are solutions of (27) as above, i.e. solutions with quadraticerror bounded by Cκ δ − (2 ν +1) k E k ρ,c, Γ kF ω k ρ,c, Γ , k ∆ − ˜∆ − DK ( θ ) α k ρ − δ,c, Γ ≤ Cκ δ − (2 ν +1) k E k ρ,c, Γ kF ω ( λ, K ) k ρ,c, Γ . In the previous estimates, the constant C depends on ρ, l, k DK k ρ,c, Γ , k Π s,c,uK ( θ ) k ρ, Γ , k ∂F λ ∂λ k ρ,c, Γ , the hyperbolicity constants and the decay function Γ but it does notdepend on c . Construction of invariant splittings out of approximately invariantones.
The main result of this section will be Proposition 4.2, which establishesthat given an approximately invariant splitting satisfying Definition 3.1, there is atruly invariant splitting nearby. Furthermore, we can estimate the distance betweenthe true invariant splitting and the approximately invariant one. This, of course,implies the usual formulation of persistence of splittings under small perturbations.The way that this result fits into the Newton scheme is that this will allow usto split the equation into different components. Compared to other estimates inthe Newton step, the construction of invariant splittings requires much less sophis-ticated analysis (it suffices to use contractions) and it does not require inductiveassumptions nor making choices (e.g. the domain loss). The subtlety of the results
UASI-PERIODIC BREATHERS 23 comes because we have to choose appropriate spaces so that the estimates are uni-form in the domains, the arrangement of the centers, etc. This uniformity of theresults will be used when we consider the limit of infinitely many frequencies.The method of proof we use is very similar to the standard proof using graphtransforms [HP70, HPS77], which adapts very well to infinite dimensions [PS99].Of course, there are several subtleties due to the infinite dimensional nature of theproblem. In particular, we make essential use of the Banach algebra propertiesof the decay functions to make sure that we obtain estimates in the same spacesof functions (it is interesting to compare this with previous results in lattice dy-namical systems). We emphasize that, in particular, the smallness conditions areindependent of the centers of the embedding. This will be crucial when we considerthe limit of a large number of centers.
Proposition 4.2.
Assume that the embedding ˜ K has a δ -invariant hyperbolicsplitting ˜ E s , ˜ E c , ˜ E u with respect to a map F (See Definition 3.2). Denote by ˜Π σ , σ = s, c, u the projections corresponding to this splitting.There exists δ > depending on k N k ρ, Γ , k DF ◦ ˜ K k ρ,c, Γ , k DF − ◦ ˜ K k ρ,c, Γ and k Π s,c,u k ρ,c, Γ , µ s,c,u such that if < δ < δ there is an analytic splitting (28) T ˜ K ( θ ) M = E s ˜ K ( θ ) ⊕ E c ˜ K ( θ ) ⊕ E u ˜ K ( θ ) which is invariant under the co-cycle DF ◦ ˜ K over T ω .Let ˜Π s,c,u ˜ K ( θ ) be the projections corresponding to the splitting (28) . We furthermorehave that there exist C h , < µ , µ < , µ > such that µ µ < , µ µ < andthe characterizations (13) , (14) , (15) of the splitting hold.Moreover, there exists C > , depending on the same quantities as δ does, suchthat for < δ < δ k Π s,c,u ˜ K ( θ ) − ˜Π s,c,u ˜ K ( θ ) k ρ, Γ ≤ Cδ, | µ , , − ˜ µ , , | < Cδ. Proof.
The ideas in this proof follow the ones in [FdlLS09a]. They have been takenfrom [HPPS70]. We make sure that the estimates are uniform with respect to δ and c . We divide the proof into several steps. Step 1: Construction of the invariant spaces . The existence of the invari-ant splitting will be done through the Banach fixed point principle applied to agraph transform operator.We begin with the case of the stable bundle E s ˜ K ( θ ) . We describe the stable space E s ˜ K ( θ ) as the graph of a linear map, i.e. E s ˜ K ( θ ) = graph ( u ◦ ˜ K ), where u ◦ ˜ K maps˜ E s ˜ K ( θ ) linearly into ˜ E c ˜ K ( θ ) ⊕ ˜ E u ˜ K ( θ ) .Since the splitting (16) is approximately invariant we can write the matrix DF (cid:0) ˜ K ( θ ) (cid:1) with respect to this decomposition as DF (cid:0) ˜ K ( θ ) (cid:1) = (cid:18) a ( θ ) a ( θ ) a ( θ ) a ( θ ) (cid:19) with k a k ρ,c, Γ < Cδ, k a k ρ,c, Γ < Cδ. We also write DF (cid:0) ˜ K ( θ + ( N − ω ) (cid:1) × · · · × DF (cid:0) ˜ K ( θ ) (cid:1) = (cid:18) a N ( θ ) a N ( θ ) a N ( θ ) a N ( θ ) (cid:19) . Note that by (17), (18) and (19) we have(29) k a N k ρ,c, Γ ≤ (1 + Cδ ) µ N , k a − N k ρ,c, Γ ≤ (1 + Cδ ) µ N , and(30) k a N k ρ,c, Γ ≤ Cδ , k a N k ρ,c, Γ ≤ Cδ .
The graph condition over the co-cycle is DF ◦ ˜ K ( θ ) (cid:18) Id u ◦ ˜ K ( θ ) (cid:19) ∈ graph (cid:16) u ◦ ˜ K (cid:0) T ω ( θ ) (cid:1)(cid:17) . This gives the functional equation for the map u (31) u ◦ ˜ K (cid:0) T ω ( θ ) (cid:1) ( a + a u ◦ ˜ K )( θ ) = ( a + a u ◦ ˜ K ) ( θ ) . Denoting ˜ u = u ◦ ˜ K , (31) can be rewritten as(32) ˜ u = a − (cid:2) ˜ u ◦ T ω ( a + a ˜ u ) − a (cid:3) . Let L η be the ball of radius η in the space of linear operators from ˜ E s ˜ K ( θ ) into˜ E c ˜ K ( θ ) ⊕ ˜ E u ˜ K ( θ ) with the norm k · k Γ .Let S η be the space of analytic sections from D ρ to L η , i.e. the space of u : D ρ → L η such that u ( θ ) : ˜ E s ˜ K ( θ ) → ˜ E c ˜ K ( θ ) ⊕ ˜ E u ˜ K ( θ ) with the norm k . k ρ, Γ .We take the operator T : S η → S η defined as the right-hand side of (32). T isapproximated by T : S η → S η defined by T ˜ u = a − ˜ u ◦ T ω a . We now consider T N and T N . An elementary computation gives T N ˜ u = a − . . . a − ◦ T N − ω ˜ u ◦ T Nω a ◦ T N − ω . . . a . Moreover, taking into account that T is a degree two polynomial operator, weobtain by simple algebraic manipulations(33) kT N − T N k < Cδ and(34) Lip( T N − T N ) < Cδ . Using the Banach algebra properties of the decay norms, we have that(35) k a N ( θ ) − a (cid:0) T N − ω ( θ ) (cid:1) . . . a ( θ ) k ρ,c, Γ < Cδ , (36) k a − N ( θ ) − a − ( θ ) . . . a − (cid:0) T N − ω ( θ ) (cid:1) k ρ,c, Γ < Cδ . By (29), (35) and (36), if δ is small, T N sends S η into S η for all η ∈ (0 ,
1] and is acontraction in this domain. By (33) and (34), if δ is small T N sends S η into S η for η ∈ ( Cδ,
1] and it is also a contraction.Therefore T N has a unique fixed point u ∗ in S which belongs to S Cδ . It is clearthat T u ∗ is also a fixed point of T N , which belongs to S C ′ δ ⊂ S for some C ′ ≥ C .By uniqueness T u ∗ = u ∗ .A similar method can be applied for the center-unstable subspace. In this casethe graph condition reads DF λ ◦ ˜ K ( θ ) (cid:18) v ◦ ˜ K ( θ )Id (cid:19) ∈ graph (cid:16) v ◦ ˜ K (cid:0) T ω ( θ ) (cid:1)(cid:17) , UASI-PERIODIC BREATHERS 25 and the resulting operator T : S η → S η is(37) T ˜ v = (cid:2) (˜ a ˜ v + ˜ a )(˜ a ˜ v + ˜ a ) − (cid:3) ◦ T − ω , where ˜ v = v ◦ ˜ K . Repeating the same procedure as above, we construct the center-unstable space and obtain similar bounds.Now let us consider G = F − . We observe that the stable space associated to themap G is the unstable space associated to F . Hence, applying the above procedureto G , we construct the unstable space E u ˜ K ( θ ) and center-stable space E c,s ˜ K ( θ ) withsimilar bounds. Finally we note that E c ˜ K ( θ ) = E c,s ˜ K ( θ ) T E c,u ˜ K ( θ ) . Step 2: Estimates on the projections . We want to estimate the norm ofthe projection Π s ˜ K ( θ ) compared to the one of ˜Π s ˜ K ( θ ) .Let ξ ∈ T ˜ K ( θ ) M . Using the decomposition ξ = ( ξ s , ξ cu ) ∈ E s ˜ K ( θ ) ⊕ ( E c ˜ K ( θ ) ⊕E u ˜ K ( θ ) )we have the following representations˜Π s ˜ K ( θ ) ξ = ( ξ s , , Π s ˜ K ( θ ) ξ = ( ˜ ξ s , ˜ u ( θ ) ˜ ξ s ) , ˜Π cu ˜ K ( θ ) ξ = (0 , ξ cu ) , Π cu ˜ K ( θ ) ξ = (˜ v ( θ ) ˜ ξ cu , ˜ ξ cu ) . Then ξ s = ˜ ξ s + ˜ v ( θ ) ˜ ξ cu ,ξ cu = ˜ u ( θ ) ˜ ξ s + ˜ ξ cu or equivalently (cid:18) ˜ ξ s ˜ ξ cu (cid:19) = (cid:18) Id ˜ v ( θ )˜ u ( θ ) Id (cid:19) − (cid:18) ξ s ξ cu (cid:19) since the matrix B = (cid:18) Id ˜ v ( θ )˜ u ( θ ) Id (cid:19) is invertible because is O ( δ )-close to theidentity and moreover, by the Neumann series theorem, we can write B − = (cid:18) Id + w ( θ ) w ( θ ) w ( θ ) Id + w ( θ ) (cid:19) with k w ij k ρ, Γ < Cδ . Therefore using (cid:16) ˜Π s ˜ K ( θ ) − Π s ˜ K ( θ ) (cid:17) (cid:18) ξ s ξ cu (cid:19) = (cid:18) ˜ ξ s − ξ s ˜ u ( θ ) ˜ ξ s (cid:19) = (cid:18) ˜ v ( θ ) ˜ ξ c,u ˜ u ( θ ) ˜ ξ s (cid:19) this gives (cid:13)(cid:13) ˜Π s ˜ K ( θ ) − Π s ˜ K ( θ ) (cid:13)(cid:13) ρ,c, Γ ≤ Cδ .
Analogously one has (cid:13)(cid:13) ˜Π cu ˜ K ( θ ) − Π cu ˜ K ( θ ) (cid:13)(cid:13) ρ,c, Γ < Cδ . The estimates for the projections Π u , Π sc are obtained in a similar way. Fromthose we deduce readily the ones for Π c by noting that Π c = Π cu − Π u . Step 3: Existence of µ , µ , µ C h for the new splitting and estimates. Since the distance between the spaces E s,c,u ˜ K ( θ ) and ˜ E s,c,u ˜ K ( θ ) is bounded by Cδ , therestriction of DF ◦ ˜ K ◦ T N − ω × · · · × DF ◦ ˜ K to them is separated by a distanceless than Cδ . Therefore there exist µ , µ , µ with µ , µ < , µ > , µ µ < , µ µ < | µ , , − ˜ µ , , | < Cδ and (17) holds for ˜ µ = µ and v ∈ ˜ E s ˜ K ( θ ) . Similarly for (18) and (19).Once we have these last properties we deduce that there exists C h such that(13), (14) and (15) hold for all n ≥ (cid:3) As a consequence of Proposition 4.2 we have the following result, which showsthat the hyperbolicity constants do not deteriorate much if we change very little theembeddings K . This will be used in the iterative process. We will use it to showthat, during the iterative process, the hyperbolicity constants remain uniformlybounded. We will also deduce that when the embeddings converge, the splittingsconverge. Proposition 4.3.
Under the hypotheses of Proposition 5.1, assume that k K − ˜ K k ρ,c, Γ is small enough. Then there exists an analytic splitting T ˜ K ( θ ) M = E s ˜ K ( θ ) ⊕ E c ˜ K ( θ ) ⊕ E u ˜ K ( θ ) invariant under the co-cycle DF λ ◦ ˜ K over T ω .Furthermore, there exists C > such that (cid:13)(cid:13) Π s,c,uK ( θ ) − Π s,c,u ˜ K ( θ ) (cid:13)(cid:13) ρ, Γ ≤ C k K − ˜ K k ρ,c, Γ , | µ i − ˜ µ i | ≤ C k K − ˜ K k ρ,c, Γ , i = 1 , , | C h − ˜ C h | ≤ Cδ Proof.
The invariant splitting T K ( θ ) M = E sK ( θ ) ⊕ E cK ( θ ) ⊕ E uK ( θ ) for DF λ ◦ K is an approximate invariant splitting for DF λ ◦ ˜ K . Here we identify T ˜ K ( θ ) M with T K ( θ ) M since M is Euclidean. We then can take δ = C k K − ˜ K k ρ,c, Γ . (cid:3) Solution of the linearized equation on the center subspace.
In thissection we solve approximately the projection of equation (27) on the invariantcenter subspace provided by Proposition 4.3 and establish estimates. Projectingwith Π cK ( θ + ω ) and using the notation∆ c ( θ ) = Π cK ( θ ) ∆( θ ) , E c ( θ ) = Π cK ( θ + ω ) E ( θ ) , we obtain(38) Π cK ( θ + ω ) ∂F λ ( K ( θ )) ∂λ Λ + DF λ ( K ( θ ))∆ c ( θ ) − ∆ c ( θ + ω ) = − E c ( θ ) . Estimates on cohomology equations.
We recall the well-known small divisorslemma (see [R¨us76a], [R¨us76b], [R¨us75], [dlL01]).
Proposition 4.4.
Let M be a finite dimensional Euclidean manifold and ω ∈ D ( κ, ν ) . Assume the mapping h : D ρ → M is analytic on D ρ and has zero average.Then for any < σ < ρ the difference equation v ( θ + ω ) − v ( θ ) = h ( θ ) has a unique zero average solution v : T l → M , real analytic on D ρ − σ for any < σ < ρ . Moreover, we have the estimate (39) k v k ρ − σ ≤ Cκσ − ν k h k ρ , and where C only depends on ν and the dimension of the torus l . We have the following corollary of R¨ussmann’s result in our context.
UASI-PERIODIC BREATHERS 27
Corollary 4.5.
Let M = ℓ ∞ ( Z N ) and ω ∈ D ( κ, ν ) and assume the mapping h : T l → M belongs to A ρ,c, Γ and has zero average. Then for any < σ < ρ thedifference equation v ( θ + ω ) − v ( θ ) = h ( θ ) has a unique zero average solution v : T l → M , belonging to A ρ − σ,c, Γ for any < σ < ρ . Moreover, we have the estimate (40) k v k ρ − σ,c, Γ ≤ Cκσ − ν k h k ρ,c, Γ , where C only depends on ν and linearly on the dimension of the torus l . Remark 4.6.
An important fact of the previous statement is that, since we considerthe supremum norm on M and the equation is solved component by component,the estimates are independent on the dimension of M . Proof.
We write the equation in coordinates i ∈ Z N to get v i ( θ + ω ) − v i ( θ ) = h i ( θ )with v i and h i mapping T l into M . We then apply the previous finite dimensionalresult Proposition 4.4 to each of the components.We also observe that the partial derivatives of the functions also satisfy ∂ θ k v i ( θ + ω ) − ∂ θ k v i ( θ ) = ∂ θ k h i ( θ )and that ∂ θ k h i has zero average. Then, ∂ θ k v i ( θ ) will be the zero average solutionobtained applying Proposition 4.4. Multiplying the estimates afforded by Proposi-tion 4.4 by Γ − ( i − c j ) and taking the supremum in i and the infimum in j we getthe desired result. (cid:3) Isotropic character of the torus.
One important issue is the approximateisotropic character of the approximate torus K ( T l ). In our context the two-form Ω ∞ is formal but K ( T l ) is finite dimensional, therefore asking the forms to be isotropicamounts to ask that the pull-back K ∗ Ω ∞ vanishes. By the decay properties of K (see Lemma 2.9), this last object is a true form on T l and we can write K ∗ Ω ∞ ( θ ) ( ξ, η ) = h ξ, L ( θ ) η i , ξ, η ∈ R l . The isotropic character of the torus is then equivalent to L ( θ ) ≡ DK ( θ ) ⊤ J ∞ (cid:0) K ( θ ) (cid:1) DK ( θ ) = 0for all θ ∈ T l . Notice that L is a l × l − matrix.We first consider the case when K is a solution of (11). Lemma 4.7.
Let ( M , Ω ∞ = dα ∞ ) be the lattice manifold. Assume that F λ ◦ K = K ◦ T ω , F λ is symplectic, ω is rationally independent and K ∈ A ρ,c, Γ . Then L ( θ ) is identically zero.Proof. Since K ∈ A ρ,c, Γ , using that F λ is symplectic (see Appendix B). we have K ∗ Ω ∞ = K ∗ F ∗ λ Ω ∞ = ( K ◦ T ω ) ∗ Ω ∞ . By the condition on ω, T ω is ergodic and therefore K ∗ Ω ∞ is constant. Hence L isalso constant. Moreover, the fact that M is formally exact symplectic shows that K ∗ Ω ∞ = d ( K ∗ α ∞ ), where now, d is the differential on the torus (see Appendix B). In coordinates this means that L ( θ ) has the form DL ( θ ) ⊤ − DL ( θ ) for somefinite dimensional vector L ( θ ). Since the average of derivatives is zero we get that L is zero on T l . (cid:3) Geometric considerations on the center bundle E cK ( θ ) in the exact case. Inthis section, we show how to construct geometrically a very natural basis of thecenter subspace E cK ( θ ) when K satisfies (11). Recall first that we are assuming that E cK ( θ ) is finite dimensional with dimension 2 l . We start with the following lemma. Lemma 4.8.
The restriction Ω cK ( θ ) of Ω ∞ to the finite-dimensional space E cK ( θ ) isa symplectic form on E cK ( θ ) .Proof. To prove the claim, it is enough to show that the form Ω cK ( θ ) is non degen-erate. Assume that u, v ∈ T K ( θ ) M . Then we haveΩ ∞ ( u, v ) = Ω ∞ ( DF n ( K ( θ ) u, DF n ( K ( θ ) v ) , n ∈ Z . Since the torus is invariant, we have that DF ◦ K ◦ T n − ω ( θ ) × · · · × DF ◦ K ( θ ) = DF n ◦ K ( θ ) . We deduce, sending n → ±∞ and using the hyperbolic conditions (expan-sion/contraction properties), that Ω ∞ ( u, v ) = 0 in the following cases • u, v ∈ E sK ( θ ) , • u, v ∈ E uK ( θ ) , • u ∈ E sK ( θ ) ∪ E uK ( θ ) and v ∈ E cK ( θ ) , • u ∈ E cK ( θ ) and v ∈ E sK ( θ ) ∪ E uK ( θ ) . Assume that: let ˜ c ∈ E cK ( θ ) Ω cK ( θ ) ( c, ˜ c ) = 0 , ∀ c ∈ E cK ( θ ) . By the previous argument, we have that for every u ∈ E uK ( θ ) and v ∈ E sK ( θ ) Ω cK ( θ ) (˜ c, u ) = Ω ∞ ( c, ˜ c ) = Ω ∞ ( c + u + v, ˜ c ) = 0Since Ω ∞ is non-degenerate, this leads to the desired result. (cid:3) Now define ˆ L = DK ⊥ J c ( K ) DK.
For every ν , ν ∈ T θ T l , ν T DK ( θ ) J c ( K ( θ )) DK ( θ ) ν = Ω cK ( θ ) ( DK ( θ ) ν , DK ( θ ) ν ) = Ω ∞ ( DK ( θ ) ν , DK ( θ ) ν ) . Hence ν T DK ( θ ) J c ( K ( θ )) DK ( θ ) ν = ν ⊥ DK ( θ ) ⊥ J ∞ ( K ( θ )) DK ( θ ) ν = ν ⊥ Lν = 0 , hence ˆ L = 0Since range DK ( θ ) is the tangent space of the torus K ( T l ) and the dynamicson the torus is conjugated to a rotation, DK ( θ ) R l is contained in E cK ( θ ) . More-over we have that J c ( K ( θ )) − DK ( θ ) R l also is contained in E cK ( θ ) . Instead of J c ( K ( θ )) − DK ( θ ) we will consider the matrix J c ( K ( θ )) − DK ( θ ) N ( θ ) where N ( θ )is the normalization l × l -matrix N ( θ ) = [ DK ( θ ) ⊤ DK ( θ )] − previously introduced.Both have the same range because N ( θ ) is non-singular. The role of N is to providesome normalization for the symplectic conjugate. UASI-PERIODIC BREATHERS 29
Now we check that the range of [Π cK ( θ ) DK ( θ ) , J c ( K ( θ )) − DK ( θ ) N ( θ )] is 2 l -dimensional. Indeed, let { e j } be the canonical basis of R l and assume that there isa linear combination that vanishes on E cK ( θ ) f = l X j =1 α j Π cK ( θ ) DK ( θ ) e j + l X j =1 β j J c ( K ( θ )) − DK ( θ ) N ( θ ) e j = 0 . Then, for 1 ≤ k ≤ l , using the isotropic character of T K ( θ ) K ( T l )0 = l X j =1 β j e ⊤ k DK ( θ ) ⊤ J c ( K ( θ )) J c ( K ( θ )) − DK ( θ ) N ( θ ) e j = l X j =1 β j h e k , e j i = β k . This calculation shows that f reduces to P lj =1 α j DK ( θ ) e j . Moreover, for 1 ≤ k ≤ l l X j =1 α j e ⊤ k N ( θ ) ⊤ DK ( θ ) ⊤ J c ( K ( θ )) −⊤ Π cK ( θ ) J c ( K ( θ )) DK ( θ ) e j = − l X j =1 α j h e k , e j i = − α k . Hence α j = β j = 0 for all j = 1 , ..., l . We conclude that range [Π cK ( θ ) DK ( θ ) , J c ( K ( θ )) − ( K ( θ )) − DK ( θ ) N ( θ )] = E cK ( θ ) . The treatment of equation (27) on the center subspace is greatly facilitated byobserving that the preservation of the symplectic structure imposes that the systemis close to a diagonal structure.In our context, the dimension of the center subspace is 2 l and E cK ( θ ) ∼ R l .In [FdlLS09a], the authors studied the case when M is finite dimensional (withsymplectic structure J ). They used the set of vectors (cid:26) ∂K ( θ ) ∂θ j , J − ( K ( θ )) ∂K ( θ ) ∂θ j (cid:27) j =1 ,...,l to perform a transformation which allows to approximately solve up to quadraticerror the projected equation on the center subspace.We consider the map ˜ M ( θ ) given in matrix notation by(41) ˜ M ( θ ) = h Π cK ( θ ) DK ( θ ) , J c ( K ( θ )) − DK ( θ ) N ( θ ) i . We will see that this map is a very convenient change of coordinates in the linearizedequations which makes them easily solvable.
Remark 4.9.
It is worth noting that all the quantities defined above (such as N ,for instance) make perfect sense even if we are manipulating “infinite dimensional”matrices. This is due to the fact that we are considering maps in L Γ ( ℓ ∞ ( Z N )). Forinstance, since K is assumed to be in A ρ,c, Γ , one has for i, j = 1 , . . . , lN ( θ ) − ij = X k ∈ Z N ∂K k ∂θ i ∂K k ∂θ j . Therefore, for θ ∈ D ρ this leads to | N ( θ ) − ij | ≤ k DK k ρ,c, Γ min ≤ p,q ≤ R X k ∈ Z N Γ( k − c p )Γ( k − c q ) ≤ k DK k ρ,c, Γ min ≤ p,q ≤ R Γ( c p − c q ) ≤ k DK k ρ,c, Γ . This gives that k N k ρ, Γ < ∞ . Remark 4.10.
It is also worth noticing that we have the following estimates forthe generalized symplectic matrix J ∞ : k J ∞ k Γ = Γ − (0) k J k , k J − ∞ k Γ = Γ − (0) k J − k since ( J ∞ ) ij = Jδ ij , where δ ij is the Kr¨onecker symbol.4.3.4. Representation of DF λ on the center subspace. In this section we study asuitable representation of DF λ applied to the basis of the center subspace given bythe columns of ˜ M ( θ ). We begin by considering the case when K is a solution of(11). Lemma 4.11.
Let K be a solution of equation (11) . Then there exists a l × l matrix S λ ( θ ) such that (42) DF λ ( K ( θ )) ˜ M ( θ ) = ˜ M ( θ + ω ) S λ ( θ ) , with (43) S λ ( θ ) = (cid:18) Id l A λ ( θ )0 l Id l (cid:19) . The matrix A λ ( θ ) is A λ ( θ ) = P ( θ + ω ) ⊤ [( DF λ J c ( K ( θ )) − P ( θ ) − [ J c ( K ) − P ]( θ + ω )] , where P ( θ ) = DK ( θ ) N ( θ ) .Proof. Differentiating equation (11) with respect to θ , we get DF λ ( K ( θ )) DK ( θ ) = DK ( θ + ω ) . This shows that S λ ( θ ) has the form (cid:18) Id l A λ ( θ )0 l B λ ( θ ) (cid:19) , where A λ ( θ ) , B λ ( θ ) are l × l matrices. We will now show that B λ = Id l via geometricproperties. We should have(44)[ DF λ ( K ) J c ( K ) − DK N ]( θ ) = DK ( θ + ω ) A λ ( θ ) + [ J c ( K ) − DK N ]( θ + ω ) B λ ( θ ) . By the isotropic character of K ( T l ) we have DK ⊤ J c ( K ) DK = 0 and the definitionof N , we have(45) [ DK ⊤ J c ( K )]( θ + ω )[ DF ( K ) J c ( K ) − DK N ]( θ ) = B λ ( θ ) . Also by the symplecticness of F λ J c ( K ( θ + ω )) DF λ ( K ( θ )) = J c ( F λ ( K ( θ ))) DF λ ( K ( θ )) =[ DF λ ( K ) −⊤ J c ( K )]( θ ) , UASI-PERIODIC BREATHERS 31
Figure 2.
Illustration of the geometric reason for reducibility ofthe equations in the center. Note that the base of a rectangle getsmapped into the base of the other (differentiating the invariance).The preservation of the symplectic form implies that the symplecticarea of both rectangles is the same.when restricted to E cK ( θ ) . Then equation (45) becomes B λ ( θ ) = DK ⊤ ( θ + ω )[ DF λ ( K ) −⊤ DK N ]( θ ) = [ DK ⊤ DK N ]( θ ) = Id l . To obtain the expression of A λ ( θ ) we multiply (44) by ( DK N )( θ + ω ) ⊤ to get(46) A λ ( θ ) = P ( θ + ω ) ⊤ h [ DF λ ( K ) J c ( K ) − P ]( θ ) − [ J c ( K ) − P ]( θ + ω ) i . (cid:3) The matrix ˜ M ( θ ) is not invertible since it is not square. However we can derivea generalized inverse for ˜ M ( θ ). As a motivation for subsequent developments, wefirst present Lemma 4.12 which deals with the geometric cancellations in the caseof an exactly invariant torus. The case of interest for a KAM algorithm — whenthe torus is only approximately invariant — will be studied in Lemma 4.14 as aperturbation of Lemma 4.12.A straightforward calculation shows that(47) ˜ M ⊤ J c ( K ) ˜ M = (cid:18) ˆ L Id l − Id l ( N ⊤ DK ⊤ J c ( K ) −⊤ DK N (cid:19) . Lemma 4.12.
Let K be a solution of (11) . Then the matrix ˜ M ⊤ J c ( K ) ˜ M isinvertible and ( ˜ M ⊤ J c ( K ) ˜ M ) − = (cid:18) ( − N ⊤ DK ⊤ J c ( K ) −⊤ DK N − Id l Id l (cid:19) . Proof.
It follows immediately from (47) and the isotropic character of the invarianttorus, i.e. ˆ L = 0. (cid:3) Now we consider the case we are interested in, that is when K is an approximatesolution with an error E ( θ ) = F ω ( λ, K )( θ ), assumed to be small. We will need theinvertibility of ˜ M ( θ ) ⊤ J c ( K ( θ )) ˜ M ( θ ) in this case.More precisely, we introduce(48) e ( θ ) = DF λ ( K ( θ )) ˜ M ( θ ) − ˜ M ( θ + ω ) S λ ( θ ) , where S λ is given by (43). If we denote e ( θ ) = ( e ( θ ) , e ( θ )), a simple algebraiccomputation yields e ( θ ) = DE c ( θ ) e ( θ ) = [( DF λ J c ( K ) − DK N ]( θ ) − DK ( θ + ω ) A λ ( θ ) − [ J c ( K ) − DK N ]( θ + ω ) = O ( E, DE )by the choice of A λ .We first prove the approximate isotropic character of the torus Lemma 4.13.
Under the previous conditions, let K ∈ A ρ,c, Γ be a function whichsolves (11) approximately and let E = F λ ◦ K − K ◦ T ω be the corresponding error.Then for < δ < ρ/ , (49) k L k ρ − δ, Γ ≤ Cκ δ − ( ν +1) k E k ρ,c, Γ , where C depends on l, ν, R, ρ, k DK k ρ,c, Γ , k F λ k C ( B r ) , k J c ( K ) k C ( B r ) .Proof. We define the two-form on the torus T l Ω e = K ∗ Ω ∞ − ( K ◦ T ω ) ∗ Ω ∞ . The corresponding matrix is L − L ◦ T ω . Using that F λ is symplectic we have thatfor any ( ξ, η ) ∈ R l Ω e ( θ )( ξ, η ) = (cid:0) ( F λ ◦ K ) ∗ Ω ∞ − ( K ◦ T ω ) ∗ Ω ∞ (cid:1) ( θ )( ξ, η )= X i ∈ Z N h Ω (cid:16) ( F λ ) i (cid:0) K ( θ ) (cid:1)(cid:17)(cid:0) D (( F λ ) i ◦ K )( θ ) ξ, D (( F λ ) i ◦ K )( θ ) η (cid:1) − Ω (cid:0) K ◦ T ω ( θ ) (cid:1)(cid:0) D ( K i ◦ T ω )( θ ) ξ, D ( K i ◦ T ω )( θ ) η (cid:1)i . Since D (( F λ ) i ◦ K ) − D ( K i ◦ T ω ) = DE i and k DE k ρ − δ,c, Γ ≤ δ k E k ρ,c, Γ , using thedecay properties of F ◦ K and K ◦ T ω to sum the series, we obtain that(50) L − L ◦ T ω = g with k g k ρ − δ ≤ Cδ − k E k ρ,c, Γ for some C as in the statement. Now we use Propo-sition 4.4 to finish the proof. (cid:3) The next step is to ensure the invertibility of the 2 l × l -matrix ˜ M ⊤ J c ( K ) ˜ M .According to expression (47), we can write˜ M ( θ ) ⊤ J c ( K ) ˜ M ( θ ) = V ( θ ) + R ( θ ) , where V = (cid:18) l − Id l N ⊤ DK ⊤ J c ( K ) −⊤ DK N (cid:19) and R = (cid:18) ˆ L
00 0 (cid:19) . UASI-PERIODIC BREATHERS 33
We have the following lemma, providing the desired invertibility result under asmallness assumption on E , namely (51) in the next lemma. Lemma 4.14.
There exists a constant
C > such that if (51) Cκδ − ( ν +1) k E k ρ,c, Γ ≤ / for some < δ < ρ/ then the matrix ˜ M ⊤ ( θ ) J c ( K ) ˜ M ( θ ) is invertible for θ ∈ D ρ − δ and there exists a matrix ˜ V ( θ ) such that ( ˜ M ( θ ) ⊤ J c ( K ( θ )) ˜ M ( θ )) − = V ( θ ) − + ˜ V ( θ ) with ˜ V ( θ ) = (cid:16) ∞ X k =1 ( − k ( V ( θ ) − R ( θ )) k (cid:17) V ( θ ) − , where the series is absolutely convergent. Furthermore, we have the estimate (52) k ˜ V k ρ − δ, Γ ≤ C ′ κδ − ( ν +1) k E k ρ,c, Γ , where the constant C ′ > depends on l , ν , k F / ambda k C ( B r ) , k J c ( K ) k C ( B r ) , k DK k ρ,c, Γ , k N k ρ, Γ .Proof. The matrix V ( θ ) is invertible with V − = (cid:18) N ⊤ DK ⊤ J c ( K ) −⊤ DK N − Id l Id l (cid:19) . We can write ˜ M ( θ ) ⊤ J c ( K ( θ )) ˜ M ( θ ) = V ( θ )(Id l + V ( θ ) − R ( θ )) . To apply the Neumann series (and consequently justify the existence of the inverseof Id l + V − R as well as the estimates for its size), we have to estimate the term V − R . According to Lemma 4.13, we have the estimate for L k L k ρ − δ, Γ ≤ Cκδ − ( ν +1) k E k ρ,c, Γ for all δ ∈ (0 , ρ/ k V − R k ρ − δ, Γ ≤ Cκδ − ( ν +1) k E k ρ,c, Γ for 0 < δ < ρ/
2, where
C > l , ν , k DK k ρ,c, Γ , k N k ρ, Γ and k J c ( K ) c k ρ, Γ .Because of assumption (51), we have that the right-hand side of the last equationis less than 1 / l + V ( θ ) − R ( θ ) is invertible with k (Id l + V − R ) − k ρ − δ, Γ ≤ − k V − R k ρ − δ, Γ ≤ . Now the estimates follow immediately. (cid:3)
Identification of the center subspace.
In this section, we identify the centerspace as being very close (up to terms that can be bounded by the error) to therange of the matrix ˜ M . This will allow us to use the range of ˜ M in place of E cK ( θ ) without changing the quadratic character of the method. Proposition 4.15.
Denote by Γ K ( θ ) the range of ˜ M ( θ ) and by Π Γ K ( θ ) the projectiononto Γ K ( θ ) according to the splitting E sK ( θ ) ⊕ Γ K ( θ ) ⊕ E uK ( θ ) .Then there exists a constant C > such that if δ − k E k ρ,c, Γ ≤ C we have the estimate (53) k Π cK ( θ ) − Π Γ K ( θ ) k ρ − δ,c, Γ ≤ Cδ − k E k ρ,c, Γ for every δ ∈ (0 , ρ/ and where C , as usual, depends on the non-degeneracy con-stants of the problem.Proof. From (48) and Cauchy estimates (see Lemma A.8 in Appendix A), we have:dist ρ − δ,c, Γ (( DF λ ◦ K )Γ K ( θ ) , Γ K ( θ ) ◦ T ω ) ≤ Cδ − k E k ρ,c, Γ , where dist stands for the distance between two spaces at the Grassmannian level.Using again equation (48) and iterating it, we obtain for n ≥ DF λ ( K ( θ + nω )) × · · · × DF λ ( K ( θ )) ˜ M ( θ ) =˜ M ( θ + nω ) S λ ( θ + ( n − ω ) × · · · × S λ ( θ ) + R n , where k R n k ρ − δ,c, Γ ≤ C n δ − k E k ρ,c, Γ and C n depends on n .Since S λ ( θ ) is upper triangular with Id l on the diagonal, we have: S λ ( θ + ( n − ω ) × · · · × S λ ( θ ) = (cid:18) Id l A λ ( θ + ( n − ω ) + · · · + A λ ( θ )0 Id l (cid:19) . Therefore, by induction, we have for every n ∈ N k DF λ ( K ( θ + nω )) · · · DF λ ( K ( θ )) ˜ M ( θ ) k ρ − δ,c, Γ ≤ Cn + C n δ − k E k ρ,c, Γ . Identical calculations give that k DF − λ ( K ( θ − nω )) · · · DF − λ ( K ( θ )) ˜ M ( θ + ω ) k ρ − δ,c, Γ ≤ Cn + C n δ − k E k ρ,c, Γ . Note that, given any µ > n µ ≥ n ≥ n µ , we have Cn < µ n . Consequently, choosing such n µ thereexists a constant C such that if the error satisfies δ − k E k ρ,c, Γ ≤ C, we have Cn + C n δ − k E k ρ,c, Γ < µ n . In other words, the above estimates hold forall sufficiently large n , provided that we impose a suitable smallness condition on δ − k E k ρ,c, Γ .As a consequence, Γ K ( θ ) is an approximately invariant bundle, and we also havebounds on the rate of growth of the co-cycle both in positive and negative times.Using Proposition 4.2, this shows that indeed one can find a true invariant subspace UASI-PERIODIC BREATHERS 35 ˜ E K ( θ ) close to Γ K ( θ ) . Since this invariant subspace should be of the same dimensionof the center space E cK ( θ ) , we deduce that˜ E K ( θ ) = E cK ( θ ) . (cid:3) Estimates on the center subspace.
We recall the projection into the centersubspace of the linearized equation(54) Π cK ( θ + ω ) ∂F λ ∂λ ( K ( θ ))Λ + DF λ ( K ( θ ))∆ c ( θ ) − ∆ c ( θ + ω ) = − E c ( θ ) . To shorten the notation till the end of the section we will write ∂F λ ∂λ ( K ( θ ))Λ instead of Π cK ( θ + ω ) ∂F λ ∂λ ( K ( θ ))Λ.We introduce the new function W ( θ ) through(55) ∆ c ( θ ) = ˜ M ( θ ) W ( θ ) + ˆ e ( θ ) W ( θ ) , where(56) ˆ e = Π cK ( θ + ω ) − Π Γ K ( θ + ω ) which was estimated in Proposition 4.15.Substituting (55) into equation (54) we get DF ( K ( θ )) ˜ M ( θ ) W ( θ ) − ˜ M ( θ + ω ) W ( θ + ω )(57) = − E c ( θ ) − ∂F λ ∂λ ( K ( θ ))Λ + ˆ e ( θ + ω ) W ( θ + ω ) − DF ( K ( θ ))ˆ e ( θ ) W ( θ )We anticipate that the term ˆ eW will be quadratic in the error. Similarly, writing ∂F λ ∂λ ( K ( θ )) = Π Γ K ( θ + ω ) ∂F λ ∂λ ( K ( θ )) + ˆ e ∂F λ ∂λ ( K ( θ )) . we also anticipate that the term ˆ e ∂F λ ∂λ ( K ( θ ))Λ will be quadratic in the error. As aconsequence, we will ignore these two terms and the equation for W is DF λ ( K ( θ )) ˜ M ( θ ) W ( θ ) − ˜ M ( θ + ω ) W ( θ + ω ) = − E c ( θ ) − ∂F λ ∂λ ( K ( θ ))Λ . (58)Using e ( θ ) above and multiplying equation (58) by ˜ M ( θ + ω ) ⊤ J c ( K ( θ + ω )),using Lemma 4.14 (giving the invertibility of ˜ M ⊤ J c ( K ) ˜ M ) and equation (48), weend up with (cid:20)(cid:18) Id l A λ ( θ )0 l Id l (cid:19) + B ( θ ) (cid:21) W ( θ ) − W ( θ + ω ) = p ( θ ) + p ( θ )(59) − [ ˜ M ⊤ J c ( K ) ˜ M ]( θ + ω ) − [ ˜ M ⊤ J c ( K )]( θ + ω ) ∂F λ ∂λ ( K ( θ ))Λ , where(60) B ( θ ) = [ ˜ M ⊤ J c ( K ) ˜ M ]( θ + ω ) − [ ˜ M ⊤ J c ( K )]( θ + ω ) e ( θ ) , (61) p ( θ ) = − V ( θ + ω ) − [ ˜ M ⊤ J c ( K )]( θ + ω ) E c ( θ )and(62) p ( θ ) = − ˜ V ( θ + ω )[ ˜ M ⊤ J c ( K )]( θ + ω ) E c ( θ ) . The next result provides the estimates of the previously introduced quantities.
Lemma 4.16.
Assume ω ∈ D ( κ, ν ) and δ and k E k ρ,c, Γ satisfy (51) . Then usingthe linear change of variables (55) , equation (58) becomes [ S λ ( θ ) + B ( θ )] W ( θ ) − W ( θ + ω ) = p ( θ ) + p ( θ ) − [ ˜ M ⊤ J c ( K ) ˜ M ]( θ + ω ) − [ ˜ M ⊤ J c ( K )]( θ + ω ) ∂F λ ∂λ ( K ( θ ))Λ , where B , p and p are given by equations (60) , (61) and (62) respectively.Moreover the following estimates hold: for p we have (63) k p k ρ,c, Γ ≤ C k E k ρ,c, Γ , where C only depends on k J c ( K ) k ρ, Γ , k N k ρ , k DK k ρ,c, Γ . For p and B , we have (64) k p k ρ − δ,c, Γ ≤ Cκδ − ( ν +1) k E k ρ,c, Γ and (65) k B k ρ − δ, Γ ≤ Cδ − k E k ρ,c, Γ , where C depends l , ν , ρ , R , k N k ρ , k DK k ρ,c, Γ , k F λ k C ( B r ) , k J c ( K ) k C ( B r ) .Proof. We have basically to estimate ˜ M ( θ + ω ) ⊤ J c ( K ( θ + ω )) E c ( θ ) and ˜ M ( θ + ω ) ⊤ J c ( K ( θ + ω )) e ( θ ) and then use Lemma 4.14. First we bound | E ci ( θ ) | ≤ X j ∈ Z N | (Π cK ( θ + ω ) ) ij | | E j ( θ ) |≤ X j ∈ Z N k Π cK ( θ + ω ) k ρ, Γ Γ( i − j ) k E k ρ,c, Γ max k Γ( j − c k ) ≤ k Π cK ( θ + ω ) k ρ, Γ k E k ρ,c, Γ R X k =1 Γ( i − c k ) . For 1 ≤ i ≤ l we have, taking into account that J c is uncoupled,(66) | (cid:0) ˜ M ( θ + ω ) ⊤ J c ( K ) E c ( θ ) (cid:1) i | ≤ C X j ∈ Z N | ∂ θ i K j ( θ + ω ) | | E cj ( θ ) | . We estimate from above by X j ∈ Z N | D θ i K j ( θ + ω ) | | E cj ( θ ) |≤ X j k DK k ρ,c, Γ max m Γ( j − c m ) k Π cK ( θ + ω ) k ρ, Γ k E k ρ,c, Γ R X k =1 Γ( j − c k ) ≤ R k DK k ρ,c, Γ k Π cK ( θ + ω ) k ρ, Γ k E k ρ,c, Γ . (67)For l + 1 ≤ i ≤ l , one gets | (cid:0) ˜ M ( θ + ω ) ⊤ J c ( K ( θ )) E c ( θ ) (cid:1) i | ≤ (68) C | (cid:16) N ( θ + ω ) ⊤ DK ( θ + ω ) ⊤ ˜ J c (cid:0) K ( θ + ω ) ⊤ (cid:1) E c ( θ ) (cid:17) i | . We get a similar bound for (68) taking into account that N is a bounded finitedimensional matrix. Now the bounds (63) and (64) follow immediately from Lemma4.14. For the estimate on B we use Cauchy estimates for e ( θ ). From B ( θ ) = ( V ( θ + ω ) − + ˜ V ( θ + ω )) ˜ M ( θ + ω ) ⊤ e ( θ ) UASI-PERIODIC BREATHERS 37 we have k B k ρ − δ ≤ k V ( θ + ω ) − k ρ − δ k ˜ M ( θ + ω ) ⊤ e ( θ ) k ρ − δ + | ˜ V ( θ + ω ) ˜ M ( θ + ω ) ⊤ e ( θ ) k ρ − δ and using estimate (52) we end up with k B k ρ − δ, Γ ≤ Cδ − k E k ρ,c, Γ + κδ − ( ν +1) k E k ρ,c, Γ δ − k E k ρ,c, Γ . This gives the desired result. (cid:3)
Approximate solvability of the linearized equation on the center subspace.
In this section we find a solution of equation (59) up to quadratic error. Theconvergence of the Newton scheme is of course not affected (see [Zeh75]).For that we introduce the following operator L W ( θ ) = (cid:18) Id l A λ ( θ )0 l Id l (cid:19) W ( θ ) − W ( θ + ω ) . Then equation (59) can be written as L W ( θ ) + B ( θ ) W ( θ ) = p ( θ ) + p ( θ )(69) − [ ˜ M ⊤ J c ( K ) ˜ M ]( θ + ω ) − [ ˜ M ⊤ J c ( K )]( θ + ω ) ∂F λ ∂λ ( K ( θ ))Λ . We will reduce equation (69) to two small divisors equations. Generically theirright-hand sides will not have zero average, but we will use the freedom in choosingΛ and in fixing the average of the solution to solve one after the other. By Lemma4.14 we can write[( ˜ M ⊤ J c ( K ) ˜ M ) − ˜ M ⊤ J c ( K )]( θ + ω ) ∂F λ ( K ( θ )) ∂λ Λ = H ( θ )Λ + q ( θ )Λ , where the 2 l × l matrix H is H ( θ ) = V ( θ + ω ) − ˜ M ( θ + ω ) ⊤ J c ( K ( θ + ω )) ∂F λ ( K ( θ )) ∂λ and q satisfies for all δ ∈ (0 , ρ/ k q k ρ − δ, Γ ≤ Cκδ − ( ν +1) (cid:13)(cid:13)(cid:13) ∂F λ ( K ( θ )) ∂λ (cid:13)(cid:13)(cid:13) ρ,c, Γ k E k ρ,c, Γ , where the constant C depends on l , ν , ρ , R , k N k ρ , k DK k ρ,c, Γ , k F k C ( B r ) , k J c ( K ) k C ( B r ) .We will take as an approximate solution the solution v of(70) L v ( θ ) = p ( θ ) − H ( θ )Λobtained from (69) by removing the terms containing B, p and q . Proposition 4.17.
Assume ω ∈ D ( κ, ν ) and ( λ, K ) is a non-degenerate pair (i.e. ( λ, K ) ∈ N D loc ( ρ, Γ) ). If the error k E k ρ,c, Γ satisfies (51) , there exist a mapping v ∈ A ρ − δ,c, Γ for any < δ < ρ/ and a vector Λ ∈ R l solving equation (70) .Moreover there exists a constant C > depending on ν, ρ, l, R, k K k ρ,c, Γ , | avg ( Q λ ) | − , | avg ( A λ ) | − , k N k ρ and k J c ( K ) k ρ such that (71) k v k ρ − δ,c, Γ < Cκ δ − ν k E k ρ,c, Γ and | Λ | < C k E k ρ,c, Γ . Proof.
We denote T ( θ ) the right-hand side of equation (70), i.e. we have to solve(72) L v ( θ ) = T ( θ ) , with T = p − H Λ . We now decompose equation (72) into two equations. Writing v = ( v , v ) ⊤ , T ( θ ) =( T ( θ ) , T ( θ )) ⊤ equation (72) is equivalent to v ( θ ) + A λ ( θ ) v ( θ ) = v ( θ + ω ) + T ( θ ) , (73) v ( θ ) = v ( θ + ω ) + T ( θ ) . (74)A simple computation shows that T ( θ ) = − [ DK ⊤ J c ( K )] ◦ T ω ◦ ( E c + ∂F λ ( K ( θ )) ∂λ Λ)We begin by solving equation (74). To apply Proposition 4.4 we choose Λ ∈ R l such that avg ( T ) = 0 . This condition is equivalent toavg (cid:16) DK ⊤ ( ω + θ ) J c ( K ( ω + θ ))( E c ( θ ) + ∂F λ ( K ( θ )) ∂λ Λ) (cid:17) = 0 . This leads toavg (cid:16) ( DK ⊤ ( ω + θ ) J c ( K ( ω + θ )) ∂F λ ( K ( θ )) ∂λ (cid:17) Λ= − avg (cid:16) DK ⊤ ( ω + θ ) J c ( K ( ω + θ )) E c ( θ ) (cid:17) . Note that the matrix which applies to Λ is the average of Q λ which, by hypothesis,is invertible. This gives | Λ | < C k E k ρ,c, Γ .From the expression of T and the value of Λ obtained above, we have that thereexists a constant C such that k T i k ρ,c, Γ ≤ C k E k ρ,c, Γ , for i = 1 , v on D ρ − δ witharbitrary average and(75) k v k ρ − δ,c, Γ ≤ Cκδ − ν k T k ρ,c, Γ + | avg ( v ) | . Now we come to equation (73). To apply Proposition 4.4 we choose avg ( v ) suchthat avg ( T − A λ v ) = 0. This condition is equivalent toavg ( v ) = avg ( A λ ) − (avg ( T ) − avg ( A λ v ⊥ )) , where v = v ⊥ + avg ( v ). This is possible since by the twist condition avg ( A λ ) isinvertible.We have that | avg ( v ) | ≤ Cκδ − ν k E k ρ,c, Γ . Then we take v as the unique analytic solution of (73) with zero average. Fur-thermore, we have the estimate k v k ρ − δ,c, Γ ≤ Cκδ − ν k T − A λ v k ρ − δ,c, Γ . Collecting the previous bounds we get the result. (cid:3)
UASI-PERIODIC BREATHERS 39
We now come back to the solutions of (38). The above procedure allows us toprove the following proposition, providing an approximate solution of the projectionof D λ,K F ω ( λ, K )(Λ , ∆) = − E on the center subspace. Proposition 4.18.
Let (Λ , W ) be as in Proposition 4.17 and assume the hypothe-ses of that proposition hold. Define ∆ c ( θ ) = ˜ M ( θ ) W ( θ ) . Then, equation (38) isapproximately solvable and we have the following estimates (76) k ∆ c k ρ − δ,c, Γ ≤ Cκ δ − ν k E k ρ,c, Γ , | Λ | ≤ C k E k ρ,c, Γ , where the constant C depends on ν, ρ, l, R, | avg ( Q λ ) | − , | avg ( A λ ) | − , k N k ρ , k ∂F λ ( K ) ∂λ k ρ,c, Γ and k J c ( K ) k ρ and (77) k D λ,K F ω ( λ, K )(Λ , ∆ c ) + E c k ρ − δ,c, Γ ≤ Cκ δ − (2 ν +1) ( k E k ρ,c, Γ + k E k ρ,c, Γ | Λ | ) , where the constant C depends on l , ν , ρ , R , k F k C ( B r ) , k DK k ρ,c, Γ , k N k ρ , | avg ( A λ ) | − , | avg ( Q λ ) | − and k ∂F λ ( K ) ∂λ k ρ,c, Γ .Proof. For the first estimate we take θ ∈ D ρ − δ and write W = ( W , W ). Then∆ c ( θ ) = DK ( θ ) W ( θ ) + J c ( K ( θ )) − DK ( θ ) N ( θ ) W ( θ ) . We have (cid:12)(cid:12) ( DK ( θ ) W ( θ )) i (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) l X j =1 D θ j K i ( θ ) W ,j ( θ ) (cid:12)(cid:12)(cid:12) ≤ k DK k ρ,c, Γ max k Γ( i − c k ) k W k ρ − δ . Also, since J ∞ is uncoupled and N is finite dimensional, (cid:12)(cid:12)(cid:12)(cid:0) J c ( K ( θ )) − DK ( θ ) N ( θ ) W ( θ ) (cid:1) i (cid:12)(cid:12)(cid:12) ≤ k J ( K ) k ρ (cid:12)(cid:12)(cid:12)(cid:0) DK ( θ ) N ( θ ) W ( θ ) (cid:1) i (cid:12)(cid:12)(cid:12) ≤ k J ( K ) k ρ k N k ρ k DK k ρ,c, Γ max Γ( i − c k ) k W k ρ − δ . Now using (71), we obtain (76).For (77), using the previous notations and Lemma 4.16, D λ,k F ω ( λ, K )(Λ , ∆ c )( θ ) + E c ( θ )= ∂F λ ∂λ ( K ( θ ))Λ + ˜ M ( θ + ω ) (cid:2) S λ ( θ ) W ( θ ) − W ( θ + ω ) (cid:3) + e ( θ ) W ( θ ) + E c ( θ ) = ∂F λ ∂λ ( K ( θ ))Λ+ ˜ M ( θ + ω ) (cid:2) p ( θ ) − Q ( θ )Λ (cid:3) + e ( θ ) W ( θ ) + E c ( θ ) . (78)Note that˜ M ( θ + ω ) p ( θ ) =˜ M ( θ + ω ) (cid:2) − ([ ˜ M ⊤ J c ( K ) ˜ M ) − ]( θ + ω )[ ˜ M ⊤ J c ( K )]( θ + ω ) E c ( θ ) − p ( θ ) (cid:3) and also that ˜ M ( ˜ M ⊤ J c ( K ) ˜ M ) − ˜ M ⊤ J c ( K ) is symmetric and˜ M ˜ M ⊤ J c ( K ) ˜ M ) − ˜ M ⊤ J c ( K ) − Idmaps the vectors of the center subspace to zero because it is generated by thecolumns of ˜ M . Also we have˜ M ( θ + ω ) Q λ ( θ )Λ =˜ M ( θ + ω ) h [ ˜ M ⊤ J c ( K ) ˜ M ) − ˜ M ⊤ J c ( K )]( θ + ω ) ∂F λ ∂λ ( K ( θ ))Λ − q ( θ )Λ i . We recall that here the derivative of F λ with respect to λ actually means theprojection of it into the center subspace.Therefore (78) becomes˜ M ( θ + ω )[ q ( θ )Λ − p ( θ )] + e ( θ ) W ( θ )and (77) follows. (cid:3) Solution of the equation in the hyperbolic subspaces.
In this section,we study the projection of the Newton equation (27) on the the hyperbolic spaces.According to the splitting (12), there exist projections on the linear spaces E sK ( θ ) and E uK ( θ ) . The analytic regularity of the splitting implies the analytic dependenceof these projections in θ . We denote Π sK ( θ ) (resp. Π uK ( θ ) ) the projections on thestable (resp. unstable) invariant subspace.We project equation (27) on the stable and unstable subspaces to obtain(79) Π sK ( θ + ω ) (cid:16) ∂F λ ( K ( θ )) ∂λ Λ + DF λ ( K ( θ ))∆( θ ) − ∆( θ + ω ) (cid:17) = − Π sK ( θ + ω ) E ( θ ) , (80) Π uK ( θ + ω ) (cid:16) ∂F λ ( K ( θ )) ∂λ Λ + DF λ ( K ( θ ))∆( θ ) − ∆( θ + ω ) (cid:17) = − Π uK ( θ + ω ) E ( θ ) . The invariance of the splitting readsΠ sK ( θ + ω ) DF λ ( K ( θ ))∆( θ ) = DF λ ( K ( θ ))Π sK ( θ ) ∆( θ )for the stable part andΠ uK ( θ + ω ) DF λ ( K ( θ ))∆( θ ) = DF λ ( K ( θ ))Π uK ( θ ) ∆( θ )for the unstable one.We define ∆ s,u ( θ ) = Π s,uK ( θ ) ∆( θ ) . Using the notation θ ′ = θ + ω , equations (79)-(80) become(81) DF λ ( K ) ◦ T − ω ( θ ′ )∆ s ( T − ω ( θ ′ )) − ∆ s ( θ ′ ) = − ˜ E s ( θ ′ , Λ) , where ˜ E s ( θ ′ , Λ) = Π sK ( θ ′ ) (cid:16) ∂F λ ( K ( T − ω ( θ ′ ))) ∂λ Λ (cid:17) + Π sK ( θ ′ ) E ◦ T − ω ( θ ′ )and(82) DF λ ( K ) ◦ T − ω ( θ ′ )∆ u ( T − ω ( θ ′ )) − ∆ u ( θ ′ ) = − ˜ E u ( θ ′ , Λ) , where ˜ E u ( θ ′ , Λ) = Π uK ( θ ′ ) (cid:16) ∂F λ ( K ( T − ω ( θ ′ ))) ∂λ Λ (cid:17) + Π uK ( θ ′ ) E ◦ T − ω ( θ ′ ) . Contrary to the projections on the center subspace, the projections on the hy-perbolic subspaces can be solved exactly. The key point for the estimates is theBanach algebra property of the decay functions.
UASI-PERIODIC BREATHERS 41
Proposition 4.19.
Fix ρ > . Then equation (81) (resp. (82) ) admits a uniqueanalytic solution ∆ s : D ρ → E sK ( θ ) (resp. ∆ u : D ρ → E uK ( θ ) ). Furthermore thereexists a constant C depending only on the hyperbolicity constant µ (resp. µ ), thenorm of the projector k Π sK ( θ ) k ρ, Γ (resp. k Π uK ( θ ) k ρ, Γ ) and k ∂F λ ( K ) ∂λ k ρ,c, Γ such that (83) k ∆ s,u k ρ,c, Γ ≤ C ( k E k ρ,c, Γ + | Λ | ) . Proof.
We only give the proof for the stable case, the unstable case being verysimilar. Using equation (81), we claim(84) ∆ s ( θ ′ ) = ∞ X k =0 ( DF λ ( K ) ◦ T − ω ( θ ′ ) × · · · × DF λ ( K ) ◦ T − kω ( θ ′ )) ˜ E s ( T − kω ( θ ′ ) , Λ) . We introduce the map F co ( k, θ ′ ) = DF λ ( K ) ◦ T − ω ( θ ′ ) × · · · × DF λ ( K ) ◦ T − kω ( θ ′ ) . From the definition of ˜ E s we have k ˜ E s ( T − kω ( θ ′ ) , Λ) k ρ,c, Γ ≤ C (cid:0) k E k ρ,c, Γ + | Λ | (cid:1) .Using(13) we obtain that the k term in (84) is bounded by kF co ( k, θ ′ ) ˜ E s ( T − kω ( θ ′ ) , Λ) k ρ,c, Γ ≤ C h µ k k ˜ E s ( T − kω ( · ) , Λ) k ρ,c, Γ and therefore k ∆ s k ρ,c, Γ ≤ C h k ˜ E s k ρ,c, Γ ∞ X k =0 µ k ≤ C (cid:0) k E k ρ,c, Γ + | Λ | (cid:1) , since µ < (cid:3) Remark 4.20.
It is perhaps interesting to compare the method of proof of thispaper with that of [FdlLS09a]. Both papers use the invariance equations and formu-late a quasi-Newton method that can be solved using techniques from hyperboliclore and some geometric identities. One of the strengths of the set-up based ondecay functions is that we can obtain estimates independent on the number andpositions of the centers of activity by methods that resemble the finite dimensionalmethods.Both [FdlLS09a] and the present paper use a counterterm to adjust some of theconstants and then prove a vanishing lemma. In [FdlLS09a], the counterterm isobtained adding J − ◦ K DK λ . In this paper, we consider a family of symplecticmaps. This allows us to use the vanishing lemma only once at the end of the proof,whereas in [FdlLS09a], the vanishing lemma had to be used at each iterative step.We also deal in a different way with the invertibility of the linear change ofvariables M . In the present paper, we obtain the invertibility on the range usingsome geometric identities, whereas in [FdlLS09a], we used some easier argumentbased on finite dimensional arguments.One geometric aspect that required several changes (due in part to the changesin the counterterm and to the infinite dimensional character) is the estimates onthe difference between the center space and the range of the change of variables M .5. Iteration of the modified Newton method, convergence and proofof Theorem 3.5
This section is devoted to the iteration of the Newton method. We derive firstthe usual KAM estimates for convergence. We assume that we are under the assumptions of Theorem 3.5. We note that with the decay norms the estimates arevery similar to the ones we obtained in the finite dimensional case (see [FdlLS09a]).5.1.
Iteration of the method.
Let ( λ , K ) be an approximate solution of (11)(i.e. a solution of the linearized equation with error E ). Following the Newtonscheme we define the following sequence of approximate solutions K m = K m − + ∆ K m − , m ≥ ,λ m = λ m − + Λ m − , m ≥ , where (Λ m − , ∆ K m − ) is a solution of D λ,K F ω ( λ m − , K m − )(Λ m − , ∆ K m − ) = − E m − with E m − ( θ ) = F ω ( λ m − , K m − )( θ ). The next lemma states a classical result inKAM theory: the approximation to the solution at step m has an error which isbounded in a smaller complex domain by the square of the norm of the error atstep m − Proposition 5.1.
Assume ( λ m − , K m − ) ∈ N D loc ( ρ m − , Γ) is an approximatesolution of equation (11) and that the following holds r m − = k K m − − K k ρ m − ,c, Γ < r . If E m − is small enough such that lemma 4.18 applies then there exists a function ∆ K m − ∈ A ρ m − − δ m − ,c, Γ for any < δ m − < ρ m − / and a vector Λ m ∈ R l suchthat (85) k ∆ K m − k ρ m − − δ m − ,c, Γ ≤ ( C m − + C m − κ δ − νm − ) k E m − k ρ m − ,c, Γ , (86) | Λ m | ≤ C k E m − k ρ m − ,c, Γ (87) k D ∆ K m − k ρ m − − δ m − ,c, Γ ≤ ( C m − δ − m − + C m − κ δ − (2 ν +1) m − ) k E m − k ρ m − ,c, Γ , where C m − , C m − depend only on ν , l , | F λ | C ( B r ) , k DK m − k ρ m − ,c, Γ , || Π cK m − ( θ ) || ρ m − , Γ , || Π sK m − ( θ ) || ρ m − , Γ , || Π uK m − ( θ ) || ρ m − , Γ , | avg ( Q λ m − ) | − and | avg ( A λ m − ) | − . More-over, if K m = K m − + ∆ K m − and r m − + C m − + (cid:16) C m − κ δ − νm − (cid:17) k E m − k ρ m − ,c, Γ < r then we can redefine C m − and C m − and all previous quantities such that the error E m ( θ ) = F ω ( λ m , K m )( θ ) satisfies (88) k E m k ρ m ,c, Γ ≤ C m − κ δ − νm − k E m − k ρ m − ,c, Γ , where we take ρ m = ρ m − − δ m − . Remark 5.2.
The estimate (88) showing that the norm of the error at step m is essentially bounded by the square of the norm of the error at step m − Proof.
Taking into account that ∆ K m − ( θ ) is the sum of its three projectionson the stable, center and unstable subspaces estimates (85) and (86) follow from UASI-PERIODIC BREATHERS 43
Proposition 4.18 and Proposition 4.19. Estimate (87) follows from estimate (85)and Cauchy’s inequalities. Define the remainder of the Taylor expansion R ( λ ′ , λ, K ′ , K ) = F ω ( λ, K ) − F ω ( λ ′ , K ′ ) − D λ,K F ω ( λ ′ , K ′ )( λ − λ ′ , K − K ′ ) . Then putting λ = λ m and λ ′ = λ m − K = K m − and K ′ = K m − , we have E m ( θ ) = E m − ( θ ) + D λ,K F ω ( λ m − , K m − ( θ ))(Λ m − , ∆ K m − ( θ ))+ R ( λ m − , λ m , K m − , K m )( θ ) . According to estimate (77) and since the equations on the hyperbolic subspace are exactly solved, we have k E m − + D λ,K F ω ( λ m − , K m − )(Λ m − , ∆ K m − ) k ρ m ,c, Γ ≤ c m − κ δ − (2 ν +1) m − k E m − k ρ m − ,c, Γ . Estimate (88) then follows from Taylor’s remainder. (cid:3)
In the following, we derive the changes in the non-degeneracy conditions duringthe iterative step.For the twist condition, we have the following lemma, which is proved easilynoting that we are just perturbing finite dimensional matrices.
Lemma 5.3.
Assume that the hypothesis of Proposition 5.1 hold. If k E m − k ρ m − ,c, Γ is small enough, then • If DK ⊤ m − DK m − is invertible with inverse N m − then DK ⊤ m DK m is invertible with inverse N m and we have k N m k ρ m ≤ k N m − k ρ m − + C m − κ δ − (2 ν +1) m − k E m − k ρ m − ,c, Γ . • If avg ( A λ m − ) is non singular then, avg ( A λ m ) is non-singular and we havethe estimate | avg ( A λ m ) | − ≤ | avg ( A λ m − ) | − + C ′ m − κ δ − (2 ν +1) m − k E m − k ρ m − ,c, Γ . • If avg ( Q λ m − ) is non singular then, avg ( Q λ m ) is non-singular and we havethe estimate | avg ( Q λ m ) | − ≤ | avg ( Q λ m − ) | − + C ′′ m − κ δ − (2 ν +1) m − k E m − k ρ m − ,c, Γ . Iteration of the Newton step and convergence.
Once we have the esti-mates for a step of the iterative method, following the standard scheme in KAM the-ory one can prove the convergence of the method. One takes 0 < δ < min(1 , ρ / , δ m = δ / ρ m = ρ m − − δ m − . From (88) we have that(89) k E m k ρ m ,c, Γ ≤ C m − κ δ − ν ν ( m − k E m − k ρ m − ,c, Γ . Moreover the constants C m are bounded uniformly in m by a constant C . Using(89) iteratively one obtains k E m k ρ m ,c, Γ ≤ (cid:16) Cκ δ − ν ν k E k ρ ,c, Γ (cid:17) m . Then, if k E k ρ ,c, Γ is small enough the iteration converges to a pair ( λ ∞ , K ∞ ) ∈ N D loc ( ρ ∞ , Γ) , with K ∞ ∈ A ρ ∞ ,c, Γ such that F λ ∞ ◦ K ∞ = K ∞ ◦ T ω . Vanishing lemma and existence of invariant tori. Proof ofTheorem 3.6
In this context of infinite dimensional lattices, one could ask for the existenceof a vanishing lemma, which would ensure that the translated tori are actuallyinvariant ones. The issue is that we do not have a true symplectic form on thewhole manifold M Z N but just a formal one. The proof we give is inspired by theone in [FdlLS09a] but we have to take into account that formal forms make senseonly via their pull-backs to T l .The following lemma, called the vanishing lemma, is more or less equivalent toshowing that some averages cancel, but is somewhat easier to implement. Lemma 6.1.
Assume F λ is analytic, smooth in λ ∈ R l and maps M into itself.Assume ω ∈ D ( κ, ν ) and let ( λ, K ) ∈ N D loc ( ρ, Γ) , where K ∈ A ρ,c, Γ is a solutionof F λ ◦ K = K ◦ T ω . Assume furthermore: • F is exact symplectic, F λ is symplectic for λ = 0 and F λ is constructed asin Appendix C. • F λ extends analytically to a neighborhood of K ( T l ) . • We have | λ | ≤ λ ∗ , where λ ∗ depends only on derivatives of F and thesymplectic structure J .Then λ = 0 . Proof.
We will write equation (11) as(90) F ◦ K = R λ ◦ K ◦ T ω , where R λ = F ◦ F − λ . We denote(91) ˆ θ i = ( θ , . . . , θ i − , θ i +1 , . . . , θ l ) ∈ T l − and similarly ˆ ω i = ( ω , . . . , ω i − , ω i +1 , . . . , ω l ) ∈ R l − . We also denote σ i, ˆ θ i : T → T l the path given by(92) σ i, ˆ θ i ( η ) = ( θ , . . . , θ i − , η, θ i +1 , . . . , θ l ) . We will compute the integral R T l − R σ i, ˆ θi K ∗ F ∗ α ∞ in two different ways. Notethat the quantity R σ i, ˆ θi K ∗ F ∗ α ∞ is well-defined since K has decay on M .Using that F is exact symplectic we have: Z σ i, ˆ θi +ˆ ωi K ∗ F ∗ α ∞ = Z σ i, ˆ θi +ˆ ωi ( K ∗ α ∞ + dW K )= Z σ i, ˆ θi +ˆ ωi K ∗ α ∞ = Z K ◦ σ i, ˆ θi +ˆ ωi α ∞ . (93)Similarly, we have Z σ i, ˆ θi ( R λ ◦ K ◦ T ω ) ∗ α ∞ = Z σ i, ˆ θi T ∗ ω ( R λ ◦ K ) ∗ α ∞ = Z σ i, ˆ θi +ˆ ωi ( R λ ◦ K ) ∗ α ∞ . (94) UASI-PERIODIC BREATHERS 45
Since the averages over T l − are the same, one gets0 = Z T l − Z σ i, ˆ θi [ K ∗ α ∞ − ( R λ ◦ K ) ∗ α ∞ ]= − Z T l − Z σ i, ˆ θi K ∗ ( R ∗ λ α ∞ − α ∞ ) . (95)Now, we estimate K ∗ (cid:16) R ∗ λ α ∞ − α ∞ (cid:17) . We know that R λ = F ◦ F − λ . Therefore, we have K ∗ (cid:16) R ∗ λ α ∞ − α ∞ (cid:17) = K ∗ (cid:16) ( F − λ ) ∗ ( F ∗ α ∞ − F ∗ λ α ∞ ) (cid:17) . This gives, using the exact symplecticness of F Z T l − Z σ i, ˆ θi K ∗ ( F − λ ) ∗ ( α ∞ − F ∗ λ α ∞ ) . By the construction of the map F λ , we have( F − λ ) ∗ ( α ∞ − F ∗ λ α ∞ ) = 12 ( F − λ ) ∗ n X j ∈J l X k =1 λ jk F ∗ δ jk + dβ o , where β is a smooth function on the torus and ( δ jk ) k =1 ,...,l is a basis of H ( T l ). Thisgives ( F − λ ) ∗ ( α ∞ − F ∗ λ α ∞ ) = 12 X j ∈J l X k =1 λ jk R ∗ λ δ jk + d ¯ β. Therefore, one gets 12 X j ∈J l X k =1 λ jk Z T l − Z σ i, ˆ θi K ∗ R ∗ λ δ jk = 0 . By the smoothness of F λ with respect to λ we can write R λ = Id + O ( | λ | ) . This gives 0 = 12 X j ∈J l X k =1 λ jk Z T l − Z σ i, ˆ θi K ∗ R ∗ λ δ jk = 12 X j ∈J l X k =1 λ jk Z T l − Z σ i, ˆ θi K ∗ δ jk + ( | λ | ) . Since K is an embedding, { δ kj } ≤ j ≤ l is a basis of H ( K ( T l )). Consequently, themap v R T l − R σ i, ˆ θi K ∗ δ jk v is invertible and the smallness assumption on λ (whichis satisfied in particular by the KAM theorem) ensures, by the Implicit FunctionTheorem, that λ = 0. (cid:3) s s ss ^ ^^ ^ w ^T T w ^ K ( ) B l lK K o o l ) o s ^ ( + G o TT KF F K o ( ) + G l ) (T l ) Figure 3.
Illustration of the proof of the vanishing lemma, Lemma 6.1.Once we have the vanishing lemma, Theorem 3.6 follows directly from Theorem3.5 by using the construction in Appendix C. Indeed, starting with the exact sym-plectic map F we construct the family F λ to which we apply Theorem 3.5, with anapproximate solution ( λ = 0 , K ), to obtain ( λ ∞ , K ∞ ) such that F λ ∞ ◦ K ∞ = K ∞ ◦ T ω . The vanishing lemma implies that λ ∞ = 0.7. Uniqueness results
In this section, we prove Theorem 3.9. We closely follow the proof in [FdlLS09a].It is based on showing that the operator D F ω ( K ) has an approximate left inverse(as in [Zeh75, Zeh76a]). Notice first that the composition on the right by everytranslation of a solution of (1) is also a solution. Therefore, one cannot expect astrict uniqueness result. Moreover, the second statement in Lemma 4.1 and thecalculation on the hyperbolic directions show that, roughly speaking, two solutionsof the linearized equation differ by their average. Moreover this difference is in thedirection of the tangent space of the torus. The idea behind the local uniquenessresult is to prove that one can transfer the difference of the averages between twosolutions to a difference of phase between the two solutions.Now we assume that the embeddings K and K satisfy the hypotheses in Theo-rem 3.9, in particular K and K are solutions of (1), or (11) with λ = 0. If τ = 0 wewrite K for K ◦ T τ which is also a solution. Therefore F ω (0 , K ) = F ω (0 , K ) = 0.By Taylor’s theorem we can write0 = F ω (0 , K ) − F ω (0 , K ) = D λ,K F ω (0 , K )(0 , K − K )+ R (0 , , K , K ) . (96)Moreover, there exists C > kR (0 , , K , K ) k ρ,c, Γ ≤ C k K − K k ρ,c, ΓUASI-PERIODIC BREATHERS 47 since F ∈ C . Hence we end up with the following linearized equation D λ,K F ω (0 , K )(0 , K − K ) = −R (0 , , K , K ) . We denote ∆ = K − K .Projecting this equation on the center subspace, writing ∆ c ( θ ) = Π cK ( θ ) ∆( θ )and making the change of function ∆ c ( θ ) = ˜ M ( θ ) W ( θ ), where ˜ M is defined in (41)with K = K , we obtain DF ( K ( θ )) ˜ M ( θ ) W ( θ ) − ˜ M ( θ + ω ) W ( θ + ω )= − Π cK ( θ + ω ) R (0 , , K , K ) . (97)Applying the property DF ( K ( θ )) ˜ M ( θ ) = ˜ M ( θ + ω ) S ( θ ) for solutions of (1),multiplying both sides by ˜ M ( θ + ω ) ⊤ J c ( K ) and using that ˜ M ⊤ J c ( K ) ˜ M is invertiblewe get S ( θ ) W ( θ ) − W ( θ + ω ) = − [( ˜ M ⊤ J c ( K ) ˜ M ) − ˜ M ⊤ J c ( K )]( θ + ω )Π cK ( θ + ω ) R (0 , , K , K ) . We get bounds for W , from the fact that it solves the previous equation, using themethods in Section 4.3.7. We write W = ( W , W ). Since S is triangular we beginby looking for W . We search for it in the form W = W ⊥ + avg ( W ). We have k W ⊥ k ρ − δ,c, Γ = Cκδ − ν k K − K k ρ,c, Γ . For W we have W ( θ ) − W ( θ + ω ) =[ N DK ⊤ ]( θ )(Π cK ( θ + ω ) R (0 , , K , K )) ( θ ) − A ( θ ) W ⊥ ( θ ) − A ( θ )avg ( W ) , (98)where N = (cid:16) DK ⊤ DK (cid:17) − . The condition that the right-hand side of (98) haszero average gives | avg ( W ) | ≤ Cκδ − ν ( k K − K k ρ,c, Γ ) . Then k W − avg ( W ) k ρ − δ ≤ Cκ δ − ν k K − K k ρ,c, Γ but avg ( W ) is free. Then (cid:13)(cid:13) ∆ c − DK avg ( W ) (cid:13)(cid:13) ρ − δ,c, Γ ≤ Cκ δ − (2 ν +1) k K − K k ρ,c, Γ . The next step is done in the same way as in [dlLGJV05]. We quote Lemma 14 ofthat reference using our notation.
Lemma 7.1.
There exists a constant C such that if C k K − K k ρ,c, Γ ≤ thenthere exits a phase τ ∈ (cid:8) τ ∈ R l | | τ | < k K − K k ρ,c, Γ (cid:9) such that avg (cid:0) N DK ⊤ Π cK ( θ ) ( K ◦ T τ − K )( θ ) (cid:1) = 0 . The proof is based on the application of the Banach fixed point theorem in R l .As a consequence of Lemma 7.1, if τ is as in the statement, then K ◦ T τ is asolution of (1) such that if W = ( ˜ M ( θ + ω ) ⊤ J c ( K ) ˜ M ( θ + ω )) − ˜ M ( θ + ω ) ⊤ Π cK ( θ ) ( K ◦ T τ − K ) , for all δ ∈ (0 , ρ/
2) we have the estimate k W k ρ − δ < Cκ δ − ν kRk ρ,c, Γ ≤ Cκ δ − ν k K − K k ρ,c, Γ . This leads to on the center subspace k Π cK ( θ ) ( K ◦ T τ − K ) k ρ − δ,c, Γ ≤ Cκ δ − ν k K − K k ρ,c, Γ . Furthermore, we can show that ∆ h = Π hK ( θ ) ( K ◦ T τ − K ) satisfies the estimate k ∆ h k ρ − δ,c, Γ < C kRk ρ,c, Γ . All in all, we have proven the estimate for K ◦ T τ − K (up to a change in theoriginal constants) k K ◦ T τ − K k ρ − δ,c, Γ ≤ Cκ δ − ν k K − K k ρ,c, Γ . We are now in position to perform the same scheme used in Section 5. We can takea sequence { τ m } m ≥ such that | τ | ≤ k K − K k ρ,c, Γ and | τ m − τ m − | ≤ k K ◦ T τ m − − K k ρ m − ,c, Γ , m ≥ , and k K ◦ T τ m − K k ρ m ,c, Γ ≤ Cκ δ − νm k K ◦ T τ m − − K k ρ m − ,c, Γ , where δ = ρ/ δ m +1 = δ m / m ≥ ρ = ρ , ρ m = ρ − P mk =1 δ k for m ≥ k K ◦ T τ m − K k ρ m ,c, Γ ≤ ( Cκ δ − ν ν k K − K k ρ ,c, Γ ) m − ν (3+ m ) . Therefore, under the smallness assumptions on k K − K k ρ ,c, Γ , the sequence { τ m } m ≥ converges and one gets k K ◦ T τ ∞ − K k ρ/ ,c, Γ = 0 . Since both K ◦ T τ ∞ and K are analytic in D ρ and coincide in D ρ/ we obtain theresult. 8. Results for flows on lattices
In this section, we study vector-fields defined on lattices. The models we consider– which have appeared naturally in solid state physics and in biophysics, see [CF05]for a review – consist of a sequence of copies of an individual system arranged in alattice coupled with their neighbors.The basic result we will prove is Theorem 8.4. The proof is based on applying theresult for maps to the time-one map of the flow. This requires to study the decayproperties of flows generated by Hamiltonian systems with decay, which may havesome independent interest. Again, it is important to emphasize that the results weprove have an a posteriori format, showing that close to approximate solutions –which satisfy some hyperbolicity and twist conditions – there is a true solution. Wewill not need that the system is close to integrable.8.1.
Integrating decay vector fields.
In this section, we study properties ofvector-fields with decay. We will show in Proposition 8.1 that the flows they gener-ate are families of diffeomorphisms with decay. As we will see, this is a consequenceof the composition properties of spaces of functions with decay, which in turn isa property of the Banach algebra properties under multiplication. This, togetherwith some more delicate study of the non-degeneracy conditions, will allow us toapply the existence Theorem 8.4 in the next section to the time-one map of theflow for a model problem given by a vector-field X .We consider the equation on M = ℓ ∞ ( Z N )(99) ∂ ω K ( θ ) = X ◦ K ( θ ) , where X is a vector-field on M and K maps T l into M . UASI-PERIODIC BREATHERS 49
We note that we can deal with systems represented by formal Hamiltonianswhich are given by formal sums which do not need to converge and hence do notdefine a function, but however their partial derivatives and therefore the differentialequations they determine are well-defined. Therefore, the invariance equationsmake sense. This is one of the reasons why the present method, based on the studyof the invariance equation has advantages over the more classical methods [Zeh76a]based on transformations of the Hamiltonian function.We prove that decay vector fields generate flows { S t } t ∈ R such that all S t aredecay diffeomorphisms. Proposition 8.1.
Let X be a C r vector-field, r ≥ , on an open set B ⊂ M (recallthat we consider M endowed with the ℓ ∞ topology) and consider the differentialequation (100) x ′ = X ( x ) . Let B ⊂ B be an open set such that d ( B , B c ) = η > .Then there exist T > such that for all the initial conditions x ∈ B thereis a unique solution x t of the Cauchy problem corresponding to (100) defined for | t | < T . We denote by S t ( x ) = x t . Note that, by the uniqueness result, we have S t + s = S t ◦ S s when all the maps are defined and the composition makes sense.Moreover (1) For all t ∈ ( − T, T ) , S t : B → B is a diffeomorphism onto its image. (2) If X ∈ C r Γ ( B ) then S t ∈ C r Γ ( B ) for all t ∈ ( − T, T ) . Moreover, there exist C, µ > such that k DS t ( x ) k Γ ≤ Ce µt , x ∈ B , t ∈ ( − T, T ) . Note also that, when B = M and DX is bounded, we have T = ∞ . Remark.
When M is the complexified manifold the derivatives can be consid-ered as complex derivatives. Therefore in such a case S t is analytic according toDefinition 2.6.Moreover if 0 is an equilibrium point of X, S t (0) = 0 and hence if ψ ∈ A ρ,c, Γ , byLemma A.15, S t ◦ ψ ∈ A ρ,c, Γ . Proof.
The first claim (1) follows from the standard proof of existence, unique-ness and regularity of solutions of ordinary differential equations in Banach spaces[Hal80]. Let m = k X k C and T < η/m . Then we have that S t ( x ) is C r withrespect to ( t, x ) ∈ ( − T, T ) × B . The uniqueness implies the flow property S t + s ( x ) = S t ( S s ( x ))and this property implies that S − t = S − t .We note that the standard theory of existence of solutions, also gives that, when B = M , we have T = ∞ (recall that our definition of C r implies that the derivativesare bounded, so that X is globally Lipschitz).(2) By the general theory we also have that DS t satisfies the first order variationalequation ( DS t ( x )) ′ = DX ( S t ( x )) DS t ( x ) , DS ( x ) = Id , and by the theory of linear systems we know that DS t ( x ) is the limit of the sequencegiven by(101) Φ t ( x ) = Id , (102) Φ kt ( x ) = Id + Z t DX ( S s ( x ))Φ k − s ( x ) ds, k ≥ t ∈ ( − T, T ) , x ∈ B . To get that DS t ( x ) ∈ L Γ we make estimates of the Γ-normof Φ kt .Using (101) and (102) we obtain by induction that(1 , k ) Φ kt ( x ) ∈ L Γ , ∀ x ∈ B , ∀ t ∈ ( − T, T ) , (2 , k ) sup x ∈B (cid:13)(cid:13) Φ k +1 t ( x ) − Φ kt ( x ) (cid:13)(cid:13) Γ ≤ k + 1)! (cid:0) k X k C | t | (cid:1) k +1 , ∀ t ∈ ( − T, T ) , for k ≥ kt ( x ) is a linear operator represented by its matrix,then, so is Φ k +1 t ( x ).Writing Φ kt ( x ) = Id + k X j =1 (cid:0) Φ jt ( x ) − Φ j − t ( x ) (cid:1) , by (1 , k ) , (2 , k ) we easily obtain that Φ kt ( x ) converges in L Γ . Hence DS t ( x ) ∈ L Γ and k DS t ( x ) k Γ ≤ Γ − (0) + ∞ X j =1 j ! (cid:0) k X k C | t | (cid:1) j = Γ − (0) − (cid:0) k X k C | t | (cid:1) . The higher order derivatives of S t satisfy higher order variational equationswhich are also linear equations. By an analogous argument we get that D k S t ( x ) ∈L Γ ( M , L k − ( M , M )). We present the details for the case k = 2 and leave to thereader the adaptation of the typography for larger k . We have:( D S t ( x )) ′ = DX ( S t ( x )) D S t ( x )+ D X ( S t ( x ))( DS t ( x ) , DS t ( x )) , D S ( x ) = 0 . Let G t ( x ) = Z t D X ( S s ( x ))( DS s ( x ) , DS s ( x )) ds. By Lemma A.5, G t ( x ) ∈ L . We can write D S t ( x ) = Z t DX ( S s ( x )) D S s ( x ) ds + G t ( x ) . The sequence given byΨ t ( x ) = 0Ψ kt ( x ) = Z t DX ( S s ( x ))Ψ k − s ( x ) ds + G t ( x )converges to D S t ( x ). Similarly to the case k = 1 one proves by induction thatΨ kt ( x ) ∈ L . Since L is complete we obtain that D S t ( x ) ∈ L . (cid:3) We also remark that, using the standard argument of adding extra equations[Hal80], we can obtain the smooth dependence on parameters.
UASI-PERIODIC BREATHERS 51
Invariant tori for flows.
The following result is our main KAM theorem forvector-fields on lattices. For the sake of simplicity, we have not formulated the mostgeneral result possible but have rather stated the result for models that appear inthe Physics literature.In the following, for the sake of simplicity, we consider equations of the type(103) ˙ u = J ∞ ∇ H ( u )where • The operator J ∞ is given by J ∞ ( z ) = diag (cid:0) . . . , J, . . . (cid:1) , where J is the standard symplectic form. • The ∇ operator is the standard operator induced by the ℓ ( Z N ) metric onthe lattice.The previous equation (103) arise in the context of statistical physics as describedin the introduction. If one desires to consider more general vector-fields X , we referthe reader to the paper [FdlLS09a] where algorithms and proofs are provided inthis more general context.We first describe the non-degeneracy conditions. The linearized equation d ∆ dt = J ∞ D ∇ H ( K ( θ + ωt ))∆plays a crucial role. We denote A ( θ ) ≡ J ∞ D ∇ H ( K ( θ )) and we remark that sincethe vector field J ∞ ∇ H has decay, by Proposition 8.1, the vector field J ∞ ∇ H gen-erates an evolution operator, denoted U θ ( t ) with decay. We have ddt U θ ( t ) = A ( θ + ωt ) U θ ( t ) , and U θ (0) = Id. We now have the following definitions. Condition 8.2. (Spectral non-degeneracy condition) Given an embedding K : D ρ ⊃ T l → M we say that K is hyperbolic non-degenerate if there is an analyticsplitting T K ( θ ) M = E sK ( θ ) ⊕ E cK ( θ ) ⊕ E uK ( θ ) invariant under the linearized equation (106) in the sense that U θ ( t ) E s,c,uK ( θ ) = E s,c,uK ( θ + ωt ) . Moreover the center subspace E cK ( θ ) has dimension l . We denote Π sK ( θ ) , Π cK ( θ ) and Π uK ( θ ) the projections associated to this splitting and we denote U s,c,uθ ( t ) = U θ ( t ) | E s,c,uK ( θ ) . Furthermore, we assume that there exist β , β , β > and C h > independentof θ satisfying β < β , β < β and such that the splitting is characterized by thefollowing rate conditions: k U sθ ( t ) U sθ ( τ ) − k ρ,c Γ ≤ C h e − β ( t − τ ) , t ≥ τ, k U uθ ( t ) U uθ ( τ ) − k ρ,c, Γ ≤ C h e β ( t − τ ) , t ≤ τ, (104) k U cθ ( t ) U cθ ( τ ) − k ρ,c, Γ ≤ C h e β | t − τ | , t, τ ∈ R . Condition 8.3.
Let N ( θ ) = [ DK ( θ ) ⊤ DK ( θ )] − and P ( θ ) = DK ( θ ) N ( θ ) . Theaverage on T l of the matrix S ( θ ) = N ( θ ) DK ( θ ) ⊤ [ A ( θ ) J ∞ − J ∞ A ( θ )] DK ( θ ) N ( θ ) . is non-singular. Here A ( θ ) = J ∞ D ∇ H ( K ( θ )) . We now state our theorem
Theorem 8.4.
Let H be a formal Hamiltonian function on T ∗ M such that theassociated vector-field X = J ∞ ∇ H is a C , analytic vector-field in M . For somedecay function Γ , let ω ∈ D h ( κ, ν ) for some κ > , ν ≥ l − , ρ > and c = ( c , . . . , c R ) ∈ ( Z N ) R . Denote S t the flow associated to X .Consider the equation (105) l X i =1 ω i ∂K∂θ i ( θ ) = ( X ◦ K )( θ ) . Assume (1) X extends analytically to a complex neighborhood U of K ( D ρ ) : B r = (cid:8) z ∈ M|∃ θ ∈ T l | | Im θ | < ρ , | z − K ( θ ) | < r (cid:9) , for some r > . (2) There exists K ∈ A ρ ,c, Γ such that K ∈ N D loc ( ρ , Γ) (the embedding K is non-degenerate) in the sense that it satisfies non-degeneracy conditions8.2 and 8.3. (3) There exists a constant
C > depending on l , κ , ν , ρ , k H k C ( B r ) Γ , k DK k ρ ,c, Γ , k N k ρ , k S k ρ , | avg ( S ) | − (where S and N are as indefinitions 8.2-8.3 replacing K by K ) and k Π c,s,uK ( θ ) k ρ , Γ such that E = J ∞ ∇ H ( K ) − ∂ ω K satisfies the following estimates Cκ δ − ν k E k ρ ,c, Γ < and Cκ δ − ν k E k ρ ,c, Γ < r, where < δ < min(1 , ρ / is fixed.Then there exists an analytic embedding K ∈ A ρ − δ,c, Γ such that K ∈ N D loc ( ρ − δ, Γ) and satisfies equation (105) for all t ∈ R .Proof. We will only sketch the proof and refer the reader to [FdlLS09a] wherethe complete proofs are provided in the case of finite dimensions. In the presentframework, the Banach algebra properties of our spaces make the proofs in theinfinite dimensional case very similar to the the ones in the finite dimensionalcontext.We consider the following linearized equation:(106) d ∆ dt − A ( θ + ωt )∆ = − E ( θ + ωt ) . We first project equation (106) on the center subspace and on the hyperbolicsubspaces. On the center subspace, one has(107) ∂ ω ∆ c ( θ ) − A ( θ )∆ c ( θ ) = − E c ( θ ) . UASI-PERIODIC BREATHERS 53
Using the following proposition ([R¨us76a], [R¨us76b], [R¨us75], [dlL01]), one canprove the following reducibility property in Lemma 8.6.
Proposition 8.5.
Assume that ω ∈ D h ( κ, ν ) with κ > and ν ≥ l − , i.e. | ω · k | − ≤ κ | k | ν , for all k ∈ Z l \ { } . Let h : D ρ ⊃ T l → M be a real analytic function with zero average. Then, for any < δ < ρ there exists a unique analytic solution v : D ρ − δ ⊃ T l → M of the linearequation l X j =1 ω j ∂v∂θ j = h having zero average. Moreover, if h ∈ A ρ,c, Γ then v satisfies the following estimate k v k ρ − δ,c, Γ ≤ Cκδ − ν k h k ρ,c, Γ , < δ < ρ. The constant C depends on ν and the dimension of the torus l but is independentof c . Lemma 8.6.
Assume ω ∈ D h ( κ, ν ) with κ > and ν ≥ l − and k E k ρ,c, Γ is smallenough. Then there exist a matrix B ( θ ) and vectors p and p such that equation (108) [ ∂ ω ˜ M ( θ ) − A ( θ ) ˜ M ( θ )] ξ ( θ ) + ˜ M ( θ ) ∂ ω ξ ( θ ) = − E c ( θ ) , can be written as h (cid:18) l S ( θ )0 l l (cid:19) + B ( θ ) i ξ ( θ ) + ∂ ω ξ ( θ ) = p ( θ ) + p ( θ ) . Moreover, the following estimates hold: (109) k p k ρ,c, Γ ≤ C k E k ρ,c, Γ , where C just depends on k J ∞ ( K ) k ρ, Γ , k N k ρ , k DK k ρ,c, Γ and k Π cK ( θ ) k ρ, Γ . For p and B we have (110) k p k ρ − δ,c, Γ ≤ Cκδ − ( ν +1) k E k ρ,c, Γ and (111) k B k ρ − δ ≤ Cκδ − ( ν +1) k E k ρ,c, Γ for δ ∈ (0 , ρ/ , where C depends on l , ν , k N k ρ , k DK k ρ,c, Γ , | H | C ( B r ) Γ , | J | C ( B r ) and k Π cK ( θ ) k ρ, Γ . The solution of the reduced equations works in the same way as in the case ofmaps. We sketch the procedure and we emphasize on the differences.We write ξ = ( ξ , ξ ). Consider the equation(112) (cid:18) l S ( θ )0 l l (cid:19) ξ ( θ ) + ∂ ω ξ ( θ ) = p ( θ ) , where p = ( p , p ). Using this decomposition of E cK ( θ ) we can write equation(112) in the form S ( θ ) ξ ( θ ) + ∂ ω ξ ( θ ) = p ( θ ) ,∂ ω ξ ( θ ) = p ( θ ) . where p ( θ ) = DK ( θ ) ⊤ J ∞ DE ( θ ) . In order to be able to solve this small divisor equations, one has to ensure thatthe average on T l of DK ( θ ) ⊤ J ∞ DE ( θ ) is zero. This is the main difference withthe finite dimensional case and we perform now the computation. We use the factthat J ∞ has a special structure. Indeed, we have( J ∞ ) ij = Jδ ij and then ( DK ⊤ J ∞ DE ) ij = X k ∈ Z N ( DK ⊤ ) ik ( J ∞ DE ) kj . But, we have ( J ∞ DE ) kj = J ( DE ) kj and then ( DK ⊤ J ∞ DE ) ij = X k ∈ Z N ( DK ⊤ ) ik J ( DE ) kj . Thereofore, the average on T l of DK ⊤ J ∞ DE c amounts to compute the averageon T l of ( DK ⊤ ) ik J ( DE ) kj . Remark 8.7.
Here we have use the fact E c = ˜ M E + ˆ eE where ˆ e = π cK ( θ + ω ) − π Γ K ( θ + ω ) and the term ˆ eE being quadratic in the error, onecan omit it.By the computations in [dlLGJV05], one proves that then the average of ( DK ⊤ ) ik J ( DE ) kj is zero. Hence this gives the desired result.We now project the linearized equation (106) on the stable and unstable sub-spaces by using the projections Π sK ( θ ) and Π uK ( θ ) respectively. We denote ∆ s ( θ ) =Π sK ( θ ) ∆( θ ), ∆ u ( θ ) = Π uK ( θ ) ∆( θ ).Using the previous notation, we obtain(113) ∂ ω ∆ s ( θ ) − A ( θ )∆ s ( θ ) = − Π sK ( θ ) E ( θ )for the stable part and(114) ∂ ω ∆ u ( θ ) − A ( θ )∆ u ( θ ) = − Π uK ( θ ) E ( θ )for the unstable one.The following result provides the solution of the previous equations. Proposition 8.8.
Given ρ > , equations (113) and (114) admit unique analyticsolutions ∆ s : D ρ → E s and ∆ u : D ρ → E u respectively, such that ∆ s,u ( θ ) ∈ E s,uK ( θ ) .Furthermore there exist constants C s,u such that (115) k ∆ s,u k ρ,c, Γ ≤ C s,u k E k ρ,c, Γ , where C s,u depend on β , k Π sK ( θ ) k ρ, Γ (resp. β , k Π uK ( θ ) k ρ, Γ ) and C h but is inde-pendent of c . The proof of Theorem 8.4 processes then as in the finite dimensional case. (cid:3)
UASI-PERIODIC BREATHERS 55 Proof of Theorem 3.11
The goal of this section is to prove Theorem 3.11. We proceed in three stages:(1) In the first stage, we construct quasiperiodic breathers around one site in-dexed by a frequency ω ∈ Ξ( ε ∗ ). This will be a straightforward applicationof Theorem 3.6. See Section 9.1. We will use as initial approximation thesolutions in which one site is oscillating quasi-periodically and the othersare at the fixed point. This is an exact solution when ε = 0 and will bean approximate solution when ε is sufficiently small. Note that, since thesystem is translation invariant, the center site can be chosen to be any pointon the lattice.(2) In a second stage, carried out in Section 9.3 we show that, given two so-lutions which are centered around two groups of sites, if we displace farenough these solutions and add them, we obtain an approximate solution(for a slightly slower decay function). Then we can conclude to the exis-tence of a true solution close to them. The estimates of solutions displacedwill be the content of the coupling lemma (Lemma 9.8), which is the cen-terpiece of the argument. This second stage of coupling different solutionsrequires several new techniques. In particular, a detailed discussion of Dio-phantine vectors in infinite dimensions. It will also be crucial that many ofthe estimates that we have obtained before are uniform in the number andthe geometry of the sites.(3) Finally, in a third stage, we will show that there is a limit to this processof clustering breathers. We obtain a well defined limit if the centers areplaced far enough apart.We will need the following definition. Definition 9.1.
Given m ∈ Z N , let τ m : M → M be defined by (cid:0) τ m ( x ) (cid:1) i = x i + m , i ∈ Z N . In particular if F : M → M and k : T p → M ( τ m F ) i ( x ) = F i + m ( x ) , ( τ m k ) i ( θ ) = k i + m ( θ ) . Let S t be the flow of the system associated to the Hamiltonian in the statementof Theorem 3.11, and let ˜ S t = τ m S t τ − m , with m ∈ Z N . Both S t and ˜ S t satisfythe same initial value problem, hence they coincide wherever they are defined. Asa consequence we have, using F = S , F = τ m ◦ F ◦ τ − m . From this we deduce that if K ω : T rl → M with ω ∈ R rl is a solution of F ◦ K ω = K ω ◦ T ω then for all m ∈ Z N we have that τ m K ω is also a solution.9.1. Existence of quasi-periodic breathers centered around one site (PartA of Theorem 3.11).
Since the problem is invariant under translations, we willchoose, without loss of generality, to center the breather at the origin. We thenconsider the Hamiltonian H ε ( q, p ) = X n ∈ Z N (cid:16) p n + W ( q n ) (cid:17) + ε X j ∈ Z N X n ∈ Z N V j ( q n − q n + j ) . We note that, by Proposition 8.1 for ε small enough, we can obtain a time-1 map,which we will denote by F ε . This map will be exact symplectic by Proposition B.9in Appendix B.We also note that, for ε = 0, F is an uncoupled map( F ( x )) i = f ( x i ) , i ∈ Z N with f the time-1 map of the the flow on M corresponding to the Hamiltonian p + W ( q ).Assumption H2 of Theorem 3.11, implies that, for ω ∈ Ξ we can find anembedding k ω : T l → M such that f ◦ k ω = k ω ◦ T ω . We can then consider the embedding K : T l → M defined by: (cid:0) K ω ( θ ) (cid:1) i = ( k ω ( θ ) i = 00 i = 0 . Note that F ◦ K ω = K ω ◦ T ω . What we want to do is to check that K ω satisfiesthe hypothesis of Theorem 3.6.We start by embedding F ε into a family F ε,λ , λ ∈ R l , constructed by setting( F ε,λ ( z ) (cid:1) i = ( ( F ε ( z ) (cid:1) i + (0 , λ ) i = 0( F ε ( z ) (cid:1) i i = 0We can think of F ε,λ as the composition of the map F ε and a translation in thedirection of the action in the i = 0 component. Both maps are symplectic but thetranslation is not exact symplectic.To verify the quality of the embedding, we note that X n ∈ Z N ∂ ( K ω ) n ∂θ i ∂ ( K ω ) n ∂θ j = ∂ ( k ω ) ∂θ i ∂ ( k ω ) ∂θ j and, by assumption the later is non-degenerate uniformly in ω .We note that for the uncoupled map, the twist condition and the parameternondegeneracy conditions in Definition 3.4 reduce to the conditions for the time-onemap. Similarly, the hyperbolicity conditions (Definition 3.1) are satisfied whenever ε = 0. The stability results developed before show that these conditions remaintrue (with uniform values) for | ε | ≪ ( ε ∗ ) with uniform Diophantine constants, and chose ε ∗ accord-ingly, we obtain from Theorem 3.6 the existence of the KAM tori. The uniformityof the hyperbolicity and the non-degeneracy constants is a consequence of the per-turbation results for the non-degeneracy conditions.9.2. Number-theoretic properties of infinite sequences of frequencies.
This section is devoted to some results on infinite sequences of frequencies. Wewant to introduce the concept of Diophantine sequence (see Definition 9.2) andshow that these sequences are very abundant in the sense that they have full prob-ability with respect to several probability measures.Let us consider Ξ ⊂ [ − L, L ] l with l ∈ N and L >
0. Assume that Ξ has positiveLebesgue measure. Later we will take as Ξ to be a subset of D ( κ , ν ) such thatthere are KAM tori in the uncoupled system with this Diophantine properties (SeeAssumption H2 in Theorem 3.11.) UASI-PERIODIC BREATHERS 57
To discuss infinite products of measures, we consider the normalized probabilitymeasure meas ∗ ( · ) = meas( · )meas(Ξ ) . where meas( · ) is any measure absolutely continuous with respect to the Lebesguemeasure on R l . By a theorem of Kolmogorov [Dur96], the product set Ξ N with theproduct σ -algebra can be endowed with the product probability measure meas N ∗ .Note that, there are different sets Ξ which satisfy the assumption H2 of The-orem 3.11. Each of these choices will lead to mutually singular measures in theinfinite product. Nevertheless, we do not include Ξ in the notation for the infinitemeasure. The result will, of course, be valid for all choices.Now we introduce a notion of Diophantine sequences in ( R l ) N which is welladapted for our needs. Basically, we just require that for every r ≥ r components are Diophantine, even if the exponent and the constant change with r .The Diophantine properties of the sequence are just the sequence of Diophantineproperties of the truncations. This will be natural for us since at every stage of theargument we will be working with just a finite number of frequencies.We introduce the following notation: consider sequences ω = ( ω , ω , . . . ) ∈ Ξ N and k = ( k , k , . . . ) ∈ ( Z l ) N \ { } and denote ω ( r ) = ( ω , . . . , ω r ) and k ( r ) = ( k , . . . , k r ) the truncated sequences oflength r . Hence ω ( r ) · k ( r ) = r X i =1 ω i · k i and (cid:12)(cid:12) k ( r ) (cid:12)(cid:12) = r X i =1 | k i | , where k i = ( k i, , . . . , k i,l ) ∈ Z l and | k i | = | k i, | + · · · + | k i,l | . Also, given ω ∈ R rl and ω ∈ R l we will write ω ( ω , ω ) ∈ R ( r +1) l the concatenation of the vectors ω , ω . Definition 9.2.
We define D = [ ( κ,ν ) ∈ ( R + ) N × ( R + ) N D ( κ, ν ) , where κ = ( κ , ..., κ r , ... ) , ν = ( ν , ..., ν r , ... ) and D ( κ, ν ) = (cid:26) ω ∈ Ξ N | ∀ r ≥ , (cid:12)(cid:12) P ri =1 ω i · k i − m (cid:12)(cid:12) − ≤ κ r | k ( r ) | ν r , ∀ k ∈ ( Z N ) N s . t . k ( r ) = 0 , ∀ m ∈ Z (cid:27) . Remark 9.3.
Note that, since we are considering infinite dimensions, the notionof | k | we are using in Definition 9.2 could matter. We note however that changingthe norms only changes the sequence κ , the sequence ν remaining the same.The next result ensures that there are many sequences of Diophantine vectors. Lemma 9.4.
Let ν be a given sequence such that ν r > rl . Then, meas N ∗ (cid:16) Ξ N \ [ κ ∈ ( R + ) N D ( κ, ν ) (cid:17) = 0 . Proof.
We follow the standard argument for the finite dimensional case (see [dlL01]for a pedagogical exposition).
Notice first that L – the size of the box in R l containing our set Ξ – is fixed.We start by considering r fixed.For k ∈ Z rl , κ r ∈ R + , ν r ∈ R + , we define B k,m,κ r ,ν r = (cid:8) ω ∈ ([ − L, L ] l ) N | | ω ( r ) · k − m | < κ − r | k | − ν r (cid:9) . We note that(116) Ξ N \ D ( κ, ν ) = [ r ≥ [ k ∈ Z rl \{ } ,m B k,m,κ r ,ν r . Geometrically, the sets B k,m,κ r ,ν r are slabs of width 2 κ − r | k | − ν r − . As a conse-quence, we obtain thatmeas N ( B k,m,κ r ,ν r ∩ Ξ N ) ≤ meas r ( B k,m,κ r ,ν r ∩ Ξ r ) ≤ C r κ − r | k | − ν r − . Moreover, given k , the number of sets B k,m,κ r ,ν r intersecting Ξ r is bounded by aconstant depending on the dimension times | k | . Hence, we have that for ν r > rl ,meas N ( ∪ k ∈ Z rl \{ } ,m B k,m,κ r ,ν r ) ≤ X k ∈ Z rl \{ } meas N ( B k,m,κ r ,ν r ) C rl | k |≤ meas(Ξ ) − r rl κ − r ∞ X s =1 C ′ rl s rl − s ν r ≤ C ′′ rl,ν r κ − r , where C rl , C ′ rl and C ′′ rl,ν r are explicit constants. The right-hand side of the previousexpression can be estimated from above by P r ≥ C rl,ν r κ − r .By choosing a suitable sequence κ , the sum can be made as small as desired. (cid:3) Constructing more complicated breathers out of simpler ones. Thecoupling lemma.
The main goal of this section is to prove Lemma 9.8 that showsthat if we have two solutions of the invariant equation and put them in placesseparated sufficiently far apart, when we add them, we obtain a very approximatesolution of the invariance equations.In Lemma 9.14 we will show that these solutions obtained superimposing thetwo non-degenerate (in the sense of Definition 3.4) solutions centered around veryfar apart centers also satisfy the same non-degeneracy assumptions with only slightworse constants.We will also show in Lemma 9.13 that if the approximate solutions are (upto a bounded error η ) superpositions of centered breathers, then, they satisfy thehyperbolicity conditions of Theorem 3.5 with uniform bounds. The crucial pointof Lemma 9.13 is that the estimates on the η allowed and the non-degeneracyconstants are independent of c , the finite set of sites that we are considering. Thiswill be a relatively easy consequence of all the uniformity properties that we havedeveloped so far.9.3.1. Some elementary calculations with the decay functions in Proposition 2.3.
Theorem 3.11 is formulated with the special scale of decay functions Γ β defined byΓ β ( i ) = ( a | i | − α e − β | i | if i = 0 ,a if i = 0 , with α = α > N fixed and 0 < β ≤ β . UASI-PERIODIC BREATHERS 59
Figure 4.
Given two breathers, placing them far apart, we obtainan approximate solution. Using the a-posteriori Theorem 3.6, weobtain that there is a true solution close to it. See Lemma 9.8In the proof of Proposition 2.3 in [JdlL00] it is shown that the value of a can bechosen as any value less than some a independent of β . Actually we have a ( α ) < (cid:0) α +1 K N,α + 2 (cid:1) − , with K N,α = X j ∈ Z N \{ } | j | − α . Throughout this section, we set Γ = Γ β In the definition of both Γ and Γ β wetake the value of a = min (cid:0) a ( α ) , a (2 α ) (cid:1) . With this choice we have the following properties:(1) if ˜ β < β then Γ β ( i ) ≤ Γ ˜ β ( i ) for all i ∈ Z N .(2) if ˜ β < β then lim | m |→∞ Γ β ( m )Γ ˜ β ( m ) = 0 . (3) for any β, ˜ β ≤ β (117) Γ( i ) ≤ a Γ β ( i ) Γ ˜ β ( i ) , i ∈ Z N . We will encounter the quantity r X k =1 Γ β ( i − c k ). To be able to estimate it in a conve-nient way, independently on r , we will work with sequences of sites c = ( c , c , . . . ) ∈ ( Z N ) N satisfying the property Definition 9.5.
We say that a sequence of sites c is spatially non-resonant whenfor all i ∈ Z N there exist at most two different sites c p , c q in the sequence such that | c p − i | = | c q − i | . Remark 9.6.
If we arrange the sites c k in a coordinate plane of Z N , for instance Z × { } N − , and for all k we have | ( c k +1 − c k ) | < | ( c k +1 − c k ) | , then c is spatiallynon resonant according to Definition 9.5. Lemma 9.7.
Let c be a spatially non-resonant sequence.Let β ∈ (0 , β ) . Then for every i ∈ Z N and r ≥ we have (118) r X k =1 Γ β ( i − c k ) < − e − β max k Γ β ( i − c k ) . Proof.
Let i ∈ Z N be fixed. Let k be such that | i − c k | = min k | i − c k | . By thespatially non resonant property in the sum (118) for any value Γ β ( i − c k ) there areat most two terms taking the same value. Then we can group the terms in pairs.Moreover if | i − c p | > | i − c q | thenΓ β ( i − c p ) < Γ β ( i − c q ) e − β ( | i − c p |−| i − c q | ) . Therefore r X k =1 Γ β ( i − c k ) < β ( i − c k ) + 2 ∞ X m =1 Γ β ( i − c k ) e − βm and (118) follows. (cid:3) Statement and proof of the
Coupling Lemma . Lemma 9.8. ( Coupling lemma ) Let K ω ∈ A ρ,c , Γ β ∩ N D loc ( ρ, Γ β ) , K ω ∈A ρ,c , Γ ∩ N D loc ( ρ, Γ) , β < β , be the parameterizations of two invariant tori for F , localized around c and c respectively, vibrating with frequencies ω ∈ R rl and ω ∈ R l respectively.Then, if | m | is large enough, K ω : T ( r +1) l → M defined by K ω = K ω + τ m K ω is an approximate solution of F ◦ K = K ◦ T ω in the following sense: given < ˜ β < β we have the estimate k F ◦ K ω − K ω ◦ T ω k ρ,c , Γ ˜ β ≤ max (cid:0) k K ω k ρ,c , Γ β , k K ω k ρ,c , Γ (cid:1) , Φ( m )(119) where c = ( c , c − m ) , and Φ depends on F, c , c , β, ˜ β and lim | m |→∞ Φ( m ) = 0 . UASI-PERIODIC BREATHERS 61
Remark 9.9.
Note that the approximate torus K ω is in an space of slightly slowerdecay than the space of the invariant tori K ω and K ω since the decay estimate(119) involves the weight Γ ˜ β instead of the weight Γ β .It is important to emphasize that if we choose two solutions, fix a decay functionslower than that of the two solutions, any target smallness for the error of thecoupled solution can be accomplished by setting the translated solution far enough.In other words, we can choose all the parameters of the Lemma 9.8 and adjust allthe requirements by setting the solutions far apart. Remark 9.10.
Notice also that in Lemma 9.8, the approximate torus K ω isdefined on T rl × T l = T ( r +1) l . To make the notations coherent we embed the tori T rl and T l into T ( r +1) l identifying T rl with T rl × { } and T l with { } × T l respectively.Hence if θ = ( θ , θ ) ∈ T ( r +1) l , K ω ( θ ) = K ω ( θ ) and K ω ( θ ) = K ω ( θ ) . Remark 9.11.
For simplicity, we have stated Lemma 9.8 as joining together abreather around one site to an already constructed solution. It is possible (andperhaps more natural) to prove a lemma that asserts that given two solutions (eachcontaining oscillations around many sites) one can displace them and obtain a veryapproximate solution (in a slower decay space). We leave the precise formulationand the proof to the reader.Before proving Lemma 9.8 we establish a lemma with two technical estimates.Given c = ( c , , . . . , c ,r ) ∈ ( Z N ) r and c , m ∈ Z N we introduce the sets of indices I = (cid:8) i ∈ Z N | min k | i − c ,k | < | i + m − c | (cid:9) , I = Z N \ I , and the functions B ( β, ˜ β, m ) = sup i ∈I Γ β ( i + m − c )max k Γ ˜ β ( i − c ,k ) ,B ( β, ˜ β, m ) = sup i ∈I max k Γ β ( i − c ,k )Γ ˜ β ( i + m − c ) . Lemma 9.12. If < ˜ β < β we have lim | m |→∞ B ( β, ˜ β, m ) = lim | m |→∞ B ( β, ˜ β, m ) = 0 . Proof.
First we note that if i ∈ I then(120) | i + m − c | >
12 min k | c ,k + m − c | . Indeed, let k be such that | i − c ,k | = min k | i − c ,k | . Then | i + m − c | ≥ | c ,k + m − c | − | i − c ,k | > | c ,k + m − c | − | i + m − c | and hence | i + m − c | > | c ,k + m − c | ≥
12 min k | c ,k + m − c | . Moreover, if i ∈ I , by the monotonicity of Γ ˜ β we have max k Γ ˜ β ( i − c ,k ) > Γ ˜ β ( i + m − c ). Now Γ β ( i + m − c )max k Γ ˜ β ( i − c ,k ) < Γ β ( i + m − c )Γ ˜ β ( i + m − c ) . The bound (120) shows that when | m | → ∞ , | i + m − c | goes to infinity uniformlyin i ∈ I . Hence by the second property of the scale Γ β we obtain the first limit.The second limit is proved in an analogous way, checking first that if i ∈ I min k | i − c ,k | ≥
12 min k | c ,k + m − c | and using that if i ∈ I max k Γ β ( i − c ,k )Γ ˜ β ( i + m − c ) ≤ max k Γ β ( i − c ,k )max k Γ ˜ β ( i − c ,k ) . (cid:3) Proof of the coupling lemma Lemma 9.8.
We denote E ( K ) = F ◦ K − K ◦ T ω theerror of the invariance equation for the coupled breather.We are going to estimate the i -th component of E = E ( K ω ). Note that thetorus τ m K ω is localized around the site c − m. We distinguish two cases: either i ∈ I or i ∈ I . In the first case we write E i = F i ( K ω ) + Z h DF (cid:0) K ω + sτ m K ω (cid:1) τ m K ω i i ds − (cid:2) K ω ◦ T ω (cid:3) i − (cid:2) τ m K ω ◦ T ω (cid:3) i . We recall that θ = ( θ , θ ) ∈ T rl × T l . Since K ω does not depend on θ then F ( K ω ) = K ω ◦ T ω . Therefore k E i k ρ ≤ X j k F k C Γ( i − j ) k K ω k ρ,c , Γ Γ( j + m − c )+ k K ω k ρ,c , Γ Γ( i + m − c ) ≤ (cid:0) k F k C + 1 (cid:1) k K ω k ρ,c , Γ Γ( i + m − c ) . Similarly, if i ∈ I we expand F around τ m K ω and we obtain k E i k ρ ≤ (cid:16) − e − β k F k C + 1 (cid:17) k K ω k ρ,c , Γ β max k Γ β ( i − c ,k ) . We take ˜ β < β and we compute k E k ρ,c , Γ ˜ β = max (cid:16) sup i ∈I min (cid:0) min k Γ − β ( i − c ,k ) , Γ − β ( i + m − c ) (cid:1) k E i k ρ , sup i ∈I min (cid:0) min k Γ − β ( i − c ,k ) , Γ − β ( i + m − c ) (cid:1) k E i k ρ (cid:17) ≤ C max (cid:16) sup i ∈I min k Γ − β ( i − c ,k ) Γ( i + m − c ) , sup i ∈I Γ − β ( i + m − c ) max k Γ β ( i − c ,k ) (cid:17) = C max (cid:0) B (2 β , ˜ β, m ) , B ( β, ˜ β, m ) (cid:1) , UASI-PERIODIC BREATHERS 63 where C = (cid:16) − e − β k F k C + 1 (cid:17) max (cid:16) k K ω k ρ,c , Γ β , k K ω k ρ,c , Γ (cid:17) . (cid:3) Statement and proof of Lemma 9.13. Verifying the non-degeneracy conditionof the coupled solutions.
In this section, we verify the nondegeneracy conditionsprovided that ε is small enough and that K is sufficiently close to an uncoupledsolution with all the sites far enough apart. That is, we consider situations whenwe are close to the completely uncoupled solution.The result Lemma 9.13 will be clear because all the uncoupled solutions forthe uncoupled dynamics satisfy the non-degeneracy assumptions. The change ofthe non-degeneracy assumptions between this uncoupled case can be controlled byelementary perturbation theories. Thanks to the systematic use of our framework,we have perturbation theories which are uniform on the excited sites.Let c, ω be sequences of r sites and frequencies. We consider k ω i , parameteriza-tions of invariant tori w.r.t. f , the time-one map of just one site. We denote K ∗ = r X i =1 τ c i k ω i We note that F ◦ K ∗ = K ∗ ◦ T ω and that K ∗ is uniformly non degenerate.The hyperbolic splitting for F ◦ K ∗ is(121) Π s,c,u = ⊕ i ∈ Z N Π s,c,ui where Π s,c,ui is the splitting corresponding to the i torus. If i is an index in c , thenwe have Π ci = Id R l , Π si = 0, Π ui = 0. Otherwise, one gets Π ci = 0, and Π si , Π si are the projections corresponding to the stable and unstable directions at the fixedpoint.Notice also that the rl × rl matrix DK ⊤ DK is block diagonal. The diagonal has r l × l blocks { Dk ⊤ ω i Dk ω i } ri =1 . Lemma 9.13. (Hyperbolicity conditions)
Assume the hypothesis and the no-tation in Theorem 3.11. In particular, F ε is an analytic family of exact symplecticmaps in C ( B ) . Let K ∈ A ρ,c, ˜Γ with ˜Γ < Γ β for β < β .Assume that ε , η ≡ k K − K ∗ k ρ,c, ˜Γ are smaller than a number that is in-dependent of c and of Γ – it depends only on k F k C ( B ) , k ∂ ε F k C ( B ×{| ε |≤ ε ∗ } ) , k ∂ ε F k C ( B ×{| ε |≤ ε ∗ } ) and the hyperbolicity constants of the uncoupled splitting.Then, K and F ε satisfy the non-degeneracy conditions in Definition 3.1 withuniform constants.Proof. We make the elementary remark(122) DF ε ◦ K = DF ◦ K ∗ + (cid:0) DF ◦ K − DF ◦ K ∗ (cid:1) + (cid:0) DF ε ◦ K − DF ◦ K (cid:1) and we will control the terms in parenthesis.By the estimates in composition in Section A.5, we obtain that: k DF ◦ K − DF ◦ K ∗ k ρ,c, ˜Γ ≤ C k K − K ∗ k ρ,c, ˜Γ k DF ε ◦ K − DF ◦ K k ρ,c, ˜Γ ≤ C | ε | so we obtain that k DF ε ◦ K − DF ◦ K ∗ k ρ,c, ˜Γ is small. The splitting indicated in (121) is invariant for DF . Hence, it is approximatelyinvariant for DF ε ◦ K and this satisfies the conditions for approximately invariantsplittings Definition 3.2.We note that Proposition 4.2 ensures that, if ε and η are small enough, there isan invariant splitting satisfying Definition 3.1. (cid:3) Statement and proof of Lemma 9.14. Verifying the non-degeneracy assump-tions of coupled solutions.
Lemma 9.14. (Twist conditions)
Assume that K , K are embeddings in A ρ,c , Γ β , A ρ,c , Γ β , resp. both c , c being finite sequences, and Γ β as before. Assume that K , K satisfy the non-degeneracy conditions in Definition 3.4.Then, for m sufficiently large, ˜ K ( θ , θ ) = K ( θ ) + τ m K ( θ ) satisfies the non-degeneracy assumptions in Definition 3.4. Furthermore, the non-degeneracy constants of ˜ K can be made as close to desired to the constants verifiedboth by K , K if we choose | m | large enough. We will be using the notation that n is the number of sites in c and that θ stands for all the n × l variables corresponding to all the sites in c . Similarly for K . Proof.
We introduce the notation that Φ( m ) stands for any quantity (vector, ma-trix, function, etc. ) which can be made arbitrarily small by making m large.We start by estimating the non-degeneracy condition of the embedding.We see that the l ( n + n ) × l ( n + n ) matrix D ˜ K ⊤ D ˜ K splits naturally intoblocks depending on whether we take derivatives with respect to variables in θ orin θ :(123) D ˜ K ⊤ D ˜ K = (cid:18) D θ K ⊤ D θ K D θ K ⊤ τ m D θ K τ m D θ K ⊤ D θ K D θ τ m K ⊤ D θ τ m K (cid:19) . Since D θ τ m K ⊤ D θ τ m K = D θ K ⊤ D θ K we see that the diagonal elements of D ˜ K ⊤ D ˜ K are precisely those of the uncoupledsystem and are therefore invertible.We will show that the non-diagonal elements in (123) can be made arbitrarilysmall by choosing m large enough. Then, it will follow that D ˜ K ⊤ D ˜ K is invertibleand that(124) ˜ N = ( D ˜ K ⊤ D ˜ K ) − = (cid:18) ( D θ K ⊤ D θ K ) −
00 ( D θ K ⊤ D θ K ) − (cid:19) + Φ( m ) = (cid:18) N N (cid:19) + Φ( m ) . We estimate the off-diagonal elements of (123). We observe that, we can estimatethe entries of n l × n l upper right block as follows: UASI-PERIODIC BREATHERS 65 (cid:12)(cid:12)(cid:12)(cid:0) D θ K ⊤ τ m D θ K (cid:1) p,q (cid:12)(cid:12)(cid:12) ≤ X i (cid:12)(cid:12)(cid:12) ∂K ,i ∂θ ,p ∂K ,i + m ∂ θ ,q (cid:12)(cid:12)(cid:12) ≤ k DK k ρ,c , Γ β k DK k ρ,c , Γ X i max k Γ β ( i − c ,k ) max l Γ( i + m − c ,l ) ≤ − e − β k DK k ρ,c , Γ β k DK k ρ,c , Γ max k,l Γ β ( c ,l − m − c ,k ) . Since the block is finite dimensional, making the previous elements small enoughmakes its norm small.Estimates for the lower left block are obtained just noticing that it is the trans-posed of the upper right one.Now, we turn to estimating the twist condition. Again the strategy is very similarto the one used in checking the non-degeneracy condition. We just check that thematrix we need to invert is arbitrarily close (by taking | m | large enough) to amatrix which is block diagonal and whose blocks correspond to the non-degeneracyconditions of each of the uncoupled solutions.We proceed to estimate systematically all the ingredients of A defined in (20).We have first ˜ P ( θ ) = (cid:0) D θ K , D θ [ τ m K ] (cid:1) (cid:18) N N (cid:19) + Φ( m )= ( P , P ) + Φ( m ) . Since N and N are finite dimensional matrices, P and P are also in A ρ,c , Γ and A ρ,c , Γ , respectively.Using that J ∞ is uncoupled and constant for the models (26) we are consideringnow, we can write:( DF ( J c ) − ) (cid:0) ˜ K ( θ ) (cid:1) P ( θ )= ( DF ( J c ) − ) (cid:0) K ( θ ) (cid:1) P ( θ )+ Z ( D F ( J c ) − ) (cid:0) K ( θ ) + sτ m K ( θ ) (cid:1)(cid:0) τ m K ( θ ) , P ( θ ) (cid:1) ds (125)( DF ( J c ) − ) (cid:0) ˜ K ( θ ) (cid:1) P ( θ )= ( DF ( J c ) − ) (cid:0) τ m K ( θ ) (cid:1) P ( θ )+ Z ( D F ( J c )) (cid:0) sK ( θ ) + τ m K ( θ ) (cid:1) (cid:0) K ( θ ) , P ( θ ) (cid:1) ds . (126) We denote T and T the integral terms in (125) and (126) respectively. Webound from above the i -th component of T by k ( J c ) − k X j,n (cid:12)(cid:12)(cid:12)(cid:12) ∂ F i ∂x j ∂x n (cid:12)(cid:12)(cid:12)(cid:12) ρ | ( K ) j + m | ρ | ( P ) n | ρ ≤ k ( J c ) − k k F k C k K k ρ k P k ρ,c , Γ β × X j,n min (cid:0) Γ( i − j ) , Γ( i − n ) (cid:1) max k,l Γ( j + m − c ,l ) Γ β ( n − c ,k ) . The last sum is bounded by41 − e − β max k,l Γ β ( i + m − c ,l ) · Γ β ( i − c ,k ) ≤ Φ( m ) . Indeed, let J ( i ) = { j, n ∈ Z N | | i − j | ≤ | i − n |} and J ( i ) = Z N \ J ( i ). Usingthat Γ( i ) ≤ a Γ β ( i )Γ β ( i ) the previous sum is bounded by X j,n ∈J ( i ) Γ( i − j ) max l Γ( j + m − c ,l ) max k Γ β ( n − c ,k )+ X j,n ∈J ( i ) Γ( i − n ) max l Γ( j + m − c ,l ) max k Γ β ( n − c ,k ) ≤ a X j ∈ Z N Γ β ( i − j ) max l Γ( j + m − c ,l ) X n ∈ Z N Γ β ( i − n ) max k Γ β ( n − c ,k ) . Analogously T is bounded by k ( J c ) − k k F k C k K k ρ,c , Γ β k P k ρ − e − β max k,l Γ β ( i + m − c ,l ) Γ β ( i − c ,k ) . Note that P ⊤ ( θ + ω ) T ( θ ) and P ⊤ ( θ + ω ) T ( θ ) are bounded by C max k,l Γ β ( c ,k + m − c ,l ), where C depends on k ( J c ) − k , k F k C , k K k ρ,c , Γ β , k K k ρ,c , Γ , k P k ρ,c , Γ β and k P k ρ,c , Γ .Also note that (cid:12)(cid:12)(cid:2) P ( θ + ω ) ⊤ ( J c ) − ( K ( θ )) P ( θ ) (cid:3) i (cid:12)(cid:12) ≤ k ( J c ) − k k P k ρ,c , Γ β k P k ρ,c , Γ × X i ∈ Z N max k,l Γ β ( i − c ,k ) Γ( i + m − c ,l ) ≤ C max k.l Γ β ( c ,l − m − c ,k ) . Now we consider the terms P ⊤ ( θ + ω ) ( DF ˜( J c ) − ) (cid:0) ˜ K ( θ ) (cid:1) P ( θ ) and P ⊤ ( θ + ω ) ( DF ˜( J c ) − ) (cid:0) ˜ K ( θ ) (cid:1) P ( θ ). We evaluate the first one, the other being analogous.Given n ∈ { , . . . , r } , (cid:12)(cid:12)(cid:2) P ⊤ ( θ + ω ) ( DF J c ) (cid:0) ˜ K ( θ ) (cid:1) P ( θ ) (cid:3) n (cid:12)(cid:12) ≤ X i,j (cid:12)(cid:12) ( P ) i,n (cid:12)(cid:12) ρ k ( J c ) − k (cid:13)(cid:13)(cid:13) ∂F i ∂x j (cid:13)(cid:13)(cid:13) ρ | ( P ) j | ρ ≤ k F k C k ( J c ) − k k P k ρ,c, Γ β k P k ρ,c , Γ X i,j max k,l Γ β ( i − c ,k ) Γ( i − j ) Γ( j + m − c ,l ) ≤ C max k Γ β ( c ,k + m − c ,l ) . UASI-PERIODIC BREATHERS 67
With all these previous estimates we can write: A = (cid:2) ( P , P ) ◦ T ω + Φ( m ) (cid:3) ⊤ ( DF J c ) ( ˜ K ) (cid:2) ( P , P ) + Φ( m ) (cid:3) − J c P ◦ T ω = P ⊤ ◦ T ω (cid:2) ( DF ( J c ) − ) ( K ) P (cid:3) − ( J c ) − P ◦ T ω P ⊤ ◦ T ω (cid:2) ( DF J c ) ( K ) P (cid:3) − ( J c ) − P ◦ T ω ! + Φ( m ) = (cid:18) A A (cid:19) + Φ( m )which shows that it is invertible if | m | is big enough and that the norm of theinverse of A can be bounded from above by max( | A − | , | A − | ) + ˜Φ m . (cid:3) The estimates about the non-degeneracy with respect to parameters in the con-struction are automatic since, in the construction in Section C, which is the one weuse here, the matrix Q is the identity, whose norm is bounded by 1 independentlyof the number of sites considered and independently of the K considered.9.4. Adding oscillating sites inductively.
Recall that we are assuming that ε ≤ ε ∗ and that we have a set Ξ ( ε ∗ ) ⊂ D ( ν , κ ) ⊂ R l of positive measure suchthat, for all ω ∈ Ξ ( ε ∗ ), the system (26) has a breather of frequency ω in A ρ, { } , Γ β .The non-degeneracy and hyperbolicity constants of all these solutions are uniformlybounded.The remaining part to be shown is that given a sequence ω ∈ D ∪ Ξ ∗ ( ε ∗ ) ∞ , wecan find a sequence of tori parameterized by K ω ( n ) ∈ A ρ n ,c ( n ) , Γ βn for a suitablesequence of centers c ( n ) . Here we have that ω ( n ) = ( ω , . . . , ω n )is the sequence of truncations of ω and ρ n , β n are strictly decreasing sequences sothat ρ n → ρ ∞ > β n → β ∞ > ρ n , β n , the infinite sequence of centers c ( n ) and the embeddings K ω ( n ) .The choices of ρ n , β n are almost irrelevant for our purposes, so we choose themright away. For example we take 0 < ρ ∞ < ρ , 0 < β ∞ < β and ρ n = ρ ∞ +2 − n ( ρ − ρ ∞ ), β n = β ∞ + 2 − n ( β − β ∞ ),So that now, our only task is to choose a sequence of sites c ( n ) (without loss ofgenerality, we will assume c = 0), such that, recursively, we have that taking c n +1 far apart from the previous sites, K ω ( n ) + τ − c n +1 K ω n +1 is a very approximate solu-tion of the invariance equation which, furthermore, satisfies uniform hyperbolicityand non-degeneracy conditions. Then, an application of Theorem 3.6 will producea true solution K ω ( n +1) .Of course, we will have to recover the inductive hypothesis we have made to con-struct this sequence. We will show that, we can ensure that k K ω ( n ) − K ∗ k ρ n ,c ( n ) , Γ βn ≤ η/ η > K ∗ n = K ω + τ − c K ω + · · · τ − c n K ω n .After this sequence of tori with increasing number of frequencies is produced,we will have to study the limit of the sequence and show that it solves the invari-ance equation (this will be accomplished in Section 9.5). Note that, since each stepchanges the number of centers, the convergence of the embeddings cannot be uni-form (even in a space of decay functions). Nevertheless, we will show that there is coordinatewise convergence and that this is enough to show that the limit satisfiesthe invariance equation.We note that the existence of the sequence and the study of the limit will beaccomplished because if we place the centers very far apart from the previouslyplaced ones, we can obtain that the error is small enough to beat the smallnessrequirements of Theorem 3.6, to ensure that the non-degeneracy and hyperbolicityconstants deteriorate an arbitrarily small amount and to ensure the passage to thelimit, so that, by recursively assuming that the new center is far away from allthe previously placed ones, we can ensure any smallness conditions we wish on theerror, on the increment of the distance from the uncoupled solution and on thedeterioration of the non-degeneracy and hyperbolicity constants.We start with K ω and K ω localized at the node c = 0 and c respectively andwe take | c | big enough so that˜ K = K ω + τ − c K ω is a sufficiently approximate solution of F ◦ K − K ◦ T ω (2) = 0 and satisfies both thespectral and the twist non-degeneracy conditions. Then Theorem 3.6 provides theexistence of a true invariant torus K ω (2) ∈ A ρ ,c (2) , Γ β such that it is non-degenerateand e = k K ω (2) − ˜ K k ρ ,c (2) , Γ β is small. Actually it can be made as small as we want by taking | c | sufficientlybig. Remembering that ( ω , ω ) is Diophantine (and chosen from the start of theprocedure), we see that Theorem 3.6 guarantees that, if we make the initial errorsmall enough, we can produce a solution K ω (2) of the invariance equation withfrequency ω (2) .In the n + 1 step of the process we assume we have the torus K ω ( n ) ∈ A ρ n ,c ( n ) , Γ βn localized around the nodes c ( n ) = ( c , . . . , c n ), which is non-resonant, that is K ω ( n ) ∈ N D loc ( ρ n , Γ β n )We consider the parameterization˜ K ( θ ) − K ω ( n ) ( θ ) + τ m n +1 K ω n +1 ( θ ) , θ = ( θ , θ ) ∈ T nl × T l , as an approximation for the new torus, which we will denote K ω ( n +1) , with some m n +1 ∈ Z N .By the coupling lemma (Lemma 9.8) if we take a suitable m n +1 big enough weobtain E n +1 = F ◦ ˜ K − ˜ K ◦ T ω ( n +1) as small as we want. In particular we take 0 < δ n +1 < min (cid:0) , ρ n / , ( ρ n +1 − ρ n ) / (cid:1) and we require Cκ n +1 δ − ν n +1 n +1 k E n +1 k ρ n +1 ,c ( n +1) , Γ βn +1 ≤ , and Cκ n +1 δ − ν n +1 n +1 k E n +1 k ρ n +1 ,c ( n +1) , Γ βn +1 ≤ e n − . We denote c n +1 = − m n +1 . Then Theorem 3.6 provides a true invariant torus K ω ( n +1) ∈ A ρ n +1 ,c ( n +1) , Γ βn +1 , non-degenerate and satisfying the estimate(127) k K ω ( n +1) − ˜ K k ρ n +1 ,c ( n +1) , Γ βn +1 ≤ e n − . UASI-PERIODIC BREATHERS 69
Passage to the limit (Part B of Theorem 3.11).
The issue now is tostudy the limit n → ∞ . Thanks to our weighted spaces and the fact that thesolutions we construct have bumps whose distance from each other tends to infinityfast enough, we can prove the following Lemma 9.15 which establishes that for anybounded sets in the lattice, the trajectories of the particles in this set convergeuniformly. Lemma 9.15.
The sequence (cid:8) K ω ( n ) (cid:9) n ≥ converges component-wise and uniformlyon every compact set of ( T l ) N . We denote K ω the limit obtained in this sense.Furthermore, each component of K ω is analytic from ( T l ) N into M . Remark 9.16.
Here, by analytic on the infinite dimensional torus ( T l ) N , we mean K ω writes component-wise( K ω ) i ( θ ) = X n ≥ ( H ( n ) ) i ( θ , . . . , θ n ) , where ( H ( n ) ) i ( θ , . . . , θ n ) are analytic in the usual sense on ( T l ) n and moreover wehave X n ≥ k ( H ( n ) ) i k ρ n < ∞ , where D ρ n ⊃ ( T l ) n . Proof.
We represent K ω ( θ ) aslim n →∞ K ω ( n ) ( θ )= K ω ( θ ) + ∞ X n =1 (cid:2) K ω ( n +1) ( θ , . . . , θ n +1 ) − K ω ( n ) ( θ , . . . , θ n ) (cid:3) . (128)We fix i ∈ Z N and we estimate the i -th component of K ω ( n +1) − K ω ( n ) . By thetriangle inequality (cid:12)(cid:12)(cid:2) K ω ( n +1) − K ω ( n ) (cid:3) i (cid:12)(cid:12) ρ n +1 ≤ (cid:12)(cid:12)(cid:2) K ω ( n +1) − K ω ( n ) − τ m n +1 K ω n +1 (cid:3) i (cid:12)(cid:12) ρ n +1 + (cid:12)(cid:12) τ m n +1 K ω n +1 (cid:3) i (cid:12)(cid:12) ρ n +1 . (129)The first term in the right-hand side of (129) is bounded by e n − max ≤ k ≤ n +1 Γ β n +1 ( i − c k )and the second one is bounded by (see (127)) (cid:12)(cid:12)(cid:2) τ m n +1 K ω n +1 ] i (cid:12)(cid:12) ρ = (cid:12)(cid:12)(cid:2) K ω n +1 ] i + m n +1 | ρ ≤ k K ω n +1 k ρ , , Γ Γ( i + m n +1 ) . This implies that the i -th component of the sum in (128) is bounded by(130) ∞ X n =1 e n − + ∞ X n =1 k K ω n +1 k ρ , , Γ Γ( i + m n +1 ) . Therefore the i -th component of (128) converges uniformly on compact sets of( T l ) N and K ω is analytic in the sense of Remark 9.16. (cid:3) The following result proves that K ω is a solution of the invariance equation andtherefore is an almost-periodic function of the initial system. Lemma 9.17.
The limit function K ω satisfies F ◦ K ω = K ω ◦ T ω component-wise.Proof. For every n ∈ N one has(131) F ◦ K ω ( n ) = K ω ( n ) ◦ T ω ( n ) . We fix a component i ∈ Z N . The passage to the limit in the right-hand side of(131) is immediate. For the left-hand side we take n such that | c n | > | i | for n > n .Then for n > n we have(132) (cid:12)(cid:12) F i ◦ K ω ( n ) − F i ◦ K ω (cid:12)(cid:12) ρ ∞ ≤ X j (cid:12)(cid:12)(cid:12) ∂F i ∂x j (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:2) K ω ( n ) − K ω (cid:3) j (cid:12)(cid:12) ρ ∞ . We estimate (cid:12)(cid:12)(cid:2) K ω ( n ) − K ω (cid:3) j (cid:12)(cid:12) ρ ∞ ≤ ∞ X p = n (cid:12)(cid:12)(cid:2) K ω ( p ) − K ω ( p +1) (cid:3) j (cid:12)(cid:12) ρ p +1 ≤ ∞ X p = n h e p − max ≤ k ≤ p +1 Γ β p +1 ( j − c k ) + (cid:12)(cid:12)(cid:2) τ − c p +1 K ω p +1 (cid:3) j (cid:12)(cid:12) ρ p +1 i ≤ ∞ X p = n h e p − max ≤ k ≤ p +1 Γ β ∞ ( j − c k ) + k K ω p +1 k ρ , , Γ Γ( j − c p +1 ) i . Then (132) is bounded by ∞ X p = n k F k C h e p − X j max ≤ k ≤ p +1 Γ β ∞ ( j − c k )Γ( i − j )+ k K ω p +1 k ρ , , Γ X j Γ( i − j )Γ( j − c p +1 ) i ≤ k F k C (cid:16) ∞ X p = n e p − − e − β ∞ + ∞ X p = n k K ω p +1 k ρ , , Γ Γ( i − c p +1 ) (cid:17) ≤ k F k C (cid:16) − e − β ∞ e n − + C K − e − β max p ≥ n Γ( i − c p +1 ) (cid:17) , which leads to the desired result. (cid:3) Acknowledgements
The work of E.F. has been partially supported by the spanish grant MTM-16425and the catalan grant 2009SGR67. The work of R.L. has been supported by NSFgrants. R.L. also thanks the hospitality of Univ. Polit. Catalunya and Centre deRecerca Matem`atica which enabled face to face collaboration. The work of Y.S.has been supported byt the ANR project ”KAMFAIBLE”.We thank very specially Prof. P. Mart´ın for very illuminating comments andsuggestions. We also thank many suggestions and encouragement from Profs. M.Jiang, X. Li, L. Sadun, E. Valdinoci, Drs. R. Calleja, T. Blass, D. Blazevski, X. Su.D. Blazevski gave a detailed reading to the paper which improved the exposition.
UASI-PERIODIC BREATHERS 71
Appendix A. Appendix: Decay functions
This appendix is devoted to the properties of spaces of decay functions.A.1.
Linear and k -linear maps over ℓ ∞ ( Z N ). We are going to consider linearmaps from ℓ ∞ ( Z N ) into itself such that(133) lim m →∞ sup u ∈ ℓ ∞ , | u |≤ uj =0 , | j − i |≤ m ( Au ) i = 0 , ∀ i ∈ Z M . The condition (133) is equivalent to the fact that A can be written in the form(134) ( Au ) i = X j ∈ Z N A ij u j , i ∈ Z N , where A ij are linear maps, u ∈ ℓ ∞ ( Z N ) and the series are convergent.We denote by L ( ℓ ∞ ( Z N )) the space of linear maps that satisfy (134). Thisis a non-trivial assumption, since ℓ ∞ ( Z N ) is not reflexive. Furthermore, the space L ( ℓ ∞ ( Z N )) is a strict subspace of the space of bounded linear operators on ℓ ∞ ( Z N ).The second assumption we will make is that there exists C > | A ij | ≤ C Γ( i − j ) , for all ( i, j ) ∈ ( Z N ) .In this case,(135) X j ∈ Z N | A ij u j | ≤ X j ∈ Z N C Γ( i − j ) | u j | ≤ X j ∈ Z N C Γ( i − j ) k u k ∞ ≤ C k u k ∞ . Then we define L Γ ( ℓ ∞ ( Z N )) = n A ∈ L ( ℓ ∞ ( Z N )) | sup i,j ∈ Z N Γ( i − j ) − | A ij | < ∞ o and we endow it with the norm(136) k A k Γ = sup i,j ∈ Z N Γ( i − j ) − | A ij | . The following Lemma has a simple proof that can be found in [FdlLM11a]. Themost subtle point is that we need to verify that the linear operators are given bythe matrix.
Lemma A.1.
The space L Γ ( ℓ ∞ ( Z N )) is a Banach space. From the definition of the norm of A and the inequalities (135) we deduce that k Au k ∞ ≤ k A k Γ k u k ∞ for all u ∈ ℓ ∞ ( Z N ). Remark A.2.
The previous definition will also be used for matrices. If one con-siders a finite set of indexes I × J ⊂ Z N × Z N , we will use k A k Γ = sup i ∈ I,j ∈ J | A ij | Γ − ( i − j ) . This is just a way to say that the set of tensors of order 2 with finite indexes arenaturally embedded into the set of tensors of order 2 on Z N by just setting all theremaining values to 0. Similarly to the previous definition, we define L k ( ℓ ∞ ( Z N )) as the space of k -linear maps on ℓ ∞ ( Z N ) which are represented by a multilinear matrix.(137) B ( u , . . . , u k ) i = X ( i ,...,i k ) ∈ ( Z N ) k B i,i ,...,i k u i . . . u ki k , where i, i , . . . , i k ∈ Z N , ( u , . . . , u k ) ∈ ( ℓ ∞ ( Z N )) k and B i,i ,...,i k ∈ L k ( M, M ).Given a decay function Γ, we define L k Γ ( ℓ ∞ ( Z N )) as the space of maps in L k ( ℓ ∞ ( Z N )) such that | B i,i ,...,i k | ≤ C min(Γ( i − i ) , . . . , Γ( i − i k )) , for some C ≥ k B k Γ = sup i,i ,...,i k ∈ Z N | B i,i ,...,i k | max(Γ − ( i − i ) , . . . , Γ − ( i − i k )) . We have the following lemma (see [FdlLM11a]).
Lemma A.3.
The space L k Γ ( ℓ ∞ ( Z N )) is a Banach space. The next result provides the Banach algebra property of L Γ ( ℓ ∞ ( Z N )). Thisproperty will be crucial for our estimates. It makes it possible to work with infinitedimensional systems in a way which is not very different from the finite dimensionalcase. Lemma A.4. If A, B ∈ L Γ ( ℓ ∞ ( Z N )) then AB ∈ L Γ ( ℓ ∞ ( Z N )) and we have theestimate k AB k Γ ≤ k A k Γ k B k Γ . Proof.
It is easy to verify that if A and B can be represented by matrices, so is theproduct. Since A, B ∈ L Γ ( ℓ ∞ ( Z N )), we have | A ij | ≤ Γ( i − j ) k A k Γ , | B jk | ≤ Γ( j − k ) k B k Γ . Therefore, we have: k AB k Γ = sup n,m ∈ Z N | ( AB ) nm | Γ − ( n − m ) ≤ sup n,m ∈ Z N X k ∈ Z N | A nk | | B km | Γ − ( n − m ) ≤ k A k Γ k B k Γ sup n,m ∈ Z N X k ∈ Z N Γ( n − k )Γ( k − m )Γ − ( n − m ) . Using property (2) of Definition 2.2 we obtain the desired result. (cid:3)
By induction on k the same result holds for k − linear maps. See [FdlLM11a]. Lemma A.5.
Let A ∈ L k Γ ( ℓ ∞ ( Z N )) and B j ∈ L n j Γ ( ℓ ∞ ( Z N )) for ≤ j ≤ k . Thenthe composition AB . . . B k ∈ L n + ··· + n k Γ ( ℓ ∞ ( Z N )) and k AB . . . B k k Γ ≤ k A k Γ k B k Γ . . . k B k k Γ . UASI-PERIODIC BREATHERS 73
A.2.
Spaces of differentiable and analytic functions on lattices.
We nowdefine the space of C r functions, based on the previous weighted norms. Given anopen set B ⊂ M we define C ( B ) = (cid:26) F : B → M | F ∈ C ( B ) , DF ( x ) ∈ L Γ ( ℓ ∞ ( Z N )) , sup x ∈B k F ( x ) k < ∞ , sup x ∈B k DF ( x ) k Γ < ∞ (cid:27) . We endow C ( B ) with the norm k F k C = max (cid:0) sup x ∈B k F ( x ) k , sup x ∈B k DF ( x ) k Γ (cid:1) . In the definition of C ( B ), when M is complex, the deriva-tive has to be understood as complex derivative.We emphasize that the definition of C includes that DF ( x ) ∈ L Γ ( ℓ ∞ ( Z N )) and,in particular, that the derivative of the function is given by the matrix of its partialderivatives. Concretely if F ∈ C then we have the following formula: DF i ( x ) v = X j ∈ Z N ∂F i ∂x j ( x ) v j , where x j is the variable in M , the j-th component of M .Now we proceed to define the space of finite differentiable maps. In [JdlL00,FdlLM11a], one can find definitions for H¨older spaces, which is useful for otherapplications (such as thermodynamic formalism). Definition A.6.
Given B an open subset of M and r ∈ N C r Γ ( B ) = (cid:26) F : B → M | F ∈ C r ( B ) , D j F ( x ) ∈ C ( B ) , ≤ j ≤ r − (cid:27) . A.3.
Spaces of embeddings from C l to M with decay properties. In thissection, we will consider embeddings from finite dimensional tori into the phasespace. We should think of these embeddings as describing some oscillations centeredaround some sites.We define the complex strip D ρ = (cid:8) z ∈ C l / Z l | | Im z i | < ρ, i = 1 , . . . , l (cid:9) . Let R ≥ c ∈ ( Z N ) R , i.e. c = ( c , . . . , c R ) . Given f : D ρ → M , we introduce the following quantity k f k ρ,c, Γ = sup i ∈ Z N min j =1 ,...,R Γ − ( i − c j ) k f i k ρ , where k f i k ρ = sup θ ∈ D ρ | f i ( θ ) | . Definition A.7.
We denote (138) A ρ,c, Γ = (cid:26) f : D ρ → M | f ∈ C ( D ρ ) , f analytic in D ρ , k f k ρ,c, Γ < ∞ (cid:27) . This space, with the norm k · k ρ,c, Γ , is a Banach space.The parameter c is the location of the centers of the oscillations of the map f : T l → M . As the argument of f changes, the range of the embedding, willoscillate mainly on the sites in neighborhoods of c , . . . , c R .The next result is a version of the Cauchy estimates in our context. Lemma A.8.
Let f : D ρ → M be an analytic function. Then for all δ ∈ (0 , ρ ) ,the following holds k D θ f k ρ − δ,c, Γ ≤ lδ − k f k ρ,c, Γ . Proof.
Consider the components f i of f , for i ∈ Z N . Each f i maps D ρ into M andwe have the standard Cauchy estimates for k = 1 , . . . , l k ∂ θ k f i k ρ − δ ≤ δ − k f i k ρ . The result then follows by just multiplying this inequality by Γ − ( i − c j ) whichis positive, summing with respect to k and taking the supremum for i and theminimum for j . (cid:3) Remark A.9.
Note that in the previous Cauchy estimate the bound depends linearly on the dimension of the torus l . This will be important when we considerthe limit of many dimensions.On the other hand, taking the supremum over components, makes it clear thatthe bounds are independent of the dimension of the range. In particular, we candiscuss mappings into infinite dimensions. Note that, if we had taken another norm,the constants would have depended on the dimension of the range.If we consider a map A from D ρ into the set of linear maps L Γ ( ℓ ∞ ( Z N )), theassociated norm is k A k ρ, Γ = sup i,j ∈ Z N sup θ ∈ D ρ Γ − ( i − j ) | A ij ( θ ) | = sup θ ∈ D ρ k A ( θ ) k Γ . Remark A.10.
The previous definition of the space of analytic maps in the stripcan be generalized to any open subset B of the complex extended manifold M C .We define A B ,c, Γ = (cid:26) F : B → M | F ∈ C ( B ) , F analytic in B , k F k B ,c, Γ < ∞ (cid:27) , where k F k B ,c, Γ = sup i ∈ Z N min j =1 ,...,R Γ − ( i − c j ) sup z ∈B | F i ( z ) | . Note that the space A B ,c, Γ is a closed subspace of C ( B ) for the C topology.Here, the derivatives are understood as complex derivatives.A.4. Spaces of localized vectors.
The space of localized vectors ℓ ∞ c, Γ definedbelow, plays a role as the space of infinitesimal deformation of the space of localizedembeddings defined above. We also isolate a class of linear operators L c, Γ whichsend ℓ ∞ into ℓ ∞ c, Γ .The key property (Proposition A.12) is that L c, Γ is an ideal of the Banachalgebra L Γ . It will be important for future developments that the bounds obtainedare independent of the parameter c . This will be an easy consequence of the Banachalgebra properties of the decay functions. Definition A.11.
Given a decay function Γ and a finite (or infinite) collection ofsites c = { c k } k ∈K ⊂ Z N with K ⊂ N we define (139) k v k c, Γ = sup i ∈ Z N inf k ∈K | v i | Γ( i − c k ) − . We denote ℓ ∞ c, Γ = { v ∈ ( R l ) Z N | k v k c, Γ < ∞} . UASI-PERIODIC BREATHERS 75
We denote by L c, Γ the space of linear operators on ℓ ∞ such that ( Av ) i = X j ∈ Z N A ij v j , | A ij | ≤ C min(sup k ∈K Γ( i − c k ) , Γ( i − j )) . We denote by k A k c, Γ the best constant C above, i.e. k A k c, Γ = max (cid:16) sup i,j ∈ Z N | A ij | Γ − ( i − j ) , sup i,j ∈ Z N | A ij | Γ − ( i − j ) , sup i,j ∈ Z N inf k ∈K | A ij | Γ − ( i − c k ) (cid:17) . Note that we use k · k c, Γ both for the norm in a linear space and the norm inthe space of operators. This will not cause any confusion since in this space we willnot use the norm of operators from ℓ ∞ c, Γ to itself.The following is an easy exercise. Proposition A.12.
We have the following results: • a) The space ℓ ∞ c, Γ endowed with k · k c, Γ is a Banach space.The embedding ℓ ∞ c, Γ ֒ → ℓ ∞ is continuous. ℓ ∞ c, Γ is a closed subspace of ℓ ∞ . • b) The space L c, Γ endowed with k · k c, Γ is a Banach space.The embedding L c, Γ → L Γ is continuous. L c, Γ is a closed subspace of L Γ . • c) Ideal character: If A ∈ L c, Γ , B ∈ L Γ , we have AB ∈ L c, Γ , k AB k c, Γ ≤ k A k c, Γ k B k Γ ,BA ∈ L c, Γ , k BA k c, Γ ≤ k A k c, Γ k B k Γ (140) As a consequence of the above, if A ∈ L c, Γ , B ∈ L c, Γ , we have (141) k AB k c, Γ , k BA k c, Γ ≤ k A k c, Γ k B k c, Γ . Note also that if x ∈ M and A ∈ L c, Γ then Ax ∈ ℓ ∞ c, Γ . A.5.
Regularity of the composition operators.
The following propositions(see [JdlL00]) establish the regularity of composition operators and provide esti-mates for the composition.
Proposition A.13.
The mapping defined by C ( G, h ) = G ◦ h is locally Lipschitz when considered as C : C × C C , and we have the estimate kC ( G, h + ¯ h ) − C ( G, h ) k C ≤ k G k C k ¯ h k C (1 + k h k C ) , Furthermore, when considered as a mapping from C × C into C , we have theformula (142) D C ( G, h )∆ = ( DG ◦ h ) ∆ . We will also need the following estimate on the composition operator.
Lemma A.14.
Consider two functions
G, h ∈ C . Then we have k G ◦ h k C ≤ C max( k G k C , k G k C k h k C ) . Proof.
Clearly, we have k G ◦ h k C ≤ k G k C . We now estimate the norm of the derivatives: D j ( G ◦ h ) i = X k ∈ Z N D k G i ◦ hD j h k . This leads | D j ( G ◦ h ) i | ≤ X k ∈ Z N Γ( i − k )Γ( k − j ) k G k C k h k C ≤ Γ( i − j ) k G k C k h k C . This ends the proof. (cid:3)
The next lemma gives an estimate on the composition of a mapping defined onthe manifold and an embedding.
Lemma A.15.
Let
B ⊂ M be a star-like from the origin open set such that ∈ B .Suppose that F ∈ C ( B ) and is analytic. Let K : D ρ → M belong to A ρ,c, Γ , with c ∈ ( Z N ) R and such that K ( D ρ ) ⊂ B . (1) Assume that F (0) = 0 . Then F ◦ K ∈ A ρ,c, Γ , and (143) k F ◦ K k ρ,c, Γ ≤ R k F k C k K k ρ,c, Γ . (2) Let
J ⊂ Z N be a finite set of indexes and assume that F j (0) = 0 for j ∈ Z N − J . Then F ◦ K ∈ A ρ,c, Γ and (144) k F ◦ K k ρ,c, Γ ≤ k F k C (cid:0) C + R k K k ρ,c, Γ (cid:1) , where C depends on c and J .In both cases (145) k D ( F ◦ K ) k ρ,c, Γ ≤ R k F k C k DK k ρ,c, Γ . Proof.
By Definition 2.6 of analytic functions, we have that F ◦ K is analytic. Toestimate k F ◦ K k ρ,c, Γ take i ∈ Z N and j ∈ { , . . . , R } . If F i (0) = 0 we can write F i (cid:0) K ( θ ) (cid:1) = Z DF i (cid:0) sK ( θ ) (cid:1) K ( θ ) ds = Z X p ∈ Z N ∂F i ∂x p (cid:0) sK ( θ ) (cid:1) K p ( θ ) ds . Taking norms | F i (cid:0) K ( θ ) (cid:1) | ≤ X p ∈ Z N k DF k Γ Γ( i − p ) k K k ρ,c, Γ max ≤ j ≤ R Γ( p − c j ) ≤ R k F k C k K k ρ,c, Γ max ≤ j ≤ R Γ( i − c j )and then, if F (0) = 0, (143) follows.In the second case F m (0) = 0 for m ∈ J , we have | F m ( K ( θ )) | ≤ | F m (0) | + R k F k C k K k ρ,c, Γ max ≤ j ≤ R Γ( i − c j ) UASI-PERIODIC BREATHERS 77 and then we obtain (144) with C = max m ∈J min ≤ j ≤ R Γ − ( m − c j ) . The estimate (145) follows from the chain rule and the definitions of the norms. (cid:3)
Appendix B. Appendix: Symplectic geometry on lattices
A symplectic structure on an infinite dimensional manifold is not easy to define(see [CM74], [Bam99]). Fortunately, the KAM theory presented here uses only veryfew properties of symplectic geometry.The aim of the next sections is to develop such ideas and give precise definitionsof the theory of symplectic forms we will need. Note that we do not need to developa systematic geometry. We just need to deal with the standard symplectic form in M , its primitives, its push-forward and perform just a few operations. This can bereadily justified in spite of the difficulties with more sophisticated material.B.1. Forms on lattices.
Remember that a form is just an antisymmetric realvalued multilinear operator on the tangent space.We just need to study local forms which are the product of forms in each of theambient spaces.We introduce π i : M → M , the projection π i ( x ) = x i for i ∈ Z N . Given acollection of smooth k -forms γ i ∈ Λ k ( M ), such that sup i k γ i k < ∞ , we define aformal form in M as follows(146) γ = X i ∈ Z N π ∗ i γ i , that is γ ( x )( u , . . . , u k ) = X i ∈ Z N γ i ( π i ( x ))( π i u , . . . , π i u k )for x ∈ M and ( u , . . . , u k ) ∈ ( T x M ) k . We denote¯Λ k ∞ ( M ) = n γ = X i π ∗ i γ i o the set of such forms.Of course, this form (146) in general does not define a multilinear function onbounded vector fields, so that it should be understood only formally. Nevertheless,we will show that there are several operations among forms that can be made senseof in the infinite dimensional setting.Roughly, we will see that these formal forms make sense acting on vectors thatdecay away from a finite set of centers. We can also push them forward by adecay diffeomorphism and pull them back by a decay embedding. They can also beintegrated and, in some weak sense, differentiated. These will be all the operationsthat we will need. Moreover, we will only need k = 1 , k = 2, if each of the γ i are uniformly non-degenerate, we can define anidentification operator defined by(147) ( J ∞ u ) i = J i π i u, i ∈ Z n , where J i is the operator of identification on the i copy of the manifold i.e. γ i ( x )( ξ, η ) = h ξ, J i ( x ) η i , ∀ ξ, η ∈ T x M i . We emphasize that, given the formula (147), it is clear that when the γ i are uni-formly non-degenerate (i.e. k J i ( x ) k , k J − i ( x ) k are bounded uniformly in i, x ) wehave that the operator J ∞ is bounded and its inverse is also bounded. Note that inthe KAM method of [dlLGJV05, FdlLS09a, FdlLS09b], the symplectic propertiesappear mainly through J, J − and their invariance properties.In the main application to the construction of almost periodic solutions, whenthe system has translation invariance, all the γ i are identical. Nevertheless we donot assume that the γ i are given in the standard form. This is useful e.g. in dealingwith oscillators, or chemical molecules whose action angle variables are singular.Let γ ∈ ¯Λ k ∞ ( B ), F : B → M with F ( B ) ⊂ B . We define the pull-back F ∗ γ by(148) F ∗ γ = X i ∈ Z N F ∗ γ i , that is(149) F ∗ γ ( x )( u , . . . , u k ) = X i ∈ Z N γ i ( F ( x ))( DF ( x ) u , . . . , DF ( x ) u k ) . For a general diffeomorphism, the sums in (148), (149) are purely formal. Onthe other hand, when F ∈ C , the sums for F ∗ γ ◦ π j ( π i u , . . . π i k u k ) make senseand converge uniformly.If ψ : D ρ ⊃ T l → M is a smooth map we define ψ ∗ γ in the analogous way. Itwill be important to emphasize for future applications that when γ is a formal formand ψ has decay, then ψ ∗ γ is a smooth form in D ρ .An easy computation shows that if F ∈ C ∞ ( B ) and G ∈ C ∞ ( B ), with F ( B ) ⊂B , then ( G ◦ F ) ∗ = F ∗ ◦ G ∗ . Also, if ψ ( D ρ ) ⊂ B we have(150) ( ψ ◦ F ) ∗ = F ∗ ◦ ψ ∗ . Again, this is a formal computation for diffeomorphisms, but, when
F, G ∈ C ,then, the calculation can be justified. Also if G ∈ C and F ∈ A ρ,c, Γ , then ( G ◦ F ) ∗ γ is a well defined form. Definition B.1.
Given γ ∈ ˜Λ k ∞ we define dγ = X i ∈ Z N dγ i . We clearly have that d γ = 0. Lemma B.2.
Let F ∈ C ( B ) , ψ : A ρ,c. Γ → M and γ ∈ ˜Λ k ∞ ( B ) so that thecomposition makes sense. F ∗ dγ = d ( F ∗ γ ) ,ψ ∗ dγ = d ( ψ ∗ γ ) . Proof.
It consists mainly in going over the formal computation, but paying attentionto the fact that all the steps can be justified by the convergence. We carry outexplicitly the first one and we let the other one to the reader.
UASI-PERIODIC BREATHERS 79 F ∗ dγ = F ∗ (cid:16) X i dγ i (cid:17) = X i F ∗ dγ i = X i d ( F ∗ γ i )= d X i F ∗ γ i = d ( F ∗ γ ) , where we have used that γ i are true differential forms. (cid:3) In the case that γ = P i ∈ Z N π ∗ i γ i , with γ i ∈ Λ k ( M ), dγ = X i ∈ Z N d ( π ∗ i γ i ) = X i ∈ Z N π ∗ i dγ i , where dγ i is the exterior differential of γ i in M , and if ψ : D ρ → M , ψ ∗ γ = X i ∈ Z N ψ ∗ π ∗ i γ i = X i ∈ Z N ψ ∗ i γ i . We can also define the contraction operator. Given a smooth vector field X in M and a k -form γ ∈ ˜Λ k ∞ we set( i X γ )( x )( u , . . . , u k − ) = X j ∈ Z N ( i X γ j )( x )( u , . . . , u k − ) . Hence we can also introduce the Lie derivative for formal forms by the usual formula L X γ = i X dγ + d ( i X γ ) . Lemma B.3.
Let γ = P i ∈ Z N π ∗ i γ i ∈ ¯Λ k ∞ be a formal form and ψ : D ρ ⊃ T l → M a map with decay, i.e. ψ ∈ A ρ,c, Γ with c = ( c , . . . , c R ) . Then for < δ < ρ , ψ ∗ γ is a well-defined k -form in D ρ − δ . As a consequence if F ∈ C ( B ) is analytic, ψ ∈ A ρ,c, Γ and ψ ( D ρ ) ⊂ B , byLemma A.15 and (150) we have that ψ ∗ F ∗ γ is a well-defined k -form in D ρ − δ . Proof of Lemma B.3.
By definition of γ , we have( ψ ∗ γ )( θ )( u , . . . , u k ) = X i ∈ Z N γ i ( ψ i ( θ ))( Dψ i ( θ ) u , . . . , Dψ i ( θ ) u k )for θ ∈ T l and u , . . . , u k ∈ T θ T l . Then | ( ψ ∗ γ )( θ )( u , . . . , u k ) |≤ X i ∈ Z N k γ i k X m | D m ψ i ( θ )( u ) m | · · · X m k | D m ψ i ( θ )( u k ) m k |≤ X i ∈ Z N k γ i k max j Γ( i − c j ) | Dψ |k u k · · · max j Γ( i − c j ) | Dψ |k u k k≤ R k k γ kk Dψ k ρ − δ,c, Γ k u k · · · k u k k . We have used that P i ∈ Z N max j Γ( i − c j ) ≤ R . This proves that the series isabsolutely convergent and so ψ ∗ γ is well-defined on T ∗ T l . (cid:3) Therefore, by the previous construction, by pulling back formal forms on thelattice to the torus, one obtains well-defined quantities.
Lemma B.4.
For every function ψ ∈ A ρ,c, Γ , we have ψ ∗ dγ = d ( ψ ∗ γ ) . Proof.
By the definition of dγ and the convergence of the series, we have ψ ∗ ( dγ ) = ψ ∗ (cid:16) X i dπ ∗ i γ i (cid:17) = X i ψ ∗ dπ ∗ i γ i . But by the definition of the exterior differentiation, we have X i ψ ∗ dπ ∗ i γ i = X i dψ ∗ π ∗ i γ i = d X i ψ ∗ π ∗ i γ i = dψ ∗ γ. (cid:3) B.2.
Some symplectic geometry on lattices.
In this section we discuss theelements of symplectic geometry that we will need. This will play a role in thevanishing lemma Lemma 6.1 in Section 6.Consider a finite dimensional exact symplectic manifold ( M, Ω = dα ) and theassociated lattice M = ℓ ∞ ( Z N ) . Let α ∞ and Ω ∞ be defined by α ∞ = X j ∈ Z N π ∗ j α, Ω ∞ = X j ∈ Z N π ∗ j Ω . Then α ∞ ∈ ¯Λ ∞ and Ω ∞ ∈ ¯Λ ∞ . Moreover note that dα ∞ = Ω ∞ . We introduce thefollowing definitions. Definition B.5.
We say that a C function F : M → M is symplectic if thefollowing identity holds for any z ∈ M DF ⊤ ( z ) J ∞ ( F ( z )) DF ( z ) = J ∞ ( z ) Definition B.6.
We say that a C function F : M → M is exact symplectic on M if there exists a one-form ˜ α defined on T M with matrix ˜ A such that • For every j ∈ Z N , there exists a smooth function W j on M such that ˜ α j = dW j where d is the exterior differentiation on M . • The following formula holds component-wise on the lattice DF ( z ) ⊤ ˆ A ∞ ( F ( z )) = ˆ A ∞ ( z ) + ˜ A ( z ) . The previous definitions are completely equivalent to the standard definitions ofsymplectic and exact symplectic maps in the finite dimensional case, but they areamong the mildest ones that we can imagine in infinite dimensions. The followingis a straightforward result.
Lemma B.7.
Let F ∈ C be a map from M into itself. If F is exact symplecticthen it is symplectic. Remark B.8.
Through a localized embedding, this is even easier.Since F is exact symplectic, for every decay function ψ ∈ A ρ,c, Γ , there exists asmooth function W ψ defined on the torus such that ψ ∗ F ∗ α ∞ = ψ ∗ α ∞ + dW ψ . By the property of the exterior differentiation and the fact that, by the hypotheses, F ◦ ψ ∈ A ρ,c, Γ , we have( F ◦ ψ ) ∗ dα ∞ = d (( F ◦ ψ ) ∗ α ∞ ) = d ( ψ ∗ α ∞ + dW ψ ) = d ( ψ ∗ α ∞ ) = ψ ∗ dα ∞ . UASI-PERIODIC BREATHERS 81
Since Ω ∞ = dα ∞ this gives the desired result.We now turn to the symplectic geometry of vector fields. We will always beconsidering vector-fields of the form X = J ∞ ∇ H where the operator ∇ has to be understood w.r.t. the inner product on ℓ ( Z N ).The following result is proved. Proposition B.9.
Aasume that the vector-field X previously defined has decay.Then it generates flows consisting of exact symplectic diffeomorphisms.Proof. Since X has decay, the operation i X Ω ∞ makes sense and one has i X Ω ∞ = dH. The proof then follows the standard one by using the fact that decay vector fieldsgenerate decay diffeomorphisms. (cid:3)
Appendix C. Appendix : Construction of deformations of symplecticmaps which are not exact symplectic
In the construction of the invariant torus, we are going to use a family of maps F λ such that F is exact symplectic and F λ is symplectic for all λ but not exactsymplectic for λ = 0. Indeed, these maps will be used to kill some averages in theinvariance equations, so it will be important that, by choosing λ appropriately wecan obtain all the possible cohomology obstructions to exactness. The change oncohomology is more or less proportional to the change in the parameter λ .The construction of F λ will be done in this section by considering flows whichare locally but not globally Hamiltonian. We emphasize that the diffeomorphismsintroduced will be quite simple. They will just deform a finite number of siteson the lattice. In the case that the phase space is T l × R d − l endowed with thestandard symplectic form the map F λ will be given by A i → A i + λ i where A i are the variables symplectic conjugate to angles. We note that the obstruction toexactness are the integrals of the forms A i dφ i around a cycle in the torus along φ i . The rest of the section is devoted to make a geometrically natural constructionthat works in all manifolds.Consider ( M i , Ω i = dα i ) i ∈ Z N a family of finite dimensional exact symplecticmanifolds and denote M the phase space of the associated lattice map.Let J ⊂ Z N be a finite set of indexes. We denote by H ( M i ) the first de Rhamcohomology group of the manifold M i and assume that it is non-trivial. Consider( δ ik ) k =1 ,...,l a basis of H ( M i ). Since Ω i are non-degenerate, one can construct afamily of vector fields Y λi on M i with indexes i ∈ J such that i Y λi Ω i = l X k =1 λ k δ ik . Note that Y λi only depends on x i ∈ M . Now we introduce the vector-field X λ onthe lattice M defined by( X λ ) j ( x ) = (cid:26) j / ∈ J ,Y λj ( π j ( x )) if j ∈ J . By construction, we have X = 0. Furthermore, the family of vector-fields X λ issymplectic for all λ . Indeed, consider a decay function ψ and compute L X λ Ω ∞ = d X j ∈ Z N i ( X λ ) j Ω j = d X j ∈J l X k =1 λ k δ jk = 0 , where we have used that the last sum is finite by construction of X λ and ( δ ik ) k =1 ,...,l are closed forms. We obtain that X λ is symplectic. Notice also that all but a finitenumber of the components of X λ are zero and so DX λ ( x ) ∈ L Γ , i.e. X λ is a decayvector field. If x ∈ M , denote ϕ ( s, λ, x ) the flow generated by X λ . That is: dds ϕ ( s, λ, x ) = X λ ( ϕ ( s, λ, x )) , ϕ (0 , λ, x ) = x. The existence and uniqueness of ϕ ( s, λ, x ) is ensured by the theorem of existenceand uniqueness of solutions for Lipschitz differential equations in Banach spaces.See [Hal80] for instance. Here the Banach space is ℓ ∞ ( Z N ).Given an exact symplectic map F satisfying F (0) = 0 and F ∈ C , we definethe family of maps we want to construct by F λ = ϕ ( λ, λ, . ) ◦ F. We have the following easy lemma
Lemma C.1.
For all s ∈ R , we have (1) ϕ ( s, , x ) = x . (2) For all j ∈ Z N , ϕ j ( s, λ, . ) only depends on x j and ϕ j ( s, λ, x ) = x j , j / ∈ J . Proof. (1) follows directly from the fact that X = 0. The first part of (2) followsfrom the fact that ( X λ ) j only depends on x j . Moreover if j / ∈ J , ( X λ ) j = 0 andthen ϕ j is constant in s . Therefore ϕ j ( s, λ, x ) = ϕ j (0 , λ, x ) = x j . (cid:3) As a consequence we have that F = F and ( F λ ) j = F j , for λ ∈ R l , j / ∈ J . Since ϕ j is not constant for only a finite set of indexes, for λ small ϕ (1 , λ, . ) is well-defined on the range of F . Moreover, since ϕ is uncoupled, i.e. π i ϕ ( s, λ, . ) dependsonly on x i we have that ϕ ( λ, λ, . ) ∈ C . On the other hand, ( F λ ) j (0) = F j (0) = 0for λ / ∈ J . Therefore F satisfies the assumptions of Lemma A.15.Finally, the following lemma ends the details of the construction. Lemma C.2.
For all λ , the map F λ ∈ C is symplectic, but it is not exact sym-plectic for λ = 0 .Indeed, we have that if Ψ is the embedding given by the coordinates in J , and [ · ] denotes the cohomology class on the torus expressed in the basis of the forms δ k ,we have (151) [Ψ ∗ F ∗ λ α ∞ ] = λ. UASI-PERIODIC BREATHERS 83
Proof.
Let ψ ∈ A ρ,c, Γ be a decay function. We want to prove that for any λψ ∗ F ∗ λ Ω ∞ = ψ ∗ Ω ∞ . By construction of F λ we have ψ ∗ F ∗ λ α ∞ − ψ ∗ F ∗ α ∞ = ψ ∗ F ∗ ϕ ∗ ( λ, λ, . ) α ∞ − ψ ∗ F ∗ ϕ ∗ (0 , λ, . ) α ∞ = Z dds (cid:0) ψ ∗ F ∗ ϕ ∗ ( s, λ, . ) α ∞ (cid:1) ds = Z ψ ∗ F ∗ (cid:16) ( dds ϕ ∗ i ( s, λ, . ) α ∞ (cid:17) ds = Z ψ ∗ F ∗ L X λ ( ϕ ( s,λ, · )) α ∞ ds = Z ψ ∗ F ∗ [ d ( i X λ α ∞ ) + i X λ dα ∞ ] ds = d Z ψ ∗ F ∗ ( i X λ α ∞ ) ds + Z ψ ∗ F ∗ X j ∈J l X k =1 λ k δ jk ds . Since δ jk are closed forms, taking exterior differential at both sides of the previousformula we get ψ ∗ F ∗ λ Ω ∞ − ψ ∗ F ∗ Ω ∞ = 0. Finally, using that F is symplectic weget that F λ is symplectic.Moreover, if λ = 0, ψ ∗ F ∗ λ α ∞ − ψ ∗ F ∗ α ∞ = dW ∞ + E, where W ∞ = R λ ψ ∗ F ∗ ( i X λ α ∞ ) ds and E is not a differential. The formula (151),follows easily from the expression for E above. We note that that[Ψ ∗ F ∗ λ α ∞ ] = [Ψ ∗ F ∗ λ α ∞ − Ψ ∗ F ∗ α ∞ ]= [ d Z Ψ ∗ F ∗ ( i X λ α ∞ ) ds + Z Ψ ∗ F ∗ X j ∈J l X k =1 λ k δ jk ds ]= 0 + Z [Ψ ∗ F ∗ X j ∈J l X k =1 λ k δ jk ] ds = Z Ψ ∗ ( F ) ∗ X j ∈J l X k =1 λ k [ δ jk ] ds = λ (cid:3) References [AG96] G. Arioli and F. Gazzola. Periodic motions of an infinite lattice of particles withnearest neighbor interaction.
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