A KAM theorem for infinite-dimensional discrete systems

A KAM theorem for infinite–dimensional discrete systems Paolo Perfetti Dipartimento di matematica, II Universit`a degli Studi di Roma Via della Ricerca Scientifica 00133 Roma, Italy E–mail: [email protected]

Pubblicato su Mathematical Physics Electronic Journal, Vol. 9 2003

Abstract Infinite–dimesional, discrete hamiltonian systems of the type kinetic energy + potential energy over RZ × TZ are studied. The existence of many quasi–periodic motions with a maximal set of nonzero frequencies is shown.

Supported by Istituto Nazionale di Alta Matematica “Francesco Severi” and Ministero dell’Universit` a e della Ricerca Scientifica e Tecnologica, research program “Metodi variazionali ed equazioni differenziali nonlineari”

1

Introduction

We study a family of hamiltonian dynamical systems of the form kinetic energy plus potential energy (possibly formal) having an infinite number of degrees of freedom (hereafter d.o.f.). The potential energy is a multi–periodic function of infinite variables whose period is 2π and the phace–space is RZ × TZ (T is the 2π–periodic torus). We prove that under suitable conditions of growth on the kinetic part and quite general conditions on the potential functions, the systems admit quasi–periodic motions with frequencies {ωi }i∈Z such that |ωi | → +∞ as |i| → +∞. The equations of motions are A˙ i = fi (ϕ)

ϕ˙ i =

∂h(Ai ) ∂Ai

i ∈ Z,

Ai ∈ R, ϕi ∈ T,

T = R/2πZ

(1.1)

and if we think of fi as minus the gradient of a function V (ϕ) respect to ϕi , they are the canonical P equations of the (formal) hamiltonian H(A, ϕ) = i∈Z h(Ai ) + V (ϕ); H is formal because the growth of h(Ai ) makes the sum unbounded and because the function V (ϕ) may not exist as a function over TZ . What must be well defined are the functions (f1 (ϕ), f2 (ϕ), . . .). On h, which is the same for each identical . particle, we impose the condition 0 < a0 ≤ |hAA | ≤ a < ∞ for A ∈ R. If f (ϕ) = (f1 , f2 , . . .) ≡ 0, all the motions are linear in time over RZ × TZ that is (A, ϕ) = {(Ai (t), ϕi (t)} = {(Aoi , ϕoi + ωi t)}, i ∈ Z. The infinite–dimensional vector {ωi }i∈Z is the vector–frequency. If not all the functions fi are identically zero instead, almost all (in the sense that the excluded subset of motions have a small relative measure) linear motions survive and are continued into {(Aoi + αi (t), ϕoi + ωi t + βi (t))} with |αi | and |βi | small provided τ . that: 1) subsets of ω like ω (l) = (ω1 , ω2 , . . . ωl ) satisfy the Diophantine condition |ω (l) · ν|−1 ≤ γe|ν| /τ for all ν ∈ Zl \{0}, 0 < τ < 1 (actually we will take 0 < τ < 21 ; γ is a positive number), 2) |ωi | is sufficiently large. By our techniques we cannot relax the condition |hAA | ≤ a allowing | hAA | to be unbounded for |A| → +∞. The same model has been studied in [P] but a much greater lower bound on |ωi | is necessary because we impose a different Diophantine condition on the frequencies; namely |ω (l) · ν|−1 ≤ γ|ν|l ∀ ν ∈ Zl \{0}. For a model represented by (1.1) one can think of an array of rotators: point particles constrained to move circularly and centered on the sites of N. In this case the kinetic energy of a single particle is quadratic in the action–variable h(A) = 21 A2 . The problem considered here falls within the family of the problems concerning the search of almost– periodic solutions in systems with infinitely many d.o.f. (originating in statistical mechanics for instance). Viewing (1.1) as the equations of infinite particles, the existence of the solutions described in this paper would imply that energy remains confined in suitable subsets of phace–space during all the time–evolution. The drawback (up to now) is that the energy–dependence of each d.o.f. is too large (faster than any power 2 α but slower than any exponential). We obtain e(|j| ln |j|) 0 < α < 1 (see the upper bound on γΩlk before Theorem 4.7)). For more realistic rates it might be appropriate to consider a particular (but quite general) interaction {fi }. An example is fi (ϕi−1 , ϕi , ϕi+1 ) = ai cos(ϕi − ϕi−1 ) + ai+1 cos(ϕi+1 − ϕi ) (or something like it) and then to study carefully the geometry of the small denominators (see for instance [BFG] where a Nekhoroshev–type theorem is obtained for infinitely many harmonic oscillator coupled by a suitable potential or [Wa1]–[Wa3] where good estimates on the perturbative parameter are obtained for chains of a very large but finite number of rotator coupled through anelastic forces 1 . Infinite–dimensional extensions 1

Nevertheless the results of [Wa1] can be hardly extended to cover the infinite–dimensional theory because the potential is

needed to be analytic over a strip around the real axis whose size is proportional to NlnN where N is the number of particles (see (1.14) at p.314)

2

of KAM theorem have been already studied but with some differences respect to this paper. [FSW], [VB] and [P¨o] consider perturbations (the {fi }0 s) depending on the action–variables (say (I1 , I2 , I3 , . . .)) which 1+α d+α are extremely localized in phace–space (|Ij (0)| ∼ e−|j| α > 0 (e−|j| α > 0 for a d–dimensional) γ − ln |j| problem in [FSW] and [VB] while at least e with γ > 1 in [P¨o]). Moreover according to our definitions (see (2.3) and (2.4)), their models are all of short–range type 2 . A first improvement has been obtained in [CP] where the short–range condition has been relaxed allowing each particle to interact directly with each other particle “of the world.” Also our models provide perturbation functions which are independent on the actions–variables. Although closely related, the result in [P] is slightly more general than [CP] because the kinetic part of the hamiltonian, like here, can have a quadratic part plus a correction in such a way that the leading term is at most quadratric for |A| → +∞. In [P] as well as in [CP], the size of the frequencies is of order ep|j| ln |j| where p ∼ 10 (epd|j| ln |j| for a d–dimensional 2 α problem). Here the size has been lowered to e(|j| ln |j|) , 0 < α < 1 using a different Diophantine condition described at point 1) before. To reach a power law (|j|p , p ≥ 1) would be a very interesting result at least for two applications: 1) the thermodynamic limit in the context of statistical mechanics 2) the study of almost–periodic solutions of some discretized PDE’s. Another difference respect to [CP] and [P] rests in the method of proof. In [CP] as well as in [P], a “configurational” KAM theorem has been proved (technically speaking one solves a system of partial differential equations whose variables are the functions defining the torus in configuration–space) while here we perform a proof based on countable changed of action–angle coordinates in phace–space (a “classic proof”). May be analogous results could be obtained through a configurational proof. We have used a “classic” proof because we think it allows the treatment of: 1) perturbations of infinitely many harmonic oscillators (where configurational proofs can be hardly adapted, if any) 2) Nekhoroshev type problems where a background made of action–angles variables and canonical transformations seems to be necessary. Acknowledgments I thank C.Liverani for some useful discussions and the referees for their valuable comments. Moreover I thank the Editors in Chief for pointing out some typographical errors

2. Setup Denote by T the Cartesian product of infinitely many copies of the one dimensional (flat) torus T: N Z T ≡ i∈Z Ti ≡ T , Ti ≡ T ≡ R/2πZ (d being a positive integer) and endow T with the standard weak topology (see, e.g. , [Ke] and [Je]). Such topology is also induced by metrics: To any summable positive P sequence w : Z → (0, ∞) s.t. i∈Z wi < ∞ (wi > 0 ∀i) which we shall call a weight, we can associate a metric ρw by setting, ∀ ϕ, ϕ0 ∈ T (ϕ = {ϕi }i∈Z , ϕ0 = {ϕ0i }i∈Z , ϕi , ϕ0i ∈ T): ρw (ϕ, ϕ0 ) ≡

X

ρ(ϕi , ϕ0i ) wi

(2.1)

i∈Z

where ρ is the standard (flat) metric on T ≡ Ti : ρ([a], [b]) ≡ inf n∈Z |a − b + 2πn| here a, b ∈ R and [·] denote equivalence (mod. 2π) class. We shall denote by Tw the pair (Tw , ρw ) and by Bw the Banach P . space formed by the sequences a ∈ RZ having finite norm kakw ≡ i∈Z |ai | wi < ∞, ρ0w (a, b) = ka − bk. N Z We define R ≡ i∈Z R ≡ R and endow this space with the topology induced by the following metric: P 0 (1) 0 ∀ A, A ∈ R λw (A, A ) ≡ i∈Z wi arctan |Ai − A0i |. We shall denote by Rw the pair (Rw , λw ); Rw × Tw 2

our short–range models correspond in [P¨ o] to finite–range couplings, see p.357 while what we call long–range models corre-

spond to his short–range couplings (1) we don’t consider here the physical dimensions

3

is a complete metric space with metric ρw (ϕ, ϕ0 )+λw (A, A0 ). The metric over T (as well as over R) defines the compact topology with its open sets and induces a well defined notion of convergence. It turns out that the sequence {ϕ(n) } ∈ T , n = 1, 2, . . . converges respect to the metric (2.1) if and only if converges (n) respect to the following notion: ϕ(n) −n→+∞ −−−−→ϕ if ∀ i ϕi −n→+∞ −−−−→ϕi (no uniformity in the components). The same equivalence of convergences holds in the space R. We shall also work with real analytic functions o

f : U × Tn → R, U = U ⊂ Rn (n will be the d.o.f. of the intermediate hamiltonians we will consider; see before Lemma 4.2). f (y, x) can be extented to an holomorphic function, which we call f (y, x) too, on the set ∪y∈U D(y; ρy ) × ∆ξ , D(y; ρy ) = {z ∈ C n : |y − z| < ρy }, ∆ξ = {z ∈ C n : Re zi ∈ T, |Im zi | < ξ}, 0 < ξ < 1 and continuous on the closure ∪y∈U D(y; ρy ) × ∆ξ . C ω (A × B; C) is the space of complex functions holomorphic over A × B ⊂ C n × C n and C(A × B; C) is the space of the functions continuous over the same domain. The norm of functions, vector–functions and matrix–valued functions are given in Appendix 1. We now give a precise meaning to second order ODE’s on Rw × Tw : Given a continuous map (hA , f ) : . ∂h (A ) (ϕ ) ) consider the system (1.1) where (hAi , fi ) ≡ (πi i ◦ hA , πi i ◦ f ) Rw × Tw → Rw × Bw , (hAi = ∂A i (ϕ )

(A )

(πi i : Bw → Ri , πi i : Rw → Ri being the standard projections). A solution of (1.1) is just a continuous map t ∈ R → (A(t), ϕ(t)) ∈ Rw × Tw , with (Ai , ϕi ) ∈ C 1 (R, R) × C 1 (R, T), ∀i, and satisfying the system (1.1). Remarks 2.1 (i) In some sense the notion of solution we have just introduced is a “weak notion” (ii) . Let f ≡ 0 so that (A(t), ϕ(t)) = (Ao , [ωt]) ≡ ({Aoi }, {[ωi t]})i∈Z and ω = hAo ; (A(t), ϕ(t)) is a solution for any ω ∈ RZ (not necessarily in Bw ). Note in particular that t ∈ R → (Ao , [ωt]) ∈ Rw ×Tw is continuous for any ω ∈ RZ . These facts are no longer true if one considers stronger topologies; for example, if |ωi | → ∞ as |i| → ∞, [ωt] is not continuous with respect to the uniform topology [ ρuniform (ϕ, ϕ0 ) ≡ supi∈Z ρ(ϕi , ϕ0i ) ] . Thus, our interest in solutions (A(t), ϕ(t)) with (Ai (t), ϕi (t)) “close” to (Aoi , ωi t) with |ωi | → ∞ as |i| → ∞ explains the choice of the topology induced by ρw + λw . Global existence and uniqueness for the Cauchy problem associated to (1.1) with (hA , f ) Lipschitz, are elementary applications of standard contraction techniques. We just stress that the “initial velocities” can be taken to be completely arbitrary (and not necessarily in Bw ): Proposition 2.1 Let (hA , f ): Rw × Tw → Rw × Bw be a Lipschitz map (i.e. ∃ L > 0 s.t.
0 0 ρw f (ϕ), f (ϕ + λw (hA , hA0 ) ≤ L(ρw (ϕ, ϕ0 ) + λw (A, A0 )), ∀ϕ, ϕ0 ∈ Tw A, A0 ∈ Rw ). Given any ϕo ∈ Tw and any Ao ∈ RZ , there exists a unique solution, global in time, of the Cauchy problem (

ϕi (0) = ϕoi

ϕ˙ i (t) = hAi (A(t)) A˙ (t) = f (ϕ(t)) i

Ai (0) =

i

Aoi .

i∈Z

(2.2)

The proof is an application of standard techniques in ODE’s and we omit it. The existence for all times of the solutions follows by: 1) the infinite–dimensional torus Tw is a compact space 2) the space Rw , though not compact, is complete, bounded and very similar to a compact space for what concerns the behavior of the metric λw (A, A0 ) respect to the indices i ∈ Z definitively large. It is appopriate here to stress one point. As is well known, in the proof we must check the Lipschitz conP dition on λw (hA (A), hA (B)) and we are led to study i∈Z wi max(1, supAi ∈R |hAi Ai |) arctan |Ai −Bi |. We know that our KAM theorem needs {|Ai |}i∈Z unbounded and then we are forced to take supAi ∈R |hAi Ai | < ∞. This problem would be absent if the d.o.f. were finite but its presence here forbids kinetic energies like h(Ai ) = A2+δ for any δ > 0 while allows e.g. h(Ai ) = A2i + o(A2i ). i 4

As for f we consider two examples of Lipschitz maps (the force). The first one is so called short range; fix L ≥ 1 and consider, for j ∈ Z, a collection of functions gj depending on sites of the lattice within (Euclidean) distance L from j; in formulae: gj ≡ gj (ϕ(L) ) , ϕ(L) ≡ {ϕk }k∈Bj (L) , Bj (L) ≡ {k : kk − jk ≤ L}. We assume that the “localized potentials” gj are real–analytic functions from T|Bj (L)| → R and that, for some positive M : supj,ϕ(L) ∈T|Bj (L)| |gj (ϕ(L) )| ≤ M. Then we set X fi ≡ ∂ϕi gj (2.3) kj−ik≤L

The system (1.1) with such fi is called a finite range system of infinitely many coupled variables (see also [Wa1] and [VB]). A particular case, often considered, is given in d = 1 by L = 1, gj = cos(ϕj − ϕj−1 ) − cos(ϕj+1 − ϕj ). It can be noted that each variable is coupled with just a finite number of different variables and this explains the words short range The second example is so called long range as each variable is coupled with any other variable. In d = 1 it is given by X Y X fi ≡ cos ϕi aj (1 + aj+k sin ϕi+k ) , |aj | < ∞ (2.4) j∈Z

j∈Z

k6=0

The property of being Lipschitz depends of course from the metric. In fact, consider two cases: (1) aj ≡ b−|j| , b > 1; (2) aj = (1 + |j|p )−1 , p > 1. Then, in the first case (1), f is Lipschitz if we take wj ≡ c−|j| with any 1 < c < b, while in case (2) we can take wi ≡ aj . We don’t need the existence of a P∞ ∂V = fi (ϕ). For instance we could take V (ϕ) = − i=1 mg(1 − cos ϕi ) + function V : TZ → R such that − ∂ϕ i P∞ P∞ −|i−j| (1 − cos(ϕi − ϕj )) for i=1 κ(1 − cos(ϕi+1 − ϕi )) for the short range case and V (ϕ) = i,j=1 e R [I] |I| [I] the long range. We need well defined the functions g : T → R, g = g(ϕ)dµJ , I ⊂ Z, J = Z\I, N dµJ = i∈J dµi for a measurable function g: Tw → R, . In Tw there exists a unique probability measure N N (see [Ha], Section 38) defined over the σ–algebra, R, generated by the cylinders RI = i∈I Ui j6∈I Ti where Ui is an open subset of Ti and I ⊂ Z is a finite subset of Z: |I| < ∞ (| · | denoting here cardinality). Q The measure µ on R is such that µ(RI ) = i∈I µi (Ui ) where µi is the normalized “Lebesgue measure” on Ti . If |I| < ∞, g [I] is a measurable function on T|I| and g [I] → g a.e. on T as |I| → Z. For the examples in (2.3) and (2.4) the convergence is uniform (see [Je]) Definition 2.2 A continuous function f : Tw → Bw is a g-gradient if for any finite I ⊂ Z there exists a [I] C 1 (T|I| ; R) function, V (I) (ϕ), so that fi (ϕ) = −∂ϕi V (I) (ϕ), ∀ i ∈ I , ∀ ϕ ∈ T|I| . It is easy to check that the examples (2.3), (2.4) in Section 2 are g–gradients. We shall speak of Hamiltonian equations on Tw whenever f in (1.1) is a g–gradient. Let us conclude this section by introducing the (strong) regularity class we shall work with. Definition 2.3 A g-gradient f is called uniformly weakly real-analytic if there exists a real number σ > 0 such that for any finite set I ⊂ Z, V (I) (ϕ) is real-analytic on T|I| and can be analytically continued to the set {z ∈ C|I| : Re zi ∈ T |Im zi | < σ} and continuous on the closure . Remarks (i) In fact we could deal with more general classes of vector fields allowing the width of analyticity of V (I) to tend to zero as |I| → ∞ (the allowed rate of decay would then be dictated by the quantitative analysis carried out below). However well known models like (2.3) with d = 1, L = 1, gj = cos(ϕj − ϕj−1 ) − cos(ϕj+1 − ϕj ) have the same analyticity width for all the variables (ii) Example (2.4) in Section 1 is uniformly weakly analytic and as parameter σ one could take any positive number; example (2.3) is uniformly weakly analytic for some (small enough) σ > 0. 5

3. Almost–Periodic Solutions and Diophantine Sequences Definition 3.1 A sequence ω ∈ RZ is said to be rationally independent if for any finite subset I of Z and P for any ni ∈ Z i∈I ωi ni 6= 0 unless ni = 0 ∀ i ∈ I In other words, ω is rationally independent if any finite vector ω (I) ≡ {ωi }i∈I ∈ R|I| is rationally independent. Definition 3.2 A continuous real function q(t) is said to be almost–periodic over Tw (with frequency ω) if there exist a rationally independent sequence ω ∈ RZ and a continuous function Q : Tw → R such that q(t) = Q([ωt]). A solution (A(t), ϕ(t)) of (1.1) is called maximal almost–periodic if (Ai (t), ϕi (t) − [ωi t]) is, for all i, almost–periodic over Tw with frequency ω. As usual in problems involving small denominators, we have to impose on the frequencies a quantitative irrationality condition. Our aim is to get the lower bound of |ωi | as small as possible and a good condition is the following Definition 3.3 A rationally independent sequence ω ∈ RZ is called (γ, τ )–Diophantine if for any finite P set I ⊂ Z, there exist constants γ > 0 and τ ∈ (0, 1) such that for any choice of ni ∈ Z with i∈I |ni | > 0 it is: X | ωi ni | ≥ γ −1 exp(−|n|τ /τ ) (3.1) i∈I

Of course (3.1) implies that a sequence ω ∈ RZ is rationally independent: for any finite subset I of Z P and for any ni ∈ Z, i∈I ωi ni 6= 0 unless ni = 0 ∀ i ∈ I. Diophantine vectors in the sense of (3.1) form a set of full Lebesgue measure in R. The proof is standard and we omit. It can be performed along the same lines of the proof present in [CP] p.194 although the irrationality condition considered there is P | i∈I ωi ni | ≥ γ −1 |n|−|I| . Also in infinite dimensions they are rather abundant as the following Lemma Pn Pn . τ shows. For any vector v ∈ Rn let be kvk = ( i=1 vi2 )1/2 while |v| = i=1 |vi |; E(t) = et /τ Lemma 3.4 Let ω ∈ Rl−1 be (γ, τ )–Diophantine and Ω ≥ subset of [Ω, +∞) :

1 γ

be a positive number. Let’s define the

−1  . Al = Al (ω, Ω) = α ≥ Ω: (ω, α) · ν ≤ Ω−1 E(|ν|) ∀ ν ∈ Zl

νl ∈ Z\{0}



(3.2)

Then the following estimate of the Lebesgue measure of the complement Al holds Leb([Ω, +∞)\Al ) ≤ l

L(τ, l − 1)kωk, where L(τ, l − 1) = C3 2c τ

Γ( l−1 τ ) ; Γ( l−1 2 )

c and C3 are constants l, τ and ω independent

Proof |ω · ν|−1 ≤ γE(|ν|),

∀ ν ∈ Zl−1 \{0}

We need the following relation for α ≥ Ω (ω , α) · ν −1 ≤ Ω−1 E(|ν|) o

∀ν ∈ Zl \{0} νl 6= 0

(3.3)

−1 If ν = (ˆ ν , νl ) = (ˆ ν , 0) we have ω · νˆ ≤ Ω−1 E(|ˆ ν |) ≤ γE(|ˆ ν |) if we impose Ω ≥ γ1 In νˆ = ˆ0 (3.3) becomes |νl α|−1 ≤ Ω−1 E(|νl |) which is due to α ≥ Ω and |ν1l | ≤ E(|νl |). Acl is the set of α for which (3.3) is false  Acl = α ≥ Ω : ∃ ν ∈ Zl

νl 6= 0, 6

(ω, α) · ν −1 > Ω−1 E(|ν|)

(3.4)

and its Lebesgue measure is bounded by X

Leb(Acl ) ≤

X

ν ˆ∈Zl−1 νl ∈Z 6 0 ν ˆ6=0 νl =

 ω · νˆ Ω 1 Leb α ≥ Ω : (α + ) < νl |νl | E(|ν|)

(3.5)

1 ν To get (3.5) let’s consider a centered interval of lenght |νΩl | E(|ν|) around each value ω·ˆ νl (νl 6= 0). For a  Ω ν 1 is an upper bound on the measure of the set given vector ν ∈ Zl , Leb α ≥ Ω : (α + ω·ˆ νl ) < |νl | E(|ν|)

of the α0 s such that the vector (ω, α), multiplied by |ν|, has projection over ν bounded by

Ω E(|ν|)

(i.e.

Ω E(|ν|) ).

(ω, α) may be resonant with ν or non–resonant up to a size given by If for for some ν we had −1 (ω, α) · ν ≤ Ω−1 E(|ν|) , then summing over all ν ∈ Zl−1 , ν 6= 0, we are considering more vectors l than those satisfiyng (3.4) so that the sum in (3.5) is an upper bound of Leb(Acl ) X

Leb(Acl ) ≤

X

ν ˆ∈Zl−1 νl ∈Z\{0} ω·ˆ ν ν ˆ6=0 ≥Ω νl

X

X

ν ˆ∈Zl−1 ν ˆ6=ˆ 0

νl ≥1 |ν1 | a νl ≥ 2

≤

The condition

ω·ˆ ν νl

Ω 2

≥

ν − ω·ˆ νl

X 2Ω −|ν|τ /τ e ≤ |νl | l−1

X νl ∈Z\{0} Ω ω·ˆ ν ν ≥2

ν ˆ∈Z ν ˆ6=0

2

l

(3.6)

X X 4 τ 4 −kνkτ /τ Ωe ≤ Ωe−kνk /τ νl νl l−1 νl ≥1 |ν1 |≥ a 2

ν ˆ∈Z ν ˆ6=ˆ 0

ν comes from α ≥ Ω. Imposing in fact − ω·ˆ νl + Ω − |νl |E(|ˆ ν |)

ν − ω·ˆ νl

+ Ω is implied by

Leb(Acl ) ≤

X

≥ Ω(1 −

Ω |νl |E(|ˆ ν |)

1 E(1) )

≥ Ω we find a bigger value

Ω 2.

≥ Moreover rotating in such . ωo a way that RT e = e(1) = (1, 0, . . . , 0) ∈ Rl−1 , e = kω and calling a = kωΩo k , we obtain the last bound ok Pl Pl Pl in (3.6). Being ( i=1 ki2 )τ ≥ |k1 |τ + C( i=2 ki2 )τ for all k1 ≥ 0, i=2 |ki | ≥ 1 i = 2, . . . , l, and for a suitable C, we have ([ a2 ] = A) of the measure.

≥

2Ω −kνkτ /τ e ≤ |νl |

4Ωe−(|ν1 |

X

τ

+C(

Pl i=2

|νi |2 )τ )/τ

(3.7)

ν ˆ∈Zl−1 νl ≥1 ν ˆ6=ˆ 0 |ν1 |≥A

X

e−|ν1 |

τ

/τ

Z

τ

dxe−x

/τ

= 2τ

1−τ τ

e−A

τ

/τ

A

|ν1 |≥ a 2

≤ 2τ

∞

≤2

1−τ τ

−Aτ /τ

e

nZ

Z

∞

dx(x + 0

Aτ /τ

1−τ 2 dx( Aτ ) τ e−x + τ

0

Z

∞

(2x)

1−τ τ

1 τ 1−τ A ) τ e−x ≤ τ

o τ e−x ≤ CA1−τ e−A /τ =

Aτ /τ

1 a τ a = C1 [ ]1−τ e− τ [ 2 ] 2

the rest of (3.7) is bounded by

P

k∈Zl−1

e−C(

Pl−1 i=1

|ki |2 )τ /τ

l

≤ C2 2c τ

Γ( l−1 τ ) . Γ( l−1 2 )

Reuniting all the previous

bounds and estimates we get l−1 l−1 1 a τ l Γ( l Γ( a τ ) τ ) Leb(Acl ) ≤ 4ΩC1 [ ]1−τ e− τ [ 2 ] C2 2c τ l−1 ≤ C3 kωo k2c τ l−1 2 Γ( 2 ) Γ( 2 )

c and C3 are constants l and τ independent. 7

4. Main Results Theorem 4.1 Let hA = {hAi (Ai )}i∈Z : Rw → Rw satisfy 0 < a0 ≤ inf Ai ∈R |hAi Ai | ≤ supAi ∈R |hAA | ≤ a and let f : Tw → Bw be a uniformly weakly analytic g–gradient (see Definition 2.3). Then there exist uncountably many maximal almost–periodic solutions of (1.1) with Diophantine frequencies. The proof, achieved in some intermediate steps, is a corollary of theorem 4.7 and is written after it. Before writing that intermediate steps we try to explain our strategy. First of all we define a bijective map of Z onto N in such a way to have a system of variables indexed by j = 1, 2, 3, . . . . Let’s define lk = l − 1 + k, k = 0, 1, 2, . . . and the function Vlk (ϕ(lk ) ), ϕ(lk ) = (ϕ1 , . . . , ϕl−1+k ) where R [I ] [I ] −∂ϕi Vlk (ϕ(lk ) ) = fi k (ϕ(lk ) ), Ik = 1, 2, . . . , l − 1 + k and fi k (ϕ(lk ) ) = fi dµJk Jk = Z\Ik . Let’s define R . Pl−1+k a sequence of hamiltonians Hlk = i=1 h(Ai ) + Vlk (ϕ(lk ) ), Vlk (ϕ(lk ) ) = dϕl+k Vlk+1 (ϕ(lk+1 ) ). For k = 0 we take a vector–frequency

ωo µ

such that

∂h ∂Ai |Ao

=

(ωo )i µ

1 ≤ i ≤ l − 1. For |µ| small enough we

apply to Hl0 nl0 (an integer number) of canonical transformations (whose composition we call R(nl0 ) ) (n ) . (n ) (n ) so that the hamiltonian becomes Hl0 l0 = h(nl0 ) (A(nl0 ) ) + Vl0 l0 (A(nl0 ) , ϕ(nl0 ) ) and where Vl0 l0 is of nl 0

order ε2 (actually a number slightly less than 2 will be present but at the moment we leave 2). We stress that we we don’t take the limit nlo → +∞. Then we write the hamiltonian with one more d.o.f. . Pl . Pl Hl1 = i=1 h(Ai ) + Vl1 (ϕ(l1 ) ) as Hl1 = i=1 h(Ai ) + (Vl1 (ϕ(l1 ) ) − Vl0 (ϕ(l0 ) )) + Vl0 (ϕ(l0 ) ), and act on Hl1 (n )

with R(nl0 ) in such a way to obtain the new hamiltonian (h(nl0 ) (A(nl0 ) ) + Vl0 l0 ) + h(Al ) + P (ϕ(l1 ) ) where P by definition is the function (Vl1 (ϕ(l1 ) ) − Vl0 (ϕ(l0 ) )) after the action of R(nl0 ) . Now using R ∂h | o dϕl (Vl1 (ϕ(l1 ) )−Vl0 (ϕ(l0 ) )) = 0 and by means of a second canonical change of coordinate, if ωl = ∂A l A is Diophantine in the sense of (3.2) and Ω big enough, we can reduce the perturbation P to order Ω−1 . nl (n ) 0 Taking nl0 such that ε2 = O(Ω−1 ), both the terms P and Vl0 l0 are reduced to order Ω−1 and the (nl0 )

hamiltonian (h(nl0 ) (A(nl0 ) ) + Vl0

) + h(Al ) + P (ϕl0 ) is transformed into (h(nl0 ) (A(nl0 ) ) + h(Al )) +

(n ) (n ) V˜l0 l0 + P˜ (ϕ(l0 ) ) where V˜l0 l0 + P˜ are of order Ω−1 and Ω is a (fast) increasing function of the number nl 1

l of d.o.f.. Then a further canonical transformation reduces the perturbation to order Ω−2 . Of course the proceedure works if Ω is sufficiently large. At this point we can continue the procedure “adding” more and more d.o.f.. In the limit k → +∞ we get the infinite–dimensional torus whose vector frequency (Diophantine) is ( ωµo , ωl , ωl1 , . . .); |ωl | ≥ Ωl (Ωl is the previous Ω), |ωl1 | ≥ Ωl1 , |ωl2 | ≥ Ωl2 and so on. The vector ( ωµo , Ωl , Ωl1 , . . .) may not be a Diophantine vector because it may not satisfy the irrationality conditions (see Definition 3.3) even if its components are sufficiently large. A priori nothing forbids the vector ( ωµo , Ωl , Ωl1 , . . .) from being Diophantine. What we know by Lemma 3.4 (applied for each value of k = 0, 1, 2, . . .) is that outside the interval [Ωlk , Ωlk + L(τ, lk )kω (lk−1 ) k], certainly there is a value ωlk such that the vector ( ωµo , ωl , ωl1 , . . .) is Diophantine. Moreover L(τ, lk )kω (lk−1 ) k increases faster than Ωlk (L(τ, lk )kω (lk−1 ) k. Precisely it increases as

Γ( l−1 τ ) Γ( l−1 2 )

while Ωlk faster that any power of k but slower that an

exponential). This is a marked difference respect to [P] and [CP]. There what increases very fast with k ((k!)a a ∼ 10 for a d = 1–dimensional problem) is Ωlk and this forces the components of ( ωµo , ωl , ωl1 , . . .) to be very large. Here what increases faster is L(τ, lk )kω (lk−1 k) but this does not forbid the components  −τ . 1−τ ωlk from being close to Ωlk (possibly equal). Let’s define M (δ) = supt≥1 E(t)e−tδ = exp 1−τ . τ δ Pl−1 Lemma 4.2 We consider the hamiltonian Hl0 (A, ϕ) = i=1 h(Ai ) + V (ϕ): Rl−1 × Tl−1 → R where: 1 o l−1 . ω o 1) h: R → R is real–analytic, h ∈ C ω (C; C), supAi ∈C |hAi Ai | ≤ a, 2) {hAo }l−1 i=1 = { µ ωi }i=1 = µ , i

8

ω o is a (γ, τ )–Diophantine vector 3) V (ϕ) ∈ C ω (∆ξ ; C) ∩ C(∆ξ ; C), kVϕ kξ = V. There exists the canonical transformation (A, ϕ) = C(A0 , ϕ0 ), A = A0 + Ξ(A0 , ϕ0 ), ϕ = ϕ0 + ∆(A0 , ϕ0 ), Ξ and ∆ 1 belong to C ω (D(Ao ; ρ4 ) × ∆ξ−3δ ; C l−1 ) ∩ C(D(Ao ; ρ4 ) × ∆ξ−3δ ; C l−1 ), ξ − 3δ > 0, ρ = 2|µ|aγN E(N ) , N=

1 2δ

ln[(µ2 aγ 2 V)−1 ], such that l−1

X . (0) Hl0 (C(A0 , ϕ0 )) = Hl0 (A0 , ϕ0 ) = h(A0j ) + V 0 (A0 , ϕ0 )+ < V >

(4.1)

j=1


16 0 kVϕ0 k ρ ,ξ−4δ + kVA0 0 k ρ ,ξ−4δ ≤ 26 la(δρ)−1 (γVM (δ)|µ|)2 = Vx 16 16 ρ 2 2−σ  −τ (γ Va) l γVx ≤ 27 |µ|3−σ exp B2 δ 1−τ δ 4δσ

B2 (τ, σ) =

1−τ τ (2

(4.2) (4.3)

−τ

16 + σ 1−τ ), (0 < σ < 1) provided that (l − 1) ρδ γ |µ| VM (δ) ≤ 1

4 1 . max{ kΞk ρ ,ξ−3δ , k∆k ρ ,ξ−3δ } ≤ 8(l − 1) VM (δ)γ|µ| = To 4 4 ρ ρ

(4.4)

Remarks i) The lemma makes clearer (in an tractable way) that the problem is of perturbative nature. The bound Vx in (4.3) is small provided |µ| is small. ii) The kinetic part of the hamiltonian (4.1) is the same of Hl0 . This makes unnecessary in the proof the step where the frequency of the torus is “restored” iii) here δ is a fixed quantity but it will become smaller and smaller as the number of d.o.f. increases. 1 Then the quantities equivalent to |µ| (the size of the components of the frequencies following ωµo ) are forced to become greater and greater. Proof It is quite standard so we just sketch it. As usual one looks for the solution of the Hamilton–Jacobi equation X eiν·ϕ V ν Φ(A0 , ϕ) = − (4.5) iν · hA0 l−1 ν∈Z

0 0 such that

(0)

(1)

Hl0 (C(A0 , ϕ0 )) = Hl0 (A0 , ϕ0 ) =

l−1 X

h(A0j )+ < V 0 (A0 ) > +V 00 (A0 , ϕ0 )

(4.7)

j=1 0

0

0

ρ (1) ρ o ρ (A0 , ϕ0 ) ⊂ D(A(1) o ; 32 ) × ∆ξo −4δo , D(Ao ; 16 ) ⊂ D(A ; 8 )

γ

−τ 
00 32 00 24 l kVϕ0 k ρ0 ,ξ −4δ + kVA0 k ρ0 ,ξ −4δ ≤ γVx(1) ≤ 2 (γVx )2−σ exp( B2 (τ, σ)δo1−τ ) 0 o o 32 o 32 o ρ σδo

1 khA0 A0 + < V 0 (A0 ) >A0 A0 k ρ0 ≤ ao (1 + ), 32 8

∂A0 (h+ < V 0 (A0 ) >)|A(1) = o

ωo µ

1 k(hA0 A0 + < V 0 (A0 ) >A0 A0 )−1 k ρ0 ≤ ηo (1 + ) 32 8 The proof (omitted) is analogous to that of Lemma 4.2 except for one point. Here is necessary to restore Pl−1 ∂ ( j=1 h(A0j )+ < V 0 (A0 ) >) = (ωµo )i . It is an the vector–frequency of the torus and the calculation is ∂A i application of the analytic–implicit function theorem Lemma 4.3 leads to the following Theorem (repeat enough times the Lemma) Theorem 4.4

τ

Let’s consider the hamiltonian (4.7). Let be 0 < σ ≤ σo , σo < 2 − 2 1−τ  −τ (ηo ao )a1 la2 (ξo − ξn )−a3 exp c1 (ξo − ξn ) 1−τ (γVx ) ≤ 1

(4.8)

(a1 , a2 , a3 , c1 are constants depending on τ, σ and independent on the number of d.o.f. l − 1), we define . a finite number of canonical tranformations C (j) (A(j) , ϕ(j) ) j = 1, . . . , n (n = nl0 ) whose domain is D(A(j) o ; ρj ) × ∆ξj (ρj ≥ ρj+1 > 0, ξj > ξj+1 > 0) such that the hamiltonian (4.7) becomes (n ) . Hl0 l0 (A(n) , ϕ(n) ) = . = H (0) (C (1) ◦ C (2) . . . ◦ C (n) )(A(n) , ϕ(n) ) = h(n) (A(n) ) + V (n) (A(n) , ϕ(n) ) (n)

(n)

k(hA(n) A(n) )−1 kρn ≤ 2ηo

khA(n) A(n) kρn ≤ 2ao , γ

h

(n) (n)

Ao

=

(4.9)

ωo µ


n 1 (n) (n) kVϕ(n) kρn ,ξn + kVA(n) kρn ,ξn ≤ γVx(n) ≤ (QγVx )(2−σ) ρn

(4.10)

−τ   2
1−σ (ξo −ξn ) 1−τ Ko0 l−1 K e ; Ko and Ko0 are constants depending on σ and τ ; ξo − ξn > 0. Q= o (ξo −ξn )σ Moreover for each j = 1, . . . , n − 1

(A(j−1) , ϕ(j−1) ) = C (j) (A(j) , ϕ(j) ) = = (A(j) + Ξ(j) (A(j) , ϕ(j) ), ϕ(j) + ∆(j) (A(j) , ϕ(j) )) ρj = (2aj γ|µ|Nj E(Nj ))−1 , Nj = max {

1 2δj

(4.11)

ln(γVx(j) )−1

1 kΞ(j) k2ρj ,ξj−1 −3δj−1 2ρj

,

k∆(j) k2ρj ,ξj−1 −3δj−1 } ≤

−τ  16(l − 1) . (j−1) 1−σ 1−τ (γV ) exp B (τ, σ)δ = Tj ≤ x 1 j−1 2 σδj−1

10

(4.12)

Proof

It is a repeated application of Lemma 4.3 plus a standard induction argument o

Remarks i) If we were to prove the existence of l −1 dimensional tori with high frequencies ωµ ∈ Rl−1 we should make the limit n → +∞. Actually our problem is infinite–dimensional and we will fix n according to how big will be the frequency (ωl ) (see the upper bound in the third formula of (4.19)) corresponding to the next d.o.f (l). Here is a key point in the proof. We do not take limn→+∞ now because we are not interested in a finite–dimensional problem. The integer n will be high accordingly to ωl . The unboundedness of the sequence {ωi } (which is a consequence of the unboundednes of the d.o.f.) will force  −τ n to go to +∞. The upper bound on µ can be rewritten as la2 (ξo −ξn )−a3 exp c1 (ξo −ξn ) 1−τ O(µ2−σ ) ≤ 1 and we stress that the bigger is l−1 (the d.o.f.), the smaller is µ. “Adding” degrees of freedom, the number l increses and we take 0 < τ < 12 to avoid the factorial behavior µ ∼ (l!)−b with b positive which otherwise would arise Let’s call C (0) the canonical transformation of Lemma 4.2 (there is indicated with C) and let’s define (C (0) ◦ C (1) ◦ . . . ◦ C (n) ) = R(n) where C (1) ◦ . . . ◦ C (n) is the transformation written in (4.9). k∂(C (0) ◦ C (1) ◦ . . . ◦ C (n) )kρn ,ξn ≤ Πnj=0 1 + (1 +


1 )Tj ≤ Co δj

Co > 1

(4.13)

(possibly adding 1 to the exponent a3 of (4.8)). Co is a constant depending on the number of d.o.f. The next step is the important passage from l − 1 to l degrees of freedom. Let’s consider the hamiltonian Pl Hl1 (I, ψ) = i=1 h(Ii ) + Vl1 (ψ1 , . . . , ψl ): Tl × Rl → R, k(Vl1 )ψ kξ = V (l) and rewrite it as l−1 X

h(Ii ) + Vl0 (ψ1 , . . . , ψl−1 ) + h(Il ) + (Vl1 (ψ) − Vl0 (ψ1 , . . . , ψl−1 ))

(4.14)

i=1

R 2π l l−1 where Vl0 (ψ1 , . . . , ψl−1 ) = 0 dψ , 2π Vl1 (ψ). Vl0 (ψ1 , . . . , ψl−1 ) is exactly the function Vl1 (ψ), ψ ∈ T present in the hamiltonian of Lemma 4.2. To the set of variables (I1 , . . . , Il−1 , ψ1 , . . . , ψl−1 ) we apply the transformation R(n) . (Il , ψl ) undergo an identity transformation. Hl1 (I, ψ) becomes (n) ˆ (n) , ϕ(n) ), ψ ) − V (n) (A(n) , ϕ(n) )) h(n) (A(n) ) + Vl0 (A(n) , ϕ(n) ) + h(Il ) + (Vl1 (ψ(A l l0

(4.15)

. ˆ . Now let’s recall (A(n) , Il ) = (A, Al ) = A, (ϕ(n) , ψl ) = (ϕ, ˆ ϕl ) = ϕ, and rewrite the hamiltonian (4.15) as . ˜ ˜ (A) ˜ (A, ϕ) = h ˆ + h(A ) + V˜ (A, ˆ ϕ) ˆ ϕ) = ˆ ϕ) H ˆ + V˜2 (A, hl1 (A) + V˜l1 (A, l1 1 l 1 . (n) (n) ˜ (A) = h h (A ) + h(Il ), l1

(4.16)

. (n) ˆ ϕ) V˜1 (A, ˆ = Vl0 (A(n) , ϕ(n) ),

. ˆ (n) , ϕ(n) ), ψ ) − V (n) (A(n) , ϕ(n) )) ˆ ϕ) = V˜2 (A, (Vl1 (ψ(A l l0 ˜ ) (Aˆ(n) , Ao ) = ( ωo , α), Aˆ ∈ D(Aˆ(n) ; ρ ), A ∈ D(Ao ; r˜), ϕˆ ∈ ∆ , ϕ ∈ ∆ , (h l1 A o l o n l l ξn l ξ µ o Being ρn < r˜ and ξn < ξ, we restrict to D((Aˆ(n) o , Al ); ρn ) the domain of the action–variables (n) o (A ∈ D((Aˆo , Al ); ρn )) and to ∆ξn the domain of the angular–variables (ϕ ∈ ∆ξn ). Now let’s define o . o ˜ ˜ −1 (Aˆ(n) o , Al ) = A k(hl1 )AA kD(Ao ;ρn ) ≤ 2ao k(hl1 )AA kD(Ao ;ρn ) ≤ 2ηo

γ

  1 n k(V˜1 )ϕˆ kρn ,ξn + k(V˜1 )Aˆ kρn ,ξn ≤ (QγVx )(2−σ) ρn 11

n

By (4.13) and for ργn (QγVx )(2−σ) ≤ Co V (l) we have k(V˜2 )ϕkρn ,ξn 2V (l) Co and k(V˜2 )Aˆ kρn ,ξn ≤ δo is the “loss of analyticity” of Lemma 4.2 and To is defined in (4.4); Let’s recall ρn = ρ˜ and ξn = ξ˜ 1 ˜ ˜ ˆ k ˜ ≤ 4 V (l) C k(V2 )ϕ kρ, o ˜ ξ˜ + k(V2 )A ρ, ˜ξ ρ˜ ρ˜

(l) 2 ρn V Co

1 ˜ ˜ ˆ k ˜ ≤ 5 V (l) C k(Vl1 )ϕ kρ, o ˜ ξ˜ + k(Vl1 )A ρ, ˜ξ ρ˜ ρ˜

. We are now ready to state a Lemma analogous to Lemma 4.2 (let’s recall V˜l1 = V˜ ) ˜ (A) ˜ ˆ + V˜ (A, ˆ ϕ): Tl × Rl → R (see Lemma 4.6 We consider the hamiltonian H(A, ϕ) = h l1 l1 ˜ ∈ C ω (D(Ao ; ρ˜); C), k(h ˜ −1 ˜ ) k 2) (4.16)) where: 1) h ˜ ≤ 2ao , k(hl1 )AA kD(Ao ;ρ) l1 AA D(Ao ;ρ) l1 ˜ ≤ 2ηo , o

˜ ) (Ao ) = ( ω , α) which is a Diophantine vector in the sense of (3.3), (h l1 A µ ∆ξ˜; C) ∩ C(D(Aˆo ; ρ˜) × ∆ξ˜; C), ˜ 0 , ϕ0 ), defined in (A, ϕ) = C(A

ˆ ϕ) ∈ C ω (D(Aˆo ; ρ˜)× 3) V˜ (A,

1 ˜ ˜ ˆ k ˜ ≤ 5 C V (l) . There exists the canonical transformation ˜ ξ˜ + kVA ρ, ˜ξ ρ˜ kVϕ kρ, ρ˜ o  Ω  ρ˜ 0 0 o ρ˜ 1 ˜ ˜ D(A ; 4 ) × ∆ξ−3 ˜ δ˜ 3 (A , ϕ ), ξ − 3δ > 0, N = 2δ˜ ln M (δ) ˜ 8Co V (l) such that

. (0) 0 0 ˜ (A0 ) + V˜ 0 (A0 , ϕ0 ) ˜ (C(A ˜ 0 , ϕ0 )) = H Hl1 (A , ϕ ) = h l1 l1

(4.17)

˜ . 1 11 M (δ) 16 ˜ 0 ˜0 2 (Co V (l) )2 kVϕ0 k ρ˜ ,ξ−4 = V˜x(o) ˜ δ˜ + kVA0 k ρ˜ ,ξ−4 ˜ δ˜ ≤ ˜ 16 16 ρ˜ ρ˜Ω δ ρ˜

(4.18)

provided that ˜

2

2

6 M (δ)

δ˜

l (4Co V (l) ) ≤ 1, ρ˜Ω


σ ao ρ˜ ρ˜Ω exp (l) ˜ ˜ δσΩ 8M (δ)Co V



1−τ 2 τ ˜ 1−τ τ (σ δ)

 ≤ 1,

2 ˜ lγ ˜ γ (QγV )−(2−σ)n (Co V (l) ) ≤ γΩ ≤ 25 (Co V (l) )2 M (δ) x ρ˜2 δ˜ ρ˜

(4.19)

6 M (δ)

Proof Following Lemma 4.2 we construct the canonical transformation whose generating function is ˜ ˜ ˜ 0 , ϕ) and (suppress in this proof the index l from h) S(A0 , ϕ) = A0 · ϕ + Φ(A 1 Φ(A0 , ϕ) = −

X ν∈Zl

eiν·ϕ (V˜2 )ν (Aˆ0 ) ˜ 0 iν · h A

(4.20)

νl 6=0 ,0