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Field theory and KAM tori@

arXiv:chao-dyn/9503006v1 20 Mar 1993Field theory and KAM tori@G. Gallavotti,∗G. Gentile,#, V. Mastropietro&Abstract: The parametric equations of KAM tori for a l degrees of freedom quasi integrable system,are shown to be one point Schwinger functions of a suitable euclidean quantum field theory on the ldimensional torus.

KAM theorem is equivalent to a ultraviolet stability theorem. A renormalizationgroup treatment of the field theory leads to a resummation of the formal pertubation series andto an expansion in terms of l2 new parameters forming a l × l matrix σε (identified as a family ofrenormalization constants).

The matrix σε is an analytic function of the coupling ε at small ε: thebreakdown of the tori at large ε is speculated to be related to the crossing by σε of a “critical” surfaceat a value ε = εc where the function σε is still finite. A mechanism for the possible universality of thesingularities of parametric equations for the invariant tori, in their parameter dependence as well asin the εc −ε dependence, is proposed.1.

IntroductionWe consider l rotators with inertia moments J, angular momenta ⃗A = (A1, . .

. , Al) ∈Rl, and angularpositions ⃗α = (α1, .

. .

, αl) ∈Tl. Their motion will be described by the hamiltonianH = 12J−1 ⃗A · ⃗A + ε f(⃗α, ⃗A) ,⃗A ∈Rl, ⃗α ∈Tl ,f =X|⃗ν|≤Nf⃗ν( ⃗A) ei⃗ν·⃗α ,f⃗ν( ⃗A) = f−⃗ν( ⃗A) ,(1.1)with f⃗ν( ⃗A) a polynomial in ⃗A.1 Let ⃗ω0 = J−1 ⃗A0 be a rotation vector, “angular velocity vector”,verifying for C0, τ > 0 suitably chosen the diophantine propertyC0|⃗ω0 · ⃗ν| > |⃗ν|−τ ,⃗0 ̸= ⃗ν ∈Zl .

(1.2)The KAM theorem states the existence of a one parameter family ε →Tε of tori with parametricequations⃗A = ⃗A0 + ⃗H(⃗ψ) ,⃗α = ⃗ψ + ⃗h(⃗ψ) ,⃗ψ ∈Tl ,(1.3)@ Archived in mp arc@math.utexas.edu, #95-151; in chao-dyn@xyz.lanl.gov, # 9503??? ; last versionat http://chimera.roma1.infn.it∗Dipartimento di Fisica, Ia Universit`a di Roma, P.le Moro 2, 00185 Roma, Italia; partially supported by Rutgers U.and CNR-GNFM.# IHES, 91440 Bures s/Yvette, France.& Dipartimento di Matematica, IIa Universit`a di Roma, 00133 Roma, Italia; supported from CNR-GNFM.1 Analyticity of f in a domain W (⃗A0, ρ) = {⃗A ∈Rl : |⃗A −⃗A0|/|⃗A0| < ρ} would make the matter more complicateonly slightly.1

where ⃗H(⃗ψ) and ⃗h(⃗ψ) are analytic functions of ε, ψj, j = 1, . .

. , l, divisible by ε, defined for |ε|, |Im ψj|small enough.

Such tori are uniquely determined by the requirements:(a)⃗ψ →⃗ψ + ⃗ω0t solves the equations of motion ,(b)H(⃗ψ) is even in ⃗ψ ,(c)h(⃗ψ) is odd in ⃗ψ ,(1.4)Consider the four (formally) gaussian vector fields ⃗Φ ≡( ⃗Hσ,⃗hσ), σ = ±, defined on the torus Tl,and with propagators2⟨h+⃗ψ,jh−⃗ψ′,j′⟩=δj,j′X⃗νei( ⃗ψ−⃗ψ′)·⃗ν(i⃗ω0 · ⃗ν + Λ−1)2 ≡δj,j′ S2(⃗ψ −⃗ψ′) ,⟨h2+⃗ψ,jH−⃗ψ′,j′⟩=⟨H+⃗ψ,jh−⃗ψ′,j′⟩= δj,j′X⃗νei( ⃗ψ−⃗ψ′)·⃗ν(i⃗ω0 · ⃗ν + Λ−1) ≡δj,j′ S1(⃗ψ −⃗ψ′) ,(1.5)where Λ is a ultraviolet cut off.3 The other propagators are taken to be zero. The physical dimensionsof the field ⃗h+, ⃗h−, ⃗H+, ⃗H−are respectively [1], [ω−2], [ω], [ω−1] in terms of the dimension [ω] of ⃗ω0.We sall also set ⃗Φ1± ≡H± and ⃗Φ2± ≡h±.We denote by P(dΦ) the formal functional integral with respect to the above gaussian process, andconsider the field theory with ⃗Φ as free field and actionV (Φ) = −εZTl d⃗ψ J−1h−⃗ψ · ∂⃗ψf⃗ψ + h+⃗ψ, ⃗A + JH+⃗ψ−εZTl d⃗ψ H−⃗ψ · ∂⃗Af⃗ψ + h+⃗ψ, ⃗A + JH+⃗ψ+ Λ−1⃗a(ε) ·ZTl d⃗ψ h−⃗ψ(1.6)where ⃗a will be called counterterm, and its physical dimensions are [⃗ω].It is easy to check that the Schwinger functionsSn(⃗ψ1, s1, σ1; .

. .

; ⃗ψn, sn, σn) =RP(dΦ) e−V (Φ) Φs1σ1⃗ψ1. .

. Φsnσn⃗ψnRP(dΦ) e−V (Φ)(1.7)of the non polynomial formal4 action Eq.

(1.6) are well defined if the one point Schwinger functions⃗h(⃗ψ) ≡S1(⃗ψ, 2, +) =RP(dΦ) e−V (Φ) h+⃗ψR P(dΦ) e−V (Φ),⃗H(⃗ψ) ≡S1(⃗ψ, 1, +) = JRP(dΦ) e−V (Φ) H+⃗ψR P(dΦ) e−V (Φ)(1.8)are well defined. The reason is simply that the structure of the free field and that of the action implythat all the Feynman graphs of the theory must be either trees or families of disconnected trees.

Therenormalization constant ⃗a(ε) will be fixed by requiring that the average of ⃗h vanishes. As in fieldtheory one could fix ⃗a equivalently by requiring that the average of ⃗h has a prefixed value: it is onlyimportant that ⃗h is well defined when Λ →∞and the value ⃗0 for its average has no special meaning,except that it is a convenient normalization which, as we shall see, makes use of the symmetry of theproblem inherited by the fact that f has a cosine Fourier series and this simplifies some considerations.2 i.e.

linear functionals on the space of complex felds on Tl such that the moments are evaluated by using the Wickrule.3 Because ⃗ω0 · ⃗ν, ⃗ν ̸= ⃗0, can become small only for |⃗ν| large.4 Because the Φ’s are complex and f is a trigonometric polynomial.2

The case in which f is ⃗A-independent has been studied in [G3], where it has been shown that inthe limit Λ →∞the one point Schwinger functions are precisely the functions ⃗h and ⃗H defined bythe KAM theorem, provided the counterterms ⃗a are chosen ⃗0. In [G3] the ⃗a does not appear (as itis ⃗0 for symmetry reasons) so that the analysis is considerably simpler and no cut offΛ is necessary.The necessity of ⃗a ̸= ⃗0 arises only if f is ⃗A-dependent (and it is related to the twist condition thatbecomes necessary in such a case: note that in [G3] the twist condition was not required; furthermore,as a consequence, only one field, namely ⃗h ⃗ψ, was used).In this paper we study the more general case in which the action Eq.

(1.6) depends also on ⃗A. Ifthe ultraviolet cut offΛ is finite the perturbative expansion for the Schwinger functions is convergentfor ε suitably small and for any choice of the counterterms.

However, in the limit Λ →∞the series isconvergent for a unique choice of the counterterm ⃗a(ε). This is what happens generically in quantumfield theory, in which the pertubative series for Schwinger functions converge only if a unique choice ofthe counterterm is made (see for instance the case of φ4, [G1]).

Moreover the choice of the countertermswhich makes the perturbative series finite in the limit Λ →∞is such that ⃗h, ⃗H in Eq. (1.8) coincidewith the corresponding quantities in the KAM theorem.2.

The Schwinger functions expansion.The latter statement can proved by writing recursively the one point Schwinger function to order n,H(n)⃗ν,j and h(n)⃗ν,j and comparing it with a similar recursive construction of the Lindstedt series for theKAM functions ⃗H,⃗h.The exponentials in Eq. (1.7) are expanded in powers of V and the P integrals of the resultingproducts of fields are evaluated using the Wick rule leading to the familiar Feynman diagrams: thespecial form of V immediately implies that the diagrams have no loops, i.e.

they are tree diagrams.The diagrams will be described later: here it is sufficient to remark that even without using thediagram representation the evaluation of the integrals immediately leads to the following recursiverelations between the coefficients of the power series (in ε) expansion of the functions ⃗H,⃗h in Eq. (1.8), i.e.

the one field Schwinger functions of the theory described by Eq. (1.5), Eq.

(1.6):H(k)⃗ν,j = S1⃗νnX∗(−i⃗ν0)jXp,q≥01p!q!pYs=1i⃗ν0 · ⃗h(ks)⃗νsqYi=1 ⃗H(k′i)⃗ν′i· ∂⃗Af⃗ν0( ⃗A) ⃗A= ⃗A0o+ Ja(k)j δ⃗ν,⃗0 ,(2.1)and:h(k)⃗ν,j = S2⃗νnX∗(−iJ−1⃗ν0)jXp,q≥01p!q!pYs=1i⃗ν0 · ⃗h(ks)⃗νsqYi=1 ⃗H(k′i)⃗ν′i· ∂⃗Af⃗ν0( ⃗A) ⃗A= ⃗A0o+ Λa(k)j δ⃗ν,⃗0 + S1⃗νnX∗Xp,q≥01p!q!pYs=1i⃗ν0 · ⃗h(ks)⃗νsqYi=1 ⃗H(k′i)⃗ν′i· ∂⃗A∂⃗Ajf⃗ν0( ⃗A) ⃗A= ⃗A0o,(2.2)where the P∗denotes sum over the integers ks, k′i ≥1 and over the integers ⃗ν0, ⃗νs, ⃗ν′i, withpXs=1ks +qXi=1k′i = k −1,⃗ν0 +pXs=1⃗νs +qXi=1⃗ν′i = ⃗ν . (2.3)The integer vectors ⃗νs,⃗ν′i,⃗ν0,⃗ν may be ⃗0.For ⃗ν = ⃗0, from the above relations we obtainH(k)⃗0,j = ΛX(k)j+ Ja(k)j ,h(k)⃗0,j = J−1Λ[ΛX(k)j+ Ja(k)j ] + ΛY (k)j,(2.4)3

where X(k)je Y (k)jare read from Eq. (2.1) and Eq.

(2.2) for ⃗ν = ⃗0. The condition that ⃗h(k)⃗0= ⃗0determines, recursively, a(k)jand implies ⃗H(k)0= −J ⃗Y (k).The first order calculation yields⃗H(1)⃗ν= S1⃗ν(−i⃗ν) f⃗ν + J⃗a(1)δ⃗ν,⃗0 ,⃗h(1)⃗ν= J−1S2⃗ν(−i⃗ν) f⃗ν + S1⃗ν⃗a(1)δ⃗ν,⃗0 + S1⃗ν∂⃗Af⃗ν ,(2.5)and the limit as Λ →+∞is well defined if ⃗a(1) = J−1 ⃗H(1)⃗0= −∂⃗Af⃗0( ⃗A0), and it is⃗H(1)⃗ν= (−i⃗ν) f⃗ν( ⃗A0)i⃗ω · ⃗ν,⃗h(1)⃗ν= (−iJ−1⃗ν) f⃗ν( ⃗A0)(i⃗ω · ⃗ν)2+ ∂⃗Af⃗ν( ⃗A0)i⃗ω · ⃗ν,⃗ν ̸= ⃗0 ,⃗H(1)⃗0= −J∂⃗Af⃗0( ⃗A0) ,⃗h(1)⃗0= ⃗0 ,if J⃗a(1) = H(1)⃗0,(2.6)with ⃗h(1)⃗0= ⃗0 and the functions ⃗H and ⃗h respectively even and odd in ⃗ν, (as in [GM]).Then, if we want that the expressions in Eq.

(2.1), Eq. (2.2) are well defined when Λ →∞, weproceed inductively by supposing that by suitably fixing ⃗a(k) the functions ⃗H(k) and ⃗h(k) have a welldefined limit as Λ →+∞and become, respectively, even and odd in ⃗ν when the limit is taken.

Weassume this to be true for k′ ≤k −1: we see that this implies X(k)j= 0 in the first equation, and thechoice a(k)j= −Y (k)jmakes the parity and finiteness requests to be fulfilled to order k.3. The Lindsted series.The classical construction of the formal series representation for the functions ⃗H,⃗h in Eq.

(1.3)defining parametrically the KAM torus starts from the Hamilton equations of motion for Eq. (1.1).One imposes that by replacing ⃗ψ with ⃗ψ + ⃗ω0t in Eq.

(1.3) one gets an exact solution to the equationsof motion. The following equations are obtained:⃗ω0 · ⃗∂⃗ψ ⃗H(⃗ψ) = −ε ∂⃗ψf⃗ψ + ⃗h(⃗ψ), ⃗A0 + ⃗H(⃗ψ),⃗ω0 · ⃗∂⃗ψ ⃗h(⃗ψ) =J−1 ⃗H(⃗ψ) + ε ∂⃗Af⃗ψ + ⃗h(⃗ψ), ⃗A0 + ⃗H(⃗ψ).

(3.1)To make easier the comparison with the euclidean field theory of §2 we can introduce a cut offparameter Λ and consider the regularized equations(Λ−1 + ⃗ω0 · ∂⃗ψ) ⃗H(⃗ψ) = −ε ∂⃗ψf⃗ψ + ⃗h(⃗ψ), ⃗A0 + ⃗H(⃗ψ),(Λ−1 + ⃗ω0 · ⃗∂⃗ψ)⃗h(⃗ψ) =J−1 ⃗H(⃗ψ) + ε ∂⃗Af⃗ψ + ⃗h(⃗ψ), ⃗A0 + ⃗H(⃗ψ). (3.2)We can solve Eq.

(3.2) by a perturbation expansion, by writing ⃗H = P∞k=1 εk ⃗H(k) and ⃗h =P∞k=1 εk⃗h(k). If one requires ⃗h(k)⃗0= ⃗0 then it follows immediately that the recursive constructionof ⃗H(k),⃗h(k) is possible and in fact it clearly coincides with Eq.

(2.1)÷Eq. (2.6).

The existence ofsuch formal series is known (if Λ = +∞) as the Lindstedt theorem: and it goes back to Poincar´e whoextended to all orders the original proofs of Lindstedt and Newcomb.The convergence radius of the Lindstedt series (hence of the euclidean field theory of §2) is uniformin Λ. For Λ = +∞this is the KAM theorem; a proof based on bounds on the coefficients ⃗H(k),⃗h(k) isdue to Eliasson, [E].

It was recently “simplified” in various papers [G2], [GG], [GM], see also [CF] fora very similar approach. The proof in [G2], [GM] can be easily extended to cover the case Λ < +∞.4

Hence the theory is uniform in the ultraviolet cut offΛ (of course the convergence at fixed Λ < ∞isquite trivial; the uniformity as Λ →∞, on the other hand, is equivalent to KAM).4. The renormalization group and resonance resummation.The KAM theory, thus, permits us to give a meaning to the non regularized field theory with actionEq.

(1.6), a somewhat surprising fact. Therefore it is interesting to investigate in more detail thestructure of the perturbation theory.As already pointed out the model is, from the point of view of field theory, somewhat deceiving asits Feynman diagrams have no loops.

Nevertheless the model is clearly non trivial and it requires adelicate analysis of a family of cancellations that make the ultraviolet stability possible at all.With the choice of the counterterm ⃗a(ε) las in §2 the Feynman rules for the model can be summarizedas follows. Consider k oriented lines, labeled from 1 to k: the final extreme v′ of the lines will becalled the root and the other extreme v will be a vertex.

The lines, denoted v′ ←v are arranged on aplane by attaching in all possible ways the vertices of some segments to the roots of others, to form aconnected tree.In this way only one root r remains unmatched and it will be called the root of the graph whoselines will be called branches and whose vertices other than the root will be called nodes.Each node v is given a mode label ⃗νv which is one of the Fourier mode ⃗ν such that f⃗ν ̸= 0 (seeEq. (1.1)).

We define the momentum flowing on the branch going from v to v′ as ⃗ν(v) = Pw≤v ⃗νw.Furthermore each branch is regarded as composed by two halves each carrying a label H or h (sothere are four possibilities per branch).Trees that can be superposed modulo the action of the group of transformations generated by thepermutation of the branches emerging from a node will be identified.To each tree we associate a value obtained by assigning to a branch v′ ←v the following quantities,if ⃗ν(v) ̸= ⃗0,a factor−i⃗νv′ · iJ−1⃗νv(i⃗ω0 · ⃗ν(v) + Λ−1)2h ←han operatori⃗νv′ · ∂⃗Avi⃗ω0 · ⃗ν(v) + Λ−1h ←Han operator−∂⃗Av′ · i⃗νvi⃗ω0 · ⃗ν(v) + Λ−1H ←hjust0H ←Hfor all the branches distinct from the one containing the root: here the symbol to the right distiguishesthe four type of labels that can be on the line v′ ←v (the arrow tells which is the right label andwhich is the left one). To the root branch we associate, instead, the following quantities, if ⃗ν(v) ̸= ⃗0,a vector−iJ−1⃗νv(i⃗ω0 · ⃗ν(v) + Λ−1)2h ←han operator∂⃗Avi⃗ω0 · ⃗ν(v) + Λ−1h ←Ha vector−i⃗νvi⃗ω0 · ⃗ν(v) + Λ−1H ←hjust0H ←HTo each branch with ⃗ν(v) = 0 which is not the root branch we associate a factor −J∂⃗Av · ∂⃗Av′ ,5

if H ←h, and 0 otherwise, while to the root branch we associate a factor −J∂⃗Av, if H ←h, and 0otherwise.We multiply all the above operators (the factors are regarded as multiplication operators) and applythe resulting operator to the function Qv f⃗νv( ⃗Av), evaluating the result at the point ⃗Av ≡⃗A0. Thisdefines the Feynman rules: the ⃗H(k)⃗νand ⃗h(k)⃗νare given by k!−1 times the sum of all the values of allthe k branches trees with total momentum ⃗ν.

In the limit Λ →∞, the above expressions are all welldefined: this is easily checked. The expansion was developed in [G2], [GM] and it coincides essentiallywith the one used in [E] (and [CF]).Note that, in [GM], each time a line λ carries a vanishing momentum, all the subtrees of fixedorder k1 having λ as first branch are summed together to give, by construction, the value of thecounterterm ⃗a(k1).

Such a contribution is called fruit in [GM], and a line of a fruitful tree can havevanishing momentum only if it comes out from a fruit. Obviously the two ways to arrange the sumsover the trees are equivalent, and give the same result, once the sums are extended to all the possibletrees.The scaling properties of the propagators (when Λ = +∞) suggest decomposing them into compo-nents relative to various scales.Let χ1, χ be two smooth functions such that:(1) χ1(x) ≡0 if |x| < 1 and χ1(x) ≡1 for |x| ≥1.

(2) χ(x) ≡0 for |x| < 12 or for |x| ≥1, and 1 otherwise. (3) 1 ≡χ1(x) + P0n=−∞χ(2nx)Then we can write:Sa⃗ν ≡1(i⃗ω0 · ⃗ν)a = χ1(⃗ω0 · ⃗ν)(i⃗ω0 · ⃗ν)a +0Xn=−∞χ(2−n⃗ω0 · ⃗ν)(i⃗ω0 · ⃗ν)a,a = 1, 2,(4.1)and correspondingly we can break each Feynamn graph into a sum of many terms by developing thesums in Eq.

(4.1). This can be simply represented by assigning to each branch λ an extra label nλand multiplying the factor associated to such a line times χ(2−nλ⃗ω0 · ⃗ν): the value of ⃗H(k)⃗ν,⃗h(k)⃗νwillbe the sum over all possible new graphs which once deprived of the new scale labels would become“old” graphs contributing to ⃗H(k)⃗ν,⃗h(k)⃗νrespectively.The branches of the new graphs are naturally collected into connected clusters “of fixed scale”: acluster of scale n (n = 1, 0, −2, .

. .) consists in a maximal connected set of branches with scale label≥n, containing at least one line of scale n. By definition each cluster is again a tree graph.

Thelines which are not contained in a cluster, but have an extreme inside the clusters will be called theexternal lines of the cluster: if the extreme inside the resonance is the root, they will be incoming,while if the extreme is the node they will be outgoing. There can be at most one outgoing line percluster.The clusters are, by definition, hierarchically ordered and therefore they form a tree with respectto the partial ordering generated by the inclusion relation between clusters.Examining the convergence of the perturbation series it becomes clear that if one considers thesum of the contributions to ⃗H(k),⃗h(k) by all the graphs that do not contain clusters with just oneincoming and one outgoing branch which, furthermore, have the same momentum ⃗ν, then the seriesso generated converge for ε small, [E], [FT].Therefore the clusters of the latter type (with one incoming and one outgoing equal momentumbranches) are called resonances and the KAM theory can be interpreted as an analysis of the reasonwhy the resonances do not destroy the analyticity in ε at ε small, i.e.

of the cancellations that makethe resonances give a contribution much smaller than one could fear.One can imagine to consider a graph and replace each resonance together its external lines with anew simple line, which will be called dressed line. We collect togheter all the graphs which becomeidentical after such an operation.We consider here for simplicity only the case in which f is ⃗A independent; the discussion of themore general case, f = f(⃗α, ⃗A), can be carried out in the same way and it is only notationally more6

involved, so that, for simplicity’s sake, we relegate it into Appendix A2. If we multiply each graphvalue by the appropriate power of ε (equal to the number of branches of the graph) we see that thevalues of ⃗H and ⃗h can be computed by considering all the graphs without resonances and by addingresonant clusters to each of their lines.

This simply means that a line factor of scale n has to bemodified as:χ(2−n⃗ω0 · ⃗ν(v))(−i⃗νv′ · iJ−1⃗νv)(i⃗ω0 · ⃗ν(v))2→χ(2−n⃗ω0 · ⃗ν(v))(i⃗ω0 · ⃗ν(v))2(−i⃗νv′ · [(1 −σn,ε(⃗ω0 · ⃗ν(v))]−1iJ−1⃗νv)(4.2)where σn,ε(⃗ω0 ·⃗ν) is a suitable function representing the sum of all the possible insertions of a resonantcluster on the line v′ ←v. The function σn,ε(⃗ω0 · ⃗ν) ≡σn,ε(2nx) does not vanish only for x in theinterval [ 12, 1].The following result is an immediate consequence of the results in [G2], [GM2].Theorem.

The matrix σn,ε(2nx) is analytic in ε for ε small, independently on n and there is aconstant R such that ||σn,ε(2nx)|| < R|ε|.Furthermore the limit:limn→−∞σn,ε(2nx) = σε(4.3)exists and is a x–independent function of ε, analytic for ε small enough and divisible by ε.The second part of the above theorem is discussed in Appendix A1.The first part is provenin [GM2] in a version in which the χ functions are not characteristic functions as above, but aresmoothed versions at least two times differentiable. However one can easily take them to be as above:this implies that when they are differentiated their derivatives have to be interpreted as combinationsof delta functions.But one checks that most of of such terms cancel with each other with someobvious exceptions which can be easily bounded.

The possibility of using characteristic functions inthe decomposition Eq. (4.1) can also be seen from [G2], where the decomposition is done as above.The constant matrix σε will be called the resonance form factor.It is natural to consider the two parameters series ⃗H∗(⃗ψ, ε, σ),⃗h∗(⃗ψ, ε, σ) obtained from the reso-nance resummed series by replacing σn,ε by a new, independent parameter σ.

Then the above theoremand the results of [G2],[GG],[GM] imply that the functions ⃗H∗,⃗h∗are analytic both in ε and σ nearthe origin.In fact it is clear that the functions ⃗H∗,⃗h∗depend only on the variables η = ε(1 −σε)−1. Thusthe possibility arises that a singularity for ⃗H∗,⃗h∗is reached at a value εc of ε where σε is stillfinite.

It seems natural, to us, to think that the singularities of the functions ⃗h, ⃗H as ε →εc arethe same as those of ⃗h∗, ⃗H∗. If so the breakdown of the torus can be studied by using for it a muchsimpler perturbation representation, i.e.

a representation in which no resonance appears in the graphsrepresenting the ⃗H∗,⃗h∗.5. Heuristic discussion of a possible universality mechanism for the brakdown of the tori.The scalar quantity σε plays the role of a stability indicator and it would be nice to see some inde-pendent physical interpretation of it.

A numerical study of the function σε appears highly desirable,as well as that of the functions ⃗H∗,⃗h∗.The possibility that the singularities of ⃗H∗,⃗h∗, as functions both of ε and ⃗ψ, have a universal naturebecomes clear because the behaviour of the large order coefficients of ⃗H∗,⃗h∗, as series in ε, is likelyto be very mildly dependent on the actual values of the Fourier components f⃗ν. This can be seen tohappen when only the contributions to the coefficients arising from simple classes of trees are takeninto account.7

The simplest class of graphs which does not give a trivial contribution, i.e. contribution which is anentire function of ε, to the invariant tori is given by the set of trees of the form (linear chains):1234k −1kWe consider the contribution to ⃗h∗(⃗ψ, ε, σ) due to the above trees.

For simplicity we fix l = 2,⃗ω = (r, 1) with r =√5−12= golden section and the perturbation as an even function of ⃗α only asf(⃗α) = a cos α1 + b cos(α1 −α2) (“Escande Doveil pendulum”).Let us call “resonant line” the line ortogonal to ⃗ω, i.e. parallel to (1, −r).Let (pn, qn) be theconvergents for continued fraction for r (i.e.

p1 = 1, p2 = 1, p3 = 2, ... = Fibonacci sequence, andq1 = 1, q2 = 2, q3 = 3, . .

. with qn = pn+1 and pn+1 = pn + pn−1, and we set p0 = q−1 = 0 andp−1 = q0 = 1).Any integer s ≥1 can be written:s = qn + σn−2qn−2 + .

. .

+ σ1q1(5.1)if qn ≤s < qn+1 and σ1, σ2, . .

. , σn−2 = 0, 1, with the constraint σjσj+1 = 0, j = 1, .

. .

, n −3. LetΛqn be the family of self avoiding walks on the integer lattice Z2 starting at (0, 0), ending at (qn, −pn)and contained in the strip 0 < x ≤qn, except for the left extreme points.

Then a self avoiding walkjoining (0, 0) to (s, s′) with s given by Eq. (5.1) and s′ = pn + σn−2pn−2 + .

. .

+ σ1p1 can be obtainedby simply joining a path in Λqn, one in Λqn−2 if σn−2 = 1, . .

. , one in Λ1 if σ1 = 1.

The latterself-avoiding walks will define the class Λs of walks. It is clear by the construction that the aboveclass Λs of self avoiding walks contains many of the ones which have the largest products Qj1(i⃗ω·⃗ν(j))2of small divisors.

Therefore we define:⃗Z(Λqn) =Xpaths in Λqn(−iηJ−1⃗ν1)f⃗ν1(i⃗ω · ⃗ν(1))2kYj=2f⃗νvj (⃗νj−1 · ηJ−1⃗νj)(⃗ω · ⃗ν(j))2ei(qnψ1−pnψ2)(5.2)(with ⃗ν(1) = (qn, −pn) which can be always realized with the vectors ⃗ν1 = (1, 0) and ⃗ν2 = (1, −1)).We expect:⃗Z(Λqn) = ⃗ζ C(η, f)qnqδnei(qnψ1−pnψ2)(1 + O(q−1n )) ≡⃗ζZ(Λqn) = Zn ei(qnψ1−pnψ2) ,(5.3)where ⃗ζ is a suitable unit vector, C(η, f) is a suitable function of ηf and δ is a critical exponentcharacteristic of the golden section. Then the contribution to ⃗h∗due to the above classes of trees andpaths can be computed approximately, by noting that, if εn = (rpn −qn) and Z(Λs) is defined as inEq.

(5.2), Eq. (5.3), with the sum being over the paths in Λs and ⃗ν(1) = (s, −s′),Z(Λs) ≃Z(Λqn)Z(Λqn−2)σn−2 .

. .

Z(Λq1)σ1 ,s < qn+1 ,Z(Λqn+1) ≃Z(Λqn)Z(Λqn−1)εn−1εn+12,s = qn+1 ,(5.4)where we can define, for consistency, Z(Λq0) = ε−20= r−2 and Z(Λq−1) = (ε0/ε−1)2 = r2. This meansthat the contribution to ⃗h∗can be written approximately, if rn = pnqn :⃗ζ∞Xn=1Xσ1,...,σn−2=0,1Zn Zσ11 Tσ1σ2 Zσ22 Tσ2σ3 .

. .

Tσn−3σn−2Zσn−2n−2 ei(sψ1−s′ψ2)≃⃗ζ∞Xn=1Znei(qnψ1−pnψ2) Tr [Θ1Θ2 . .

. Θn−2] ,(5.5)8

where Tσσ′ is the compatibility matrix defined to be T11 = 0, T00 = T01 = T10 = 1, and Θj,j = 2, . .

. , n −2, are defined as (Θj)σσ′ = Tσσ′Zσ′jand (Θ1)σσ′ = Zσ′1 .If Tσσ′ were ≡1 the trace would be simply Qj(1+C(η, f)qjq−δj ); so that, in the above approximationthe series will become singular when |C(η, f)| = 1 and in that case Tr [Θ1Θ2 .

. .

Θn−2] can probably bereplaced by a constant, as far as the determination of the singularity in ⃗ψ is concerned (and perhapsin η or ε as well). Hence we find the following representation of the contribution to ⃗h∗that we areconsidering:⃗ζ∞Xn=1[C(η, f)ei(ψ1−rnψ2)]qnqδn= ⃗ζ∞Xn=1[C(η, f)ei(ψ1−rψ2)]qnqδne−iψ2O( 1qn ) .

(5.6)We expect that the singularities of ⃗H∗,⃗h∗, as ε grows, are the same as those of ⃗H,⃗h, and furthermorewe expect the above considered contributions to the functions ⃗H∗,⃗h∗to be the most singular. Hencewe interpret Eq.

(5.6) as saying that we should expect ⃗h, ⃗H to be, at the breakdown of the invarianttorus which corresponds to |C(η, f)| = 1, singular as functions of ψ1, ψ2 and of η (hence of ε).Furthermore the set |C(η, f)| = 1 is in the η-plane a natural boundary for the functions ⃗h∗, ⃗H∗asfunctions of η and, if η =ε1−σε is smooth in ε, or at least Lipshitz continuous, as mentioned above,when ε →ε−c , C(η, f) ≃(1 −γ(εc −ε)) so that the singularity of ⃗h(ψ1, 0) or ⃗H(ψ1, 0) in ε, ψ1 isdescribed by the singularity of a single function ξ(z), or ξ′(z), of the single variable z = e−γ(εc−ε)eiψ:ξ(z) =∞Xk=1zqkqδk,orξ′(z) =∞Xk=1zqkqδ+1k(5.7)which would mean that the critical torus has a Lipshitz continuous regularity with any exponentδ′ < δ in the ψ1–variable and δ in the ε −εc variable, [K].For instance if we fix ψ2, e.g. as ψ2 = 0 (which can be regarded as a special Poincar´e section of theinvariant torus), then ⃗h∗is a Cδ′ function and ⃗H∗, which is obtained from ⃗h by applying the operator⃗ω · ∂⃗ψ, is a C1+δ function, [K].

Note that the structure of the operator ⃗ω · ∂⃗ψ is such that when it isapplied to ⃗h as in Eq. (5.6) it generates a smoother function.

Therefore, based on the hypothesis thatthe singularity of ⃗H,⃗h and of ⃗H∗,⃗h∗are the same, see §4, and on the above heuristic discussion, thefollowing conjecture emerges.Conjecture. Consider the conjugacy to a pure rotation of the motion generated by the Poincar´e map ona circle on the critical torus.

There is δ > 0 such that it is described by two functions ⃗h, ⃗H and writtenas ⃗α = (ψ, 0)+(h1(ψ), h2(ψ)) and ⃗A = (H1(ψ), H2(ψ)) with ⃗h H¨older continuous with exponent δ′ < δand ⃗H of class C1+δ′. Furthermore the above conjugacy has a H¨older continuous regularity δ′ < δ inthe ε −εc variable.The mechanism for universality in the breakdown of the invariant tori that we propose above is,in our opinion, a refined version of an important idea in [PV]: except that we have not made herethe simplifying assumption of absence of resonances (i.e.

we allow for non zero Fourier components ofopposite wave label ±⃗ν, and find resummations that in some sense eliminate them).If one accepts that the above pendulum system has the same critical exponents for the golden meantorus in the standard map then it follows that δ = 0.7120834 by the scaling argument on p.207 of[Ma].1 The regularity of the two conjugators is in fact in that case not smoother than Cδ for theanalogue of ⃗h and of C1.9568 for the analogue of ⃗H: hence the above conjecture is in agreement withthe data and gives some independent reasons for the difference of about 1 between the regularity of1 Private communication of MacKay.9

⃗h and that of ⃗H. Unfortunately an exact computation of the regularity of ⃗H does not seem to havebee attempted yet.2Appendix A1.

The stability constant σε.We fix n and we consider the contribution to σ(k)n,ε(2nx) arising from a k-th order term correspondingto a given Feynman graph: it will be given by the sum of products of factors whose dependence onthe variable 2nx is through terms of the form:(⃗ω0 · (⃗ν0λ + 2nx))−1 ,where ⃗ν0λ is the momentum of the branch λ inside the resonance, i.e. the sum of all the modes of thevertices preceding λ contained in the resonance.

Then |⃗ν0λ| ≤kN and by the diophantine property|⃗ω0 · ⃗ν0λ| > [C0(kN)]−τ so that nλ > ˜n = −τ log(kN) −log C0, for all λ inside the resonance. Then,if k is fixed and n →−∞, the quantity |⃗ω0 · ⃗ν0λ| remains bounded from below because |⃗ω0 · ⃗ν0| ≥2˜nwhile 2nx →0 and the x–dependence is only via quantities like (⃗ω0 · ⃗νλ + 2nσx), σ = 0, 1.

Thereforethe dependence on x disappears, and we have:limn→∞σ(k)n,ε(2nx) = σ(k)ε.On the other hand, as σ(k)n,ε(2nx) is a power series in ε uniformly convergent, see [GM], and we canpass to the limit under the sign of series and the theorem is proven.Appendix A2. Resonance form factors for an action dependent interactionIn general the interaction potential depends also on the action variables.

This yields that all the linefactors introduced in §4 are possible, so that to the dressed lines we associate the following quantitiesa factorχ(2−n⃗ω0 · ⃗ν(v)) −i⃗νv′ · [1 −σsn,ε(⃗ω0 · ⃗ν(v))]−1iJ−1⃗νv(i⃗ω0 · ⃗ν(v) + Λ−1)2h ←han operatorχ(2−n⃗ω0 · ⃗ν(v))i⃗νv′ · [1 −σsn,ε(⃗ω0 · ⃗ν(v))]−1∂⃗Avi⃗ω0 · ⃗ν(v) + Λ−1h ←Han operatorχ(2−n⃗ω0 · ⃗ν(v))−∂⃗Av′ · [1 −σsn,ε(⃗ω0 · ⃗ν(v))]−1i⃗νv′i⃗ω0 · ⃗ν(v) + Λ−1H ←hjust0H ←Hwhere n is the scale label of the line, and σsn,ε(⃗ω0 ·⃗ν), s = 1, . .

. , 4, will have a different form dependingon the labels (H or h) attached to the half branches contributing to form, respectively, the outgoingand the incoming external lines of the resonant clusters whose values add to σsn,ε(⃗ω0 ·⃗ν).

The analysisin [GM] applies to all kinds of resonance, so that a result analogous to the theorem of §4 holds forall the functions σsn,ε(⃗ω0 · ⃗ν), and four resonance form factors can be shown to be well defined anddepending only on ε: the proof can be carried out exactly in the same way.Acknowlegments: We are indebted to G. Parisi for an early suggestion discussed in [G3] and toR. MacKay for explaining us the results on the standard map.This work is part of the research2 Private communication of MacKay.10

program of the European Network on: “Stability and Universality in Classical Mechanics”, # ER-BCHRXCT940460.References[CF] Chierchia, L., Falcolini, C.: A direct proof of a theorem by Kolmogorov in hamiltonian systems,Annali della Scuola Normale Superiore di Pisa, 21, 541–593, 1994. [E] Eliasson, L.H.

: Absolutely convergent series expansions for quasi periodic motions, report 2-88,Dept. of Mathematics, University of Stockholm, 1988.

[FT] Feldman, J., Trubowitz, E.: Renormalization in classical mechanics and many body quantumfield theory, Journal d’Analyse Math´ematique, 58, 213-247, 1992. [G1] Gallavotti, G.: Renormalization Theory and Ultraviolet Stability for scalar Fields via Renormal-ization Group Methods, Reviews of Modern Physics, 57, 471–562, 1985.

[G2] Gallavotti, G.: Twistless KAM Tori, Communications in Mathematical Physics, 164, 145–156,1994. [G3] Gallavotti, G.: Perturbation Theory, in “Mathematical physics towards the XXI century”, p.275–294, ed.

R. Sen, A. Gersten, Ben Gurion University Press, Ber Sheva, 1994. [GG] Gallavotti, G., Gentile, G.: Majorant series for the KAM theorem, in mp arc@math.utexas.edu,#93-229, to appear in Ergodic Theory and Dynamical Systems.

[GM]Gentile, G., Mastropietro, V.: Tree expansion and multiscale decomposition for KAM tori,Roma 2, CARR–preprint 8/9, 1994. [K] Katznelson, Y.: An introduction to harmonic analysis, Dover, 1976.

[Ma] MacKay R.: Renormalization in area preserving maps, World Scientific, London, 1993.11

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