Compensations in small divisor problemsL. Chierchia and C. FalcoliniDipartimento di MatematicaUniversit�a di Roma \Tor Vergata"via della Ricerca Scienti�ca, 00133 Roma (Italy)(Internet: chierchia@mat.utovrm.it and falcolini@mat.utovrm.it) �August 1994AbstractSeveral small divisor problems arising in the perturbative theory of Hamiltonian and La-grangian systems are considered. A general method that allows to prove compensationsamong the elementary contributions of the formal power series expansions associated toinvariant surfaces is presented.Contents1 Introduction 12 Notations and Results 23 Weighted trees 94 Compensations I (Maximal Hamiltonian tori) 104.1 Tree expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Compensations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Compensations II (Maximal Lagrangian tori) 156 Compensations III (Lower dimensional tori) 167 An example with no compensations 17

�The authors gratefully acknowledge helpful discussions with C. Liverani.i

1 IntroductionC.L. Siegel [20] was the �rst to solve a small divisor problem (linearization of germs ofanalytic functions). His solution was based on a \direct method" consisting in computingthe kth coe�cient of the formal power series solution and estimating such coe�cient by aconstant to the kth power. The solution of small divisor problems arising in perturbativeseries in the theory of Hamiltonian systems has been one of the major contributions todynamical systems of this century: the techniques employed for such a task, started byA.N. Kolmogorov in the early �fties and developed in the early sixties by V.I. Arnold andJ.K. Moser, are known as \Kolmogorov{Arnold{Moser (KAM) theory" (see e.g. [1] andreferences therein). Such methods show indirectly the convergence of the formal expan-sions of the perturbative series (longly before considered by astronomers and especiallyby H. Poincar�e [19]).In 1988 H. Eliasson proposed a direct proof of the convergence of formal expansionsin the Hamiltonian context ([9], [10], [11]). More recently direct proofs (in the Hamilto-nian context) have been reconsidered in several papers: [6], [12], [13], [14] [15], [16], [17];a small divisor problem arising in elliptic systems of PDE's has been solved by similarmethods in [7].In this paper we consider several small divisor problems and discuss a general methodthat allows to prove compensations among \elementary" contributions to the kth order ofthe associated formal expansion. Such compensations, modulo technical estimates (whichhave been thoroughly discussed in [6]), yield new proofs of the convergence of the formalpower series.The starting point in this subject is a formal power series whose coe�cients are bona�de functions determined by linear recursive equations. \Recursive" means that the kthcoe�cient can be expressed (typically by Taylor's expansions and by inversion of suitabledi�erential operatorD) in terms of the derivatives of a given function (\the Hamiltonian"or \the Lagrangian") and in terms of the hth coe�cients with h < k. If one keeps onexpressing each coe�cient in terms of the preceding ones, one will end up with a �nite(but huge) decomposition of the kth coe�cient involving only the derivatives of thegiven function or, more precisely, involving simple operations, coming from inverting D,on such derivatives; the inversion of D brings in the small divisors. We shall call theterms obtained through this procedure \elementary contributions". In writing out theelementary contributions in the most explicit way, it appears readily clear that a graphicalrepresentation is of big help. On a more formal level this is done using graph theory(more speci�cally the theory of labeled, rooted trees). Estimating the kth coe�cient byinserting absolute values within the sum of the elementary contributions leads, in general(and inevitably), to divergences (i.e. to estimates of the kth coe�cients by a power ofthe factorial k!). But if the formal series were convergent (and in many examples oneknows by KAM that this is the case) it would mean that it is possible to group togetherthe elementary contributions in such a way that huge (size k!) terms compensate amongthemselves leaving out well behaving (exponential in k) quantities. Graph theory provides1

a very useful tool in order to recognizes the families of elementary contributions thatcompensate. Most of the known small divisor problems and some new ones1 can be solvedby this approach.The small divisor problems we shall consider in this paper (always in the real{analyticcategory) are: P1) maximal quasi{periodic solutions (or \invariant tori") for nearly{integrable non{degenerate Hamiltonian systems; P2) maximal quasi{periodic solutionsfor nearly{integrable non{degenerate Lagrangians; P3) lower dimensional hyperbolic(whiskered) tori for \second order" nearly{integrable Hamiltonian systems. We shall alsoconsider a small divisor problem, P4), (lower dimensional \inde�nite" tori for second or-der Hamiltonian systems) which is \non compensable" (in the precise sense describedbelow), strongly supporting that the associated formal power series are divergent; in fact,it would be nice to give a direct proof of the divergence of such series.Our approach to P1) (integrated by the estimates of [6]) yields a direct proof of theclassical \KAM theorem" alternative to that of Eliasson [9]; the direct proofs in thecontext of P2) and P3) are new2.In x 2 we give a precise description of the models P1) � P4) and of the results obtained.In x 3 a new family of trees (simply related to the standard labeled, rooted trees) isintroduced. In x 4, x 5, x 6 the compensation mechanism for the models P1), P2), P3)is discussed. Finally, in x 7 we exhibit an example where no compensations (of the typedescribed in this paper) take place.2 Notations and ResultsLet us describe more formally what we mean by compensations in small divisor problems.Given a \nearly{integrable" dynamical system (say, as in P1) � P3)), one is interested in�nding invariant surfaces run by a linear ow or, equivalently, in �nding quasi{periodicsolutions. Quasi{periodic solutions are described in terms of functions on the standardN{dimensional torus N � N=(2�N) substituting the phase variable � 2 N with !t =(!1t; :::; !N t) where ! 2 N is a given (rationally independent) vector and t denotes time3.The system being \nearly{integrable" means that there is a perturbative parameter, say", such that for " = 0 the system is completely \solvable". It is therefore natural totry to establish the existence of formal solutions: this constitutes the �rst step. Formalquasi{periodic solutions have been considered already in the last century; for a moderndiscussion of the formal solutions of the models discussed here we refer the reader toAppendix B of [6]4.1e.g. the elliptic systems considered in [7].2The KAM theory in the context of P2) is relatively recent and is sometimes referred to as \KAM incon�guration space" (see [21], [5]). For the KAM theory in the context of P3) see [18]. The small divisorproblem P3) is particularly relevant in connection with the so{called Arnold di�usion (see [2], [8], [22]).3In the PDE case t is a multidimensional independent variable and ! is a matrix; see [7].4In [6] the formal expansion for P1) is proved; formal solutions for P2) � P4) can be proved in acompletely similar way. 2

The formal solutions we will be dealing with have the formZ � (Z1; :::; Zd) �Xk�0Zk(�)"k ; � 2 N (2.1)where, for each k � 1, the vector{valued function Zk is a real{analytic function over Nhaving an exponentially fast converging Fourier expansion of the form5Zk = Xn2N Zknein�� : (2.2)while for k = 0 Z0 is the solution of the unperturbed problem.The second step is to give Z a representation in terms of trees i.e. a representation ofthe form: Zkn = 1k! XT2T k� X�:V!N��v=n X�:V!B�(T; �; �) Yv2V v(T; �; �) ; (2.3)where, intuitively speaking, T k� is a suitable family of trees of \order k" taking care ofthe combinatorics coming out of Taylor's formula and its repeated applications (see x 3);the sum over the indices � attached to each vertex comes from expanding everythingin terms of Fourier series (hence the constrain of the total sum of such indices to be nsince on the left hand side we have the n{Fourier coe�cient of the solution); the �nalsum is �nite and encodes all possible (case{by{case depending) indices which may helpin writing out \in the most explicit way" the solution6; the addenda of these sums aresplit into two terms: �(T; �; �), which are complex vectors related to the derivativesof the given function (the Hamiltonian or the Lagrangian) ruling the evolution of thedynamical system and the products of v's which are the small divisors. More formally:T k� is a suitable family of labeled rooted trees7, V � V (T ) denotes the set of vertices ofT and the su�x (\order") k refers to the following estimates on cardinalities# T k� � k! ck1 ; # V (T ) � c2 k ; (8 T 2 T k� ) ; (2.4)(for a suitable constant ci > 0); the second sum runs over all possible functions assigningto each vertex v 2 V of a rooted tree T an integer vector �v 2 N with the constrainPv2V �v = n; the third sum runs over a suitable set of (possibly) vector{valued indicestaking a �nite number of values (#B < 1); � is a complex vector depending on T ,f�vgv2V , f�vgv2V and on the Hamiltonian (or Lagrangian); �nally v 2 are divisors thatare described as follows. Rooted trees can be naturally equipped with a partial order: wesay that v0 � v if the path joining the root r of T with v0 contains v; (obviously v0 < v5Note that we are using the su�x k with di�erent meaning as "k denotes the kth power of the number" while Zk is a vector valued function and k is used as an index. The hth power of the jth componentof Zk will be denoted (Zkj )h; the jth component of the Fourier coe�cients of Zk will be denoted Zknj .6e.g. �v will typically be related to the degree of the small divisors, see (2.7) below.7See any introductory book on graph theory, such as [3], for the standard terminology or AppendixA in [6].3

means v0 � v and v0 6= v and r � v 8 v 2 V i.e. the root r is the �rst vertex of therooted tree T ). Given a function �, we de�ne�v � �v(T; �) � Xv02V : v0�v �v : (2.5)The divisors v, which may assume arbitrarily small values, are de�ned in terms of areal{valued function h � i which satis�es the Diophantine conditionjhnij�1 � jnj� ; 8 n 2 Nnf0g (2.6)with suitable positive constants ; � .8 Then v � v(T; �; �) � ( h�vi��v if �v 6= 01 if �v = 0. (2.7)where �v is, say, the �rst component of the index �v and takes value 0, 1 or 2. In (2.3)only the second sum runs over an in�nite set of indices: therefore we assume that thereexist positive numbers �; �0; a > 0 such that, if we setakn � 1k! XT2T k� X�:V!N��v=n X�:V!B j�(T; �; �)j Yv2V e�0j�vj ; (2.8)then, for all k and n one has akn � ake��jnj : (2.9)In the models considered here, such an assumption is an immediate consequence ofthe well{posedness of the (formal) problem and of the analyticity assumptions on theHamiltonian (Lagrangian). >From (2.8) it follows at once that if one could bound theproduct of the divisors as9 Yv2T j vj � ck3 YT (1 + j�vjb) (2.10)for some c3 > 1 and b > 0, then from (2.4), (2.8) and (2.10) it would followjZknj � 1k! XT2T k� X�:V!N��v=n X�:V!B j�(T; �; �)j ck4 Yv2V (1 + j�vjb) � ck5e��jnj; (2.11)(for suitable c1 > 0) leading to \absolute" convergence (i.e. convergence without com-pensations) of the formal expansion Z. Indeed Siegel's original proof is based on a similar8Typically, in dynamical systems, hni = ! � n (where the dot denotes the standard inner product inN ) but in other situations (e.g. [7]) the function h � i might be more complicate (e.g. non linear); in thecase hni = ! � n it is well known that, if � > N � 1 , up to a set of Lebesgue measure zero, all ! 2 Nsatisfy (2.6) for some .9From now on we will adhere to the the common abuse of notation v 2 T in place of the more properv 2 V (T ) and also if vv0 = v0v denotes an edge of T , we shall denote vv0 2 T rather than the moreproper vv0 2 E(T ). 4

argument, even though the set up is slightly di�erent (and simpler). Technically Siegel'sproblem corresponds to � varying in a small (complex) ball so that Fourier series is re-placed by Taylor series and �v ranges over N+ : in such a case �v 6= �v0 whenever v > v0 andSiegel's method [20] yields the estimates (2.10)10. The problem with �v 2 N is that onecan have \resonances" i.e. �v = �v0 for v > v0 which may lead to obstinate repetitions ofparticularly small divisors. It is well known (see e.g. [6], Appendix B) that, in general,one has, for arbitrarily large k, subfamilies Fdiv � T k� and a choice of �� and �� (dependingonly on the subfamily) such that, for suitable �a;�b > 0,1k! XT2Fdiv �(T; ��; ��) Yv2V v � �akk!�b : (2.12)Such families are obtained by taking chains of resonances which are de�ned as follows.Given T 2 T k� and � (i.e. f�vgv2T ), a resonance is a subtree11 R � T such that: i) R isof degree two (i.e. R is connected to TnR by two edges); ii) if u is the �rst vertex12 inR and z is the �rst vertex following R, then �u = �z 6= 0; iii) R cannot be disconnected,by removal of one edge, into two subtrees of degree two satisfying i) and ii). It will beimportant to consider di�erent choices of the index � (i.e. f�vgv2T ): in particular givena resonance R and given � we call order of the resonance the number (see (2.7)) � � �u(u being the �rst vertex of R). A chain of resonances is a maximal series of resonancesR1; :::; Rh with Ri adjacent13 to Ri+1; given a choice of �, the order of the chain isde�ned to be �� � �1 + � � � + �h where �i � �ui , ui being the �rst vertex of Ri. >Fromthese positions it follows that if C is a chain of order ��, if n = �z where z is the �rstvertex following the chain (i.e. following the last resonance, which by convention will beRh), then Yv2C v = hni��� Yv2Cv 6=ui v (2.13)(where v 2 C means v 2 Si V (Ri)). The examples for which (2.12) holds are based onchains with �� � k and jhnij � k�1. This phenomenon may be cured through compen-sations. To be more precise we introduce the notion of \compensable chain". Considera chain C = (R1; :::; Rh), (h � 1) and �x the indices �v. Let, as above, ui be the�rst vertex in Ri, let Ri be connected to Ri+1 by the edge wiui+1 with wi 2 Ri and10For a detailed discussion, in the present language, of Siegel's methods see Appendix C of [6]; for adi�erent approach see [4].11When referred to trees the notation T 0 � T will always mean \T 0 (unrooted) subtree of T". Otherspecial conventions we are adopting are the following: a) for rooted trees the root (usually denoted r)may be identi�ed by adding an extra edge �r where � is a symbol (not a vertex of the tree) sometimescalled the \earth"; with these positions one has, for trees, #E = #V �1 and, for rooted trees, #E = #V ;b) the degree of a vertex v is the number of edges vv0 incident with v and if v = r is the root, the edge�r is included in the count; c) the degree of a subtree T 0 � T is the number of edges connecting T 0 withTnT 0; if T is rooted and the root r belongs to T 0 the edge �r must be included in the count.12Recall that T is a rooted tree and hence partially ordered (the order being such that the �rst vertexof T is always its root).13i.e. connected by one edge.5

u1 � w1 > u2 � ::: � wh; let z be the �rst vertex following the chain (i.e. following wh)and n � �z; and, �nally, let Pi be the path joining ui with wi. Consider the followingfunction of x 2�C(x;T; �; �) � hYi=1 �Ri(x;T; �; �) ; �Ri(x) � Yv2(RinPi) v Yv2Piv 6=ui � v(x) ; (2.14)where, if ui = wi (i.e. Pi = fuig), the product over Pi is absent, while if v 2 Pi 6= fuigwe set � v(x) � h Xv02Riv0�v �v0 + xni��v : (2.15)Thus, v = � v(1) and Yv2C v = hni����C(1) : (2.16)We say that a chain C � T 2 T k� (i.e. C = (R1; :::; Rh) with Ri � T ) is compensableif there exists a family of trees FC � T k� whose elements T 0 have C as common chainof resonance, and, for each T 0, there exists a choice of indices � 0 � � 0(T 0), such that thefunction14 ��C(x) � XT 02FC �(T 0; �; � 0) �C(x;T 0; �; � 0) (2.17)has a zero in x of order at least �� � 1. >From the detailed estimates in [6], it followseasily that if in a small divisor problem all chains of resonances are compensable thenthe associated formal series (2.1) is in fact convergent. In the rest of the paper we shallprove the following statements.P1) Let H = H0(y) + "H1(x; y) be a real{analytic Hamiltonian with y 2 B(y0) �N , x 2 N and (x; y) standard symplectic coordinates15; here B(y0) denotes some N{sphere centered at y0. Assume the following standard \non degeneracy conditions" onthe \integrable part" H0: i) the Hessian matrix @2yH0(y0) is invertible; ii) ! � @yH0 2 Nis a Diophantine vector i.e. there exists ; � > 0 such that16j! � nj�1 � jnj� ; 8 n 2 Nnf0g : (2.18)Then, quasi{periodic solutions with frequencies !, i.e. solutions of the form (x(t); y(t)) =Z(!t) with Z0(�) = (�; y0) and Zk : � 2 N ! Zk(�) 2 2N , satisfy the equationsDZ = J@H�Z(�)� ; (2.19)14In other words, the elements T 0 of FC are obtained from T and C by (possibly) changing the edgesconnecting TnC with R1, R1 with R2,...,Rh with TnC, and by (possibly) changing the values of theindices � on C.15This simply means that the Hamiltonian (evolution) equations are given by _x = @yH , _y = �@xH .16Compare with (2.6).6

where D � ! � @�, J is the standard symplectic matrix � 0 I�I 0� and @ is the gradient@(x;y) with respect to the variables (x; y). Then it is well known17 that there exists aunique formal solutions Z � Pk�0 "kZk of (2.19) with the normalization conditionZ N�1 � Zkd� = 0 (k � 1); (2.20)where �1 is the projection onto the �rst coordinates: �1(x; y) = x. We can then proveTheorem 2.1 There exists a tree expansion (2.3) for Z such that all chains of reso-nances are compensable.P2) The formulation for the Lagrangian case is very similar. We let L � L0(y) +"L1(x; y) be a real{analytic Lagrangian18 with Li satisfying the same hypotheses as-sumed above for Hi and !, except that now ! � y0. Quasi{periodic solutions x(t) =Z(!t) = Pk�0 "kZk(!t), where now Z0(�) = � and Zk : � 2 N ! Zk(�) 2 N , satisfy theequations D @yL(Z(�); DZ(�)) = @xL(Z(�); DZ(�)) ; (D � ! � @�) : (2.21)Mimicking the proof of Appendix B in [6] one can show that there exists a unique formalsolution Z � Pk�0 "kZk of (2.21) with the normalization condition as in (2.20) butwithout the projection �1. Then, Theorem 2.1 holds also in this case.P3) Maximal invariant tori correspond to analytic continuation (in ") of unperturbedtori having \all frequencies excited" i.e. tori run by linear ow t! !t with ! � @yH0(y0)rationally independent over N and N = # of degrees of freedom. More delicate is whathappens to unperturbed \resonant" tori19 i.e. invariant unperturbed tori for which thereexist n 2 Nnf0g such that ! � n = 0. We shall argue (see next item P4)) that in general,such tori are not analytically continuable for " 6= 0. Nevertheless we will show, undersuitable conditions, how to construct, with the (intrinsically analytic) methods outlinedabove, lower dimensional invariant (unstable) tori when " 6= 0. For simplicity, we shalldiscuss only a special case of \second order" nearly integrable Hamiltonian systems withHamiltonian function H � 12y2 + 12p2 + "f(x; q) (2.22)where (x; y) are symplectic variables as above (i.e. with x 2 N ) and so are (q; p) withq 2 M ; hence the number of degrees of freedom is N + M and we shall study N{dimensional invariant tori. The speci�cation \second order" is due to the particularform in which can be cast the Hamilton equations for H:�x = �"@xf ; �q = �"@qf : (2.23)17See [19] or Appendix B of [6].18The Lagrangian (evolution equations) are ddt@yL(x; _x) = @xL(x; _x).19Here, the word \resonant" which is related to the \resonances" of celestial mechanics, is used in atechnically di�erent meaning from the one used throughout this paper.7

For " = 0, N{dimensional invariant tori (up to a trivial linear and symplectic change ofcoordinates20) are spanned by solutions of the form (y; p) = (!; 0), (x; q) = (x0+!t; q0).>From classical transformation theory it follows that (if ! satis�es (2.18)) the evolutionequations of H in (2.22) are equivalent to the evolution equations for a Hamiltonian ofthe form y22 + p22 + "f0(q)+ o(") where f0 is the average (w.r.t. x) over N of f . Motivatedby this observation, we consider the Hamiltonian12y2 + 12p2 + "f0(q) + "2 ~f(x; q) � H0(q; y; p; ") +H1(x; q; ") ; H1 � "2 ~f : (2.24)Even though H0 is not, in general, integrable, if q0 is a critical point for f0, H0 still admitsthe invariant N{torus T0 spanned by (y; p) = (!; 0), q = q0 and x = x0+!t and we wantto study the persistence of such torus for the full Hamiltonian. To attack the problemperturbatively, we introduce a new analyticity parameter � with respect to which weshall make formal (and eventually convergent with radius of convergence greater thanone) power series expansion and consider the HamiltonianH0+�H1 (so that for � = 1 werecover the Hamiltonian (2.24)). We also make the following hyperbolicity assumption:we assume that q0 is such that the matrix "@2qf0(q0) is positive de�nite. Under thesehypotheses it is easy to prove that there exists a (unique) formal expansionZ � (Z1; :::; Zd) �Xk�0Zk(�; ")�k ; � 2 N ; d = 2(N +M) (2.25)such that t ! Z(!t) is a formal quasi{periodic solution of the Hamiltonian equationsgoverned by H0 + �H1 and such that the set fZ0(�) : � 2 Ng coincides with the un-perturbed torus T0 introduced above. Uniqueness is achieved by requiring (2.20) where�1 denotes the projection onto the x variable. Then, Theorem 2.1 holds also in thiscase. >From this result, as already remarked, it follows that the formal power series isactually convergent but, what is more interesting in this case, one can show that for" 6= 0 small enough, the radius of convergence (in �) is greater than one so that the setfZ(�) : � 2 N ; � = 1g is an invariant N{torus for the Hamiltonian (2.24). We �nallymention that from [18] it follows easily that these tori are whiskered in the sense of [2].P4) Consider again the Hamiltonian (2.22). Indeed, one can consider formal power seriesin " and one can show that if q0 is a non degenerate critical point of the x{average of fthen there exists a (unique) formal power series (2.1) (with d = 2(N +M)) such thatt! Z(!t) (! satisfying (2.18)) is a formal quasi{periodic solution for (2.22) and the setfZ0(�) : � 2 Ng coincide with the torus spanned by y = !, p = 0, q = q0, x = x0 + !t.Uniqueness, again, is enforced by the requirement (2.20). Also for such a formal series onecan write down a tree expansion completely analogous to those referred to in P1)�P3)above. However, we shall prove that there exist chains C = (R1; :::; Rh) such that if F0denotes the family of all trees with chain C thenXT 02F0 �(T 0) �C(0) 6= 0 : (2.26)20A transformation of (standard) symplectic coordinates (q; p) (of a 2d{dimensional phase space) iscalled symplectic if it preserves the (standard) two form Pdi=1 dqi ^ dpi.8

In view of this fact it seems natural to conjecture that in the present case the formalpower series Z is divergent.3 Weighted treesHere we describe the tree family T k� , which appears in the basic formula (2.3).For the models P3) and P4) introduced in x 2, T k� is simply the family of all labeled,rooted trees with k vertices21, which we will denote by T k. In this case, as is well known(see e.g. [3]), #T k = kk�1 and (2.4) is clearly satis�ed.To treat the cases P1) and P2) one has to distinguish, in the Taylor's expansion, thecontributions coming from H0 and L0 from those coming from H1 and L1. To do this weintroduce the following family of \weighted trees". Given a rooted (unlabeled) tree T wecall a function of the vertices of T� : v 2 V (T )! �v 2 f0; 1g (3.1)a weight function. A weighted rooted tree is a couple (T; �) with T a rooted tree and �a weight function. We now denote eTk the set of weighted rooted trees satisfying(i) deg v � 2 =) �v = 1 ; (ii) Xv2T �v = k : (3.2)Notice that, in particular, all �nal vertices (i.e. vertices of degree 1) have weight 1 andthat, for any T 2 eTk, #V (T ) � 2k � 1 as is easy to verify22. We now de�ne the classTk of labeled, weighted rooted trees obtained from eTk by labelling with k di�erent labelsthe k vertices with weight 1.In cases P1) and P2) we let T k� = Tk; it is easy to check that (2.4) holds also in thiscase23.The relation between Taylor's formula and trees may be based on the following operation� (that we now brie y discuss for the case of Tk; for the case T k see Appendix B of [6]).If T 2 Tk we denote by eT the tree in eTk obtained by removing the labels from T ; we alsodenote T 0 (or eT 0) the unrooted tree obtained from T (or eT ) by removing the edge �ri.e. by not distinguishing any more the root from the other vertices; �nally if T (or eT ) isan unrooted labeled (or unlabeled) tree and r is one of its vertices, we denote by Tr (oreTr) the rooted tree obtained by adding the edge �r (i.e. by decreeing that r is the root).21The basic terminology can be found in any introductory book on graphs or in Appendix A of [6].22Let Vi = fv 2 V : �v = ig, and let ki = #Vi, so that k = k1. It is well known (see, e.g. , [3]and recall our convention on degree of the root) that PV0 deg v +PV1 deg v = 2(k0 + k1) � 1. If v 2 V0then deg v � 3, thus the sum over V0 can be bounded from below by 3k0 while the sum over V1 can bebounded from below by k1. This leads to k0 � k1 � 1 which is the claim. Such inequality is optimal.23Since ([3]) #eT h � 4h and #V (T ) � 2k�1 for any T 2 eTk one sees that #eTk � 42k. Since the labelsare attached to k vertices, it is #Tk � k!42k. 9

Let � 2 f0; 1g and ` � 1, let hi be ` positive integers such that h1+ � � �+h` = k�� andpick trees Ti 2 eThi. We can form a tree T 2 eTk with root r (r being a vertex di�erentfrom the vertices of Ti, 8 i) of weight �r � � by settingT � Th1 � � � � � Th` � �T 0h1 [ � � � [ T 0h` [ frg+ Xi=1 rri�r (3.3)where ri is the root of Ti (and summing an edge e to a tree S means, obviously, to adde to E(S)). Then, one has the followingProposition 3.1 Let F be a complex valued function de�ned on trees in eTk (for any k).Then 1k! XT2Tk F ( eT ) = X�=0;1 k��X`=2�� 1! Xh1+���+h`=k�� Yi=1 1hi! XTi2Thi F ( eTh1 � � � � � eTh`) : (3.4)For the proof we refer to [6] (Corollary B.1 of Appendix B)24 .gr r r1 1 1 gr r r1 1 1 gr r1 0 11r��rQQ gr r1 0 11r��rQQ gr r r r1 0 1 1 gr r r1 0 0 r��rQQ 11Figure 1: The elements of eT3. (The numbers 0, 1 are the values ofthe index � for the corresponding vertices and the encircled vertexis the root of the tree).gr r1 1u2 u1 gr r1 1u1 u2 rg0 11 u1u2r��rQQFigure 2: The elements of T2 (u1 and u2 are the labels of T2).4 Compensations I (Maximal Hamiltonian tori)4.1 Tree expansionConsider the model introduced in P1) of x 2 and recall that there exists a unique formalsolutions Z � Pk�0 "kZk satisfying (2.19) and (2.20). Denote by Z(1)k the x{component24In [6] it is treated the case of T k (the � operation in eT k is de�ned as above, replacing � systematicallyby 1); adapting the proof to Tk is a trivial exercise.10

(i.e. the �rstN components) of the vector Zk and by Z(2)k the y{component; consistently,let @(1) � @x and @(2) � @y. We also let[ � ]k � 1k! dkd"k (�)j"=0denote the kth order operator which to a (possibly formal) power series a � P ak"kassociates its kth order coe�cient: [a]k � ak. Finally letA � @2yH0(y0) : (4.1)With these de�nitions, we can rewrite (2.19) asDZ(1)k = AZ(2)k + [@(2)H0](k�1)k + [@(2)H1](k�1)k�1 ; DZ(2)k = �[@(1)H1](k�1)k�1 ; (4.2)where the su�x (k�1) means that the argument of the function within square bracketsis, for k � 2, the polynomial in " of degree (k � 1) given byx = � + k�1Xh=1 "hZ(1)k ; y = y0 + k�1Xh=1 "hZ(2)k ; (4.3)and, for k = 1, is (x; y) = (�; y0). Since the average of [@(1)H1](k�1)k�1 vanishes (as is clearfrom the second of (4.2)) we can apply to it the operator D�1 obtained by inverting theconstant coe�cient operator25 D and rewrite (4.2) in a more compact way asDZ(�)k = (2� �)AZ(2)k + X�=0;1(�1)3��[@(3��)H�](k�1)k�� ; (� = 1; 2) : (4.4)Notice that while, by (2.20), the average of Z(1)k vanishes, the average of Z(2)k can beread (for � = 1) from (4.4) by integrating over N :Z NZ(2)k = �A�1 X�=0;1 Z N [@(2)H�](k�1)k�� : (4.5)Taking the n{Fourier coe�cient of (4.4), (4.5) one getsZ(�)kn = X�2f0;1;2g��2f0;1g hni�� n[D(�;�)H�](k�1)k�� on ; (hni � ! � n) ; (4.6)where D(�;�) is the vector{valued operator26D(�;�) � (�1)1��� i�� A��1 @(4����) ; (4.7)25If f is a (smooth) function on N with vanishing mean value we denote by D�1f the unique solutionwith vanishing mean value of the equation for g: Dg = f . Expanding in Fourier series one has g(�) =D�1f(�) = �i Pn2Nnf0g fn!�n exp(in � �), where i = p�1: the inversion of D introduces the small divisors.26For example the jth component of D(2;1) is given by D(2;1)j = NPs=1 @2H0@yj@ys (y0) @@xs .11

and the � attached to the range of � means that the following constraints have to besatis�ed: � + � 2 f2; 3g and � = 0 () n = 0 in which case we adopt the conventionthat hni� = 00 � 1. Notice that D(�;�)H� = 0 if � = 0 and � + � = 3 (as in such a case@(4����) = @x and H0 is independent of x).We shall now use Taylor's formula in the following form. If f : x 2 m ! f(x) 2 is a C1function and if a(") � Ps�1 "sa(s) is a m{valued (possibly formal) power series, thenf(a(")) � f(0) +Xh�1 "h hX=1 1! Xh1+���+h`=h1�hi�` Xj1;:::;j`1�ji�m @`f@xj1 � � �@xj` (0) a(h1)j1 � � �a(h`)j` : (4.8)Thus, if z � (x; y) and z(1) � x, z(2) � y, by (4.6) and (4.8) we get for the jth componentof the vector Z(�)kn , and for k � 2,Z(�)knj = (4.9)X�2f0;1;2g��2f0;1g k��X`=2�� 1! X1�hi�k�1�hi=k��(1�i�`) Xni2N�ni=n(0�i�`) X�1;:::;�`�i2f1;2g Xj1;:::;j`1�ji�N hni��n @` D(�;�)j H�@z(�1)j1 � � �@z(�`)j` on0 Yi=1Z(�i)hiniji ;where the derivatives of H� are evaluated at (�; y0) and then one takes the n0{Fouriercoe�cient (with respect to �); for k = 1, since [D(�;�)H0](0)1 = 0, one has the simpleformula Z(�)1nj = X�2f0;1;2g� hni��fD(�;�)j H1gn : (4.10)We are ready to prove the following tree expansion formula (recall (2.7), (2.5))Z(�0)knj0 = 1k! XTr2Tk X�:V!�v2N�v�v=n X�:V!�v2B�r=�0 Xj:V!jv2f1;:::;Ngjr=j0 Yv2Tr f�v(Tr; �)H�vg�v Yv2Tr v ;(4.11)where: the index set B, which depends on the function �v, is de�ned asB � n� = (�; �) : � 2 f0; 1; 2g; � 2 f1; 2g; s:t:� + � 2 f2; 3g ; � = 0 () �v = 0o ; (4.12)the scalar operator �v(Tr; �) is given by�v(Tr; �) � D(�v ;�v)jv Yv02Nv @(�v0 )jv0 ; Nv � fv0 < v; v0 adjacent to vg ; (4.13)(if Nv = ;, i.e. deg v = 1, the product is omitted). To recover (2.3) from (4.11), onesimply de�nes the component j0 of �(Tr; �; �) as�(Tr; �; �) � Xj:V!f1;:::;Ngjr=j0 Yv2Trf�v(�)H�vg�v : (4.14)12

The proof of (4.11) is by induction. For k = 1 (4.11) is an immediate consequence of(4.10). Assume that (4.11) holds with k replaced by 1; :::; k � 1. Given h � 1, �0, j0, n,consider the function of (unlabeled) trees Tr 2 eTh, given byF (�0)hnj0 (Tr) � X�:V!N :��v=n�2Bj:V!f1;:::;Ng:jr=j0 Yv2Tr v Yv2Trf�v(�)H�vg�v (4.15)and observe that if Tr = T1 � � � � � T` with Ti 2 eThi and h1 + � � �+ h` = h, then (�xing �,n, �0, j0)F (�0)hnj0 (T1 � � � � � T`) (4.16)= X�2f0;1;2g� Xn0+���+n`=n X�1;:::;�`j1;:::;j` hn0i��n @` D(�;�)j H�@z(�1)j1 � � �@z(�`)j` (�; y0)on0 Yi=1F (�i)hiniji (Ti)Equation (4.11) follows now from (4.9), (4.16) and Proposition 3.1.4.2 CompensationsHere, we show how to choose families of trees F and corresponding indices so that allchains of resonances are compensable (recall the de�nitions given in x 2).Given T = Tr, � and �, consider a resonance R and let u denote its �rst vertex andw0 < u the �rst vertex following R. By de�nition of resonance it is �u = �w0 6= 0 (i.e.Pv2R �v = 0) and �v 6= �w0 if u > v > w0. We shall classify resonances by assigning tothem an integer s � sR 2 f0; 1; 2g, which we shall call index of the resonance R. Then, toeach resonance R � T we shall associate a family FR of trees T 0 obtained by (possibly)changing the edges connecting R with TnR and choosing a suitable set of indices . Thefamily FR and the index � 0 � � 0(T 0) will be chosen so that (recall (2.14))XT 02FR �(T 0; �; � 0) �R(x;T 0; �; � 0) = O(xs) (4.17)and the family FC will simply be given byFC = h[i=1FRi : (4.18)Let R be a resonance and let u be its �rst vertex and w0 the �rst vertex following R. Wede�ne the index of R as sR � �u + �u � �w0 : (4.19)Thus if C = (R1; :::; Rh) is a chain of resonances and if �� denotes its order (see x 2),then hXi=1 sRi = �� + �u1 � �z (4.20)13

where z is the �rst vertex following Rh (which, by convention, is the last resonance inthe chain C). Hence, if (4.17) holds, from (4.18), (4.20), (2.14) and the de�nition of �(4.14) it follows easily thatXT 02FC �(T 0; �; � 0) �C(x;T 0; �; � 0) = O(x��+�u1��z) (4.21)which implies that the chain C is compensable.Let R � T be a resonance and let uu and ww0 be the edges connecting R with TnR,with u > u � w > w0 (hence u; w 2 R). We proceed by constructing the family FR. If�w0 6= 1 we set F 0R = fTg; if �w0 = 1 we let F 0R be the family of all trees T 0 obtainedfrom T by replacing the edge ww0 with the edge �ww0, as �w varies in R. Hence, T 2 F 0Rand if �w0 = 1, #F 0R = #R. If �u + �u 6= 3 we set F 00R = fTg; if �u + �u = 3 (i.e.(�u; �u) = (1; 2) or (�u; �u) = (2; 1)) we let F 00R be the family of all trees T 0 obtained fromT by replacing the edge uu with the edge u�u, as �u varies in R. As above, T 2 F 00R andif �u + �u = 3, #F 0R = #R. In the �rst case (i.e. for T 0 2 F 0R) we do not modify thevalues �v (i.e. � 0(T 0) � �). In the second case (i.e. for T 0 2 F 00R) we de�ne � 0 � � 0(T 0) asfollows. If v =2 R then � 0v = �v. Recall that, by de�nition, the �rst vertex of R consideredas subtree of T 0 is �u while the �rst vertex of R considered as subtree of T is u. Let �u 6= u(otherwise, obviously, we set � 0 � �) and consider the path P (u; �u) connecting u with�u. The path P will be formed by p � 2 ordered vertices that we denote vi: the order issuch that v1 = u, vp = �u and the edges of the path are v1v2,...,vp�1vp. We then set� 0vp � �v1 ;� 0vi � (�vi+1 ; 4� �vi+1 � �vi+1) ; for 1 � i � p� 1� 0v � �v ; 8 v =2 P (u; v) : (4.22)It is easy to see that this de�nition is well posed (see also (ii) of the following Remark).Finally, we de�ne FR � F 0R [ F 00R : (4.23)Remark 4.1 (i) If sR = 0, it is FR = fTg and � 0 � �.(ii) The map � ! � 0 is involutive: more precisely, if we denote by � 0(�; u; �u) themap de�ned in (4.22) (de�nition which depends on the ordered path P (u; �u)) then� 0(� 0(�; u; �u); �u; u) = �.(iii) One might say that the families F 0R and F 00R are constructed \going around" theresonance R with a \discrete curve" obtained by moving, respectively, the edge connect-ing R with w0 and the edge connecting u with R. This interpretation might explain thename \index" given to the quantity sR: FR is obtained by \going around" R exactly sRtimes.(iv) (On the de�nition of F 0R) In practice, moving around the edge �wz produces a factorproportional to � �w in (4.14) coming from the @(�z) = @(1) appearing in (4.13) (as z 2 N �w).It is easy to see that, for x = 0, summing � �R(x) over the family F 0R produces a commonfactor P �w2R � �w which vanishes by de�nition of resonance.14

(v) (On the de�nition of F 00R) The idea is similar: one wants to produce a factor propor-tional to ��u when moving around the \�rst" edge u�u connecting R with u. But now thesituation is more delicate as changing the connection u�u changes the order in R and,consequently, change the small divisors and also the structure of the derivatives (sinceboth v and �v depend on the order). Since one wants eventually to set x = 0 and collectthe factor P�u2R ��u one sees the necessity of changing the values of the indices �. In fact� 0 is de�ned in such a way that, when x = 0, one can factor out the product of divisorsand the products of the operators (derivatives) �v.With the above remarks in mind, it is not di�cult to verify that (4.17) holds, provingTheorem 2.1 in case P1).5 Compensations II (Maximal Lagrangian tori)Consider the model introduced in P2) of x 2 for a real{analytic Lagrangian L � L0(y)+"L1(x; y). Formal quasi{periodic solutions of the Lagrangian equations have the formx(t) = Z(!t) for an ! satisfying (2.18) and a function Z(�) which is the (unique) formalsolution Z � Pk�0 "kZk, with Z 2 N , of (2.21). The vector valued function Zk can beput in a form which is amazingly similar to the Hamiltonian case P1).Let us denote by Z(1)k the full vector Zk 2 N and by Z(2)k the vector DZk = DZ(1)k(where D � ! �@�) and, as before, let @(1) � @x, @(2) � @y and AL � @2yL0(y0). Equation(2.21) can then be written asALDZ(2)k = �D[@(2)L0](k�1)k �D[@(2)L1](k�1)k�1 + [@(1)L1](k�1)k�1 : (5.1)adopting the same convention used in (4.2): in particular the arguments of the functionswithin square brackets are as in (4.3). As for the previous case (see (4.6)), the n{Fouriercoe�cient of (5.1) is Z(�)kn = X�2f0;1;2g��2f0;1g hni�� n[D(�;�)L L�](k�1)k�� on (5.2)where D(�;�)L is now the vector{valued operatorD(�;�)L � (�1)1�(�+�) i�� A�1L @(4����) ; (5.3)and f0; 1; 2g� is the same set of equation (4.6).The Lagrangian problem is now in a form which is identical to the Hamiltonian case,except for the de�nition ofD(�;�)L . Therefore the tree expansion formula for the componentj0 of the n{Fourier coe�cient of Zk, with �xed values of �0 and j0 at the root r, is givenagain by (4.11) with the only proviso of replacing D(�v ;�v)jv in (4.13) with D(�v;�v)L;jv .Also, the families F of trees for which there are compensations are found exactly as inthe Hamiltonian case and we refer to x 4.2 for details.15

6 Compensations III (Lower dimensional tori)Recall the notations of x 2, P3). Because of the particular form of the Hamiltonian, theformal solution27 Z(�; �) is of the form Z � (X;Q;DX;DQ) where, as usual, D � ! �@�,� 2 N and X 2 N , Q 2 M . DenoteZ(1) � X ; Z(2) � Q ; @(1) � @x ; @(2) � @q ; A � "@2qf(q0) (> 0) : (6.1)One checks immediately that the recursive equations for Z(�)k (� = 1; 2) are��D2 + (�� 1)A�Z(�)k = X�=0;1 [@(�)H�](k�1)k�� ; � = 1; 2 (6.2)where the argument of the derivatives of H� is x = �, q = q0. From (6.2) one can seethat the average (over N) of Z(1)k vanishes, while the average of Z(2)k is as in (4.5) butwith the minus sign replaced by a plus. Taking Fourier coe�cients of (6.2) we get theanalogous of (4.6), namelyZ(�)kn = X�2f0;2g��2f0;1g hni�� n[D(�)n H�](k�1)k�� on ; (hni � ! � n) ; (6.3)which di�ers from (4.6) for the range of � and relative constraints:� 2 f0; 2g� () � + � 2 f2; 3g ; n = 0 =) � = 0 ; (6.4)and for the de�nition of the vector valued operator D(�)n :D(�)n � �hni2 + A�1�� @(�) : (6.5)Notice that the components D(�)nj are N if � = 1 and M if � = 2; we therefore letN1 � N ; N2 � M =) j 2 f1; :::; N�g : (6.6)Recall that A = "@2f0(q0) is positive de�nite, and so is a+ A for any a � 0; thusk(a+ A)�1k � bj"j (6.7)for a suitable constant b depending only on @2f0(q0). We also remark that, by (6.4), � = 2implies � = 0 (hence no divisors) while if � = 1 then � = 2 and n 6= 0, in which case thedivisors are hni2. In view of these remarks, we see that (4.11) holds also in the present27Recall that here " is a �xed real number di�erent from zero, while the (complex) perturbationparameter appearing in the formal power series is �.16

case provided we change the following items: N in the fourth sum is replaced (see (6.6))by N�v ; the index set B is de�ned asB � n� = (�; �) : � 2 f0; 2g; � 2 f1; 2g; s:t:� + � 2 f2; 3g ; �v = 0 =) � = 0o ; (6.8)�nally the operator �v now depends also on �: �v(Tr; �) in (4.11) is now replaced by�v(Tr; �; �) � D(�v)�vjv Yv02Nv @(�v0 )jv0 : (6.9)Also (4.14) is readily adapted replacing N by N�v and �v with (6.9). We can proceed tode�ne the families of trees F and relative indices � 0 which exhibit compensations. Givena resonance R (and a choice of � and �), we de�ne the index of R and the family F 00Rand relative indices � 0 exactly in the same way we did in x 4.2. Also the family F 0R isde�ned in the same way but the relative indices � 0 are now de�ned in a slightly di�erentway (due to the di�erent de�nition of D(�)n ): we let (same notations as in x 4.2)� 0vp � �v1 ;� 0vi � �vi+1 ; for 1 � i � p� 1� 0v � �v ; 8 v =2 P (u; v) : (6.10)With these de�nitions it is easy to check that (4.17) holds and hence that Theorem 2.1is valid in case P3) too.We close by a remark on the �{radius of convergence of Pk�1 �kZk. It is an easy ex-ercise to adapt the estimates in [6] to the present case and to check how the radius ofconvergence depends on ". In fact, observing that, by (6.7), one hasjhni��j j(hni2 + A)1��j � maxf 2; bj"jg maxfjnj2� ; 1g ; (6.11)leading to an estimate on the radius of convergence �0 of the type�0 � const minf" ; �2g : (6.12)Thus � = "2 (or � = "c with any c > 1) is within the domain of analyticity provided "is small enough.7 An example with no compensationsConsider the Hamiltonian (2.22) withf(x; q) =Xs�1 fsei(n(s)�x+m(s)�q) (7.1)17

where n(s) 2 N ,m(s) 2 M are given integer vectors (with jn(s)j+jm(s)j > 0) and the Fouriercoe�cients fs decay exponentially fast with jn(s)j + jm(s)j. Let q0 be a non degeneratecritical point of the x{average of f (i.e. of f0 = Ps�1 fseim(s)�q); then, as for the previouscases, there exists a (unique) formal power seriesZ � (Z1; : : : ; Zd) � Xk�0Zk(�)"k ; � 2 N (7.2)with d = 2(N +M), such that t ! Z(!t) is a formal quasi{periodic solution for (2.22)and the set fZ0(�) : � 2 Ng coincide with the torus spanned by y = !, p = 0, q = q0,x = x0 + !t, where ! 2 N satis�es condition (2.18).Expanding Zk in Fourier series also in the variable q, besides the variable x (see (2.2)),it is still possible to write Zkn as in (2.3) provided one makes the following changes.T k� = T k; B is the trivial set B = f� = � = 2g; �(T; �; �) is replaced by�(T; �; �) = X�0:V!�0v2M Yv2V f�v Yvv02E �v � �v0 (7.3)where �v � (�v; �0v) 2 N+M ; �nally v = h�vi�2 � (! �Pv0�v �v0)�2.It is then clear that in order to have compensations it would be su�cient to have reso-nances in the variable � whenever there are resonances in the variable �. In other wordscompensations take place if the Fourier modes in (7.1) are such thatXs2I n(s) = 0 =) Xs2Im(s) = 0 (7.4)8I � ; in fact in such a case one could repeat word{by{word the arguments in [6].In general, if (7.4) does not hold, compensations (of the type described in this paper) donot occur as it is shown by the following example.Take N = 2, M = 1, �x a scalar integer n 6= 0 and letf(x1; x2; q) = 2fcos x1 + cos x2 + cos(x1 + x2 � q) + cos(nx1 + x2 + q)g : (7.5)Hence, the range of � is the set f�(1; 0; 0); �(0; 1; 0); �(1; 1;�1); �(n; 1; 1)g; for de�-niteness we also �x ! = (p2;�1).gr r r r rr r r r�� ���� ���� ���� ��(n,1,1)zR1 R2 R3 R4Figure 3: Divergent contribution: the chain C = (R1; R2; R3; R4) is made of 4adjacent resonances with jRij = 3 (in this symbolic picture only two vertexare drawn). 18

We �x a tree T 2 eT k, with k = 3h + 1, and a function �v : V (T ) ! 3 so that Tcontains a chain C = (R1; :::; Rh) made of h identical resonances Ri � fv1; v2; v3g, with�v1 = (�1;�1; 1), �v2 = (0; 1; 0), �v3 = (1; 0; 0), and the last vertex z, following thechain, with �z = (n; 1; 1) (see Figure 3).Let FC be the family of all trees T 2 T k which contains the chain C. A lengthy butstraightforward computation shows that the ith component of the vector valued function��C(x) (see (2.17)) for h = 1 is:(��C(x))i � (��R1(x))i � Yv2V f�v Xu;w2R1 ei � �u � Yvv02E(R1) �v � �v0� �w � �z �R1(x)= 2 3Xj=1(�i3�j3 + x2Bij(x))�zj (7.6)where �ij is the Kronecker's symbol, ei is the vector of jth component �ij andB(x) = 0BBBBBB@ �2(6�x2)(2�x2)2 8p2+18x2�9x4+x6(2�x2)2(1�x2)2 6�x2(2�x2)28p2+18x2�9x4+x6(2�x2)2(1�x2)2 �2(3�x2)(1�x2)2 3�x2(1�x2)26�x2(2�x2)2 3�x2(1�x2)2 01CCCCCCA : (7.7)Hence ��R1(0) 6= 0. For h > 1, one simply has:��C(x) = 2h(A+ x2B)h�z ; Aij = �i3�j3 : (7.8)Thus, if x = ! � �z = p2n� 1, taking the third component of (7.8) (the component forwhich condition (7.4) is violated) and assuming that n � h, one easily checks that((A+ x2B)h�z)3 � 1 +O( 1n) (7.9)which implies that C is a non compensable chain.References[1] Arnold, V.I. (Ed.): Encyclopaedia of Mathematical Sciences, Dynamical Systems, Vol. 3,Springer{Verlag, 1988[2] Arnold, V.I.: Instability of dynamical systems with several degrees of freedom Sov. Math.Dokl. 5 581{585 (1964)[3] Bollobas, B.: Graph Theory, Springer (Graduate text in mathematics: 63), 1979[4] Brjuno, A. D.: Convergence of transformations of di�erential equations to normal form,Dokl. Akad. Nauk SSSR 165, 987{989 (1965); Analytic form of di�erential equations,Trans. Moscow Math. Soc. 25, 131{288 (1971) and 26, 199{239 (1972)19

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