COHOMOLOGY EQUATIONS NEAR HYPERBOLICPOINTS AND GEOMETRIC VERSIONS OF STERNBERGLINEARIZATION THEOREM.A. BanyagaMath Dept.Penn State Univ.University Park, PA 16802R. de la LlaveMath Dept.Univ. of TexasAustin, TX 78712C.E. WayneMath Dept.Penn State Univ.University Park, PA 16802Abstract.We prove that if two germs of di�eomorphisms preserving a volume, symplectic or1

contact structure are tangent to a high enough order and the linearization is hyperbolic,it is possible to �nd a smooth change of variables preserving the same structure thatsends one into the other. This result is a geometric version of Sternberg's linearizationtheorem which we recover as a particular case.An analogous result is also proved for ows.

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1. Introduction and statement of results.The celebrated Sternberg linearization theorem states that, given a local di�eomorphismwith a �xed point, if the eigenvalues of its linearization satisfy certain non-resonanceconditions, it is possible to �nd a di�erentiable change of variables making it linear. Thatis, given f , f(0) = 0, it is possible to �nd h in such a way that h�1 � f �h(x) = Df(0)x.In many applications, f preserves a geometric structure | symplectic, volume orcontact | and it is natural to require that h also does.In the case that f preserves a geometric structure, it necessarily violates severalof the non-resonance conditions and, there are easy examples where it is impossible toreduce to the linear part. Nevertheless, very frequently, one can hope to reduce to amuch simpler form | usually called a \normal form."Typically, it is not di�cult to �nd polynomial germs hk of degree k in such a waythat h�1k � f � hk(x) = Nk(x) + o(kxkk)with Nk a much simpler di�eomorphism. (These eliminations usually entail only powermatchings.)The main goal of this paper is to prove a theorem stating that if the formal elim-inations can be carried to a large enough order, and that Df(0) is hyperbolic, there isa di�eomorphism h, which reduces f to exactly the normal form Nn. In the case thatf preserves a symplectic or volume form or a contact structure so does h.We also discuss an analogue for ows.Such theorems were sketched in [St3]. A proof by other methods appeared in [Ch].The method of proof that we use is based on the deformation method of singularitytheory [Ma]. This method is ideally suited to discussing conjugacy problems in which ageometric structure is preserved. In principle, the preservation of a geometric structureis a non-linear non-local problem, but with the use of deformations, the preservationof the geometric structure is implemented by considering equations in an appropriatelinear space. 3

In a �rst section, we describe the basic formalism of deformations for general dif-feomorphisms as well as for di�eomorphisms preserving symplectic or volume forms orcontact structures. The basic idea of the deformation method is to embed the probleminto a family of problems, that includes also a trivially solvable one. Then, we study thederivatives of the quantities involved. The advantage is that the equations involved arealways linear cohomology equations. (If we think of derivatives as in�nitesimal quanti-ties, it is clear that the only equations we can form among in�nitesimal quantities arelinear.) The method is also well suited for geometric problems since the non-linear andnon-local constraints which are imposed by the preservation of the geometric structurealso become linear constraints, which can be implemented by considering the cohomol-ogy equations among linear spaces of local objects.In a second section, we provide estimates for the cohomology equations of theprevious sections and, establish the main theorems of this paper.We point out that from the point of view of group theory, the problem just dis-cussed is the problem of classifying the conjugacy classes of the group of germs ofdi�eomorphisms preserving a geometric structure, or classifying the orbits under thenatural action.The reduction of the problem to the study of cohomology equations in the Liealgebra is quite standard in �nite dimensional Lie groups, so in a certain sense, themethod considered here is the extension to an in�nite dimensional situation of methodsthat had been successful in the �nite dimensional situation. This point of view isemphasized in [St2].For problems of conjugacy, the deformation method was introduced in the contextof singularity theory. Some early refences are [Ma] and the notes of the lectures of R.Thom by H. Levine [Le].For the symplectic case, a discussion of the formal normal forms using this methodcan be found in [Mo]. For problems like ours, the main technical tools of the methodare Ck estimates for solutions of cohomology equations. In a global context, theseestimates were introduced in [LMM] and, in a context very similar to ours in [BLW].The paper [Il] uses the deformation method to prove the Sternberg theorem for generaldi�eomorphisms without regard to geometric structures.4

We point out that the deformation method can also be used to study the formaleliminations and the classi�cation of normal forms. For example, the papers [Ta] and[Ro] used the deformation methods to classify germs of vector �elds, forms, and di�eo-morphisms to a �nite order and the paper [Mo] discusses normal forms of symplecticdi�eomorphisms using the deformation method. The advantages of the method appearnot only in theoretical treatments, but it also allows e�ective numerical implementations[DL].In relation with the problem of convergence of symplectic normal forms of hyper-bolic maps we also point out that for analytic, two dimensional, symplectic mappingsa proof of the existence of analytic changes of variables reducing to a normal form hasalso been presented in [Mo3]. The method of proof is, nevertheless quite di�erent andmany of the techniques we use such as cut-o� functions etc. clearly do not apply foranalytic regularities.Remark. We will consider the volume preserving problem only on manifolds of dimen-sion 3 or bigger. The case when the dimension is 1 is completely trivial and, for the casein which the dimension is 2 volume preserving is the same as preserving a symplecticstructure. The results for symplectic structures are sharper than those we obtain forgeneral volume preserving case particularized for dimension 2.More precisely, we will prove:Theorem 1.1. Let f;N be Cr di�eomorphisms of IRn, f(0) = N(0) = 0, and let Aand B be as de�ned below.Assumei) Dif(0) = DiN(0) i = 0; : : : ; k < r � 1,ii) SpecDf(0) � fz 2 jC j ��1� � jzj � �+g [ fz 2 jC j ��1� � jzj � �+gfor some 0 < ��1� < �+ < 1 < ��1� < �+:Then, provided that 1 � ` < kA � B, for some integer `, we can �nd a C` dif-feomorphism h such that h�1 � f � h = N on a neighborhood of the origin, h(0) = 0,Dh(0) = Id. 5

If both f;N preserve symplectic, contact or volume structures h can be chosen topreserve the same structure.Remark. In this paper we will show in detail that, in all the cases we can takeA = j ln�+jln�+ j ln��j(ln�� + j ln��j) :Morever, we can take :B = Bno�structure � (ln�+)2 + j ln��j(ln�� � j ln�+j)(ln�+)(ln�� + j ln��j)in the case that there is no geometric structure to be preserved.B = Bsymplectic � 1� 2 j ln�+j(ln�+) j ln��j(ln�� + j ln��j)when we require preservation of a symplectic structure ;B = Bvolume � 1 + (n� 2) ln�� � j ln��j(ln�+)(ln�� + j ln��j) + (n� 2) ln�+(ln�� + j ln��j)for volume preserving case andB = Bcontact � 1 +Bno�structurein the contact case.Analogously, we have for vector �elds:Theorem 1.2. Let X;Y be Cr vector �elds of IRn, X(0) = Y (0) = 0.Assumei) DiX(0) = DiY (0) i = 0; : : : ; k < r � 1ii) SpecDX(0) � fz 2 jC j ��� � Re(z) � �+g [ fz 2 jC j ��� � Re(z) � �+gfor some ��� < �+ < 0 < ��� < �+ 6

Then, provided that ` = kA�B � 1 we can �nd a C` di�eomorphism h such thath�X = Y on a neighborhood of the origin, h(0) = 0, Dh(0) = Id.In case that both X;Y preserve a volume, symplectic or contact structure h canbe chosen to preserve the same structure.Remark. In all the above cases, we can take the values of A and B to be equal to theirvalues in the di�eomorphism case, except that we omit all logarithms. For example,A = �+�+ j��j(�� + j��j) ;B = Bno�structure � (�+)2 + j��j(�� � j�+j)(�+)(�� + j��j) ;and so on.Remark. The values of A and, especially, of B that we have obtained are not optimaleven using the method of proof in this paper. At the end of the proof we indicate inremarks how the results could have been improved by using more complicated spacesto measure the regularity. For example, one can obtain better results if one keepstrack separately of the derivatives with respect to parameters and with respect to thevariables in the manifold. A further improvement can be obtained if one keeps trackof the derivatives along stable and unstable manifolds in the way that it was done in[LMM]. We also point out that in the contact and volume cases, it is possible to obtainmore precise formulas that involve the eigenvalues of the derivative at 0 and not justbounds on their size. Since these improvements require a lot of technical complicationand are not central to the main goal of this paper, we have just relegated them toremarks along the side of the main proof.Remark. Notice that the expressions for B in Theorem 1.1 and Theorem 1.2 change ifwe substitute for f and N , f�1, N�1 respectively. The h obtained in one case wouldwork in the other. Given a particular pair f;N we could choose whichever of theexpressions gives the more favorable values of A and B.Remark. The method of proof we present carries over to in�nite dimensions. Theonly properties of IRn which are used is that it is a Banach space and that it has7

the \approximation property" | that is, that there exist C1 functions with boundedsupport and taking the value 1 in an open ball.Even if the generalization of Theorem 1.2 and Theorem 1.1 to Banach spaces satis-fying the approximation property is quite straightforward, and natural when there areno geometric structures present, we point out that generalization of geometric structuresto in�nite dimensions is somewhat delicate. There is no natural de�nition of volumeforms and, even if it is possible to de�ne symplectic and contact forms by extendingstraightforwardly the �nite dimensional de�nitions, these extensions are not natural inmany applications (see e.g., [CM], [M] for a discussion of natural extensions of symplec-tic forms to in�nite dimensions).2. The deformation methodThe basic idea of the deformation method is to consider a family of problems thatinterpolates between a trivial one and the problem we want to solve.Assuming that the problem is solved for all values of the deformation parameter,we can derive equations between the in�nitesimal deformations of the problem and thesolution.From a heuristic point of view, the deformation method can be considered as an\in�nitesimal" version of induction. Assuming that the problem is solved for all valuesof the parameter between [0; �] we study what it would take to solve it for [0; � +in�nitesimal]. (This amounts to �rst order perturbation theory.)The advantage of this procedure is that the equations between the in�nitesimaldeformations are linear. More importantly from our point of view, they are geometricallynatural and it is possible to perform geometrically natural constructions on them thatforce the preservation of geometric structures. The preservation of geometric structuresis, in principle, a non-linear and non-local problem. Nevertheless, by considering thein�nitesimal equation, we can formulate it as a problem about quantities belonging toappropriate linear spaces.We will be concerned with the so-called hyperbolic di�eomorphisms and ows.The following de�nition is quite standard. 8

De�nition 2.1. We say that a di�eomorphism f : IRn ! IRn with f(0) = 0 ishyperbolic at 0 if the spectrum of Df(0) does not intersect the unit circle. We say thata vector �eld L(x) in IRn with L(0) = 0, is hyperbolic if the spectrum of DL(0) doesnot intersect the imaginary axis.As is well known, (see e.g. [Ni]) f is a hyperbolic di�eomorphism if and only if onecan write IRn = Es � Eu, and, denoting A = Df(0), we have AEs = Es, AEu = Euand, for some norm kAjEsk < 1; k(AjEu)�1k < 1.Similarly, a ow is hyperbolic if, denoting A = DL(0) we have IRn = Es � Eu,AEs = Es, AEu = Eu and k exp(AjEs)k < 1, k exp(�AjEu)k < 1, for some norm.This characterization makes it clear that a vector �eld is hyperbolic at zero if andonly if its time t map de�nes a hyperbolic map.In this paper, we will consider mainly two problems:Problem 2.2. Given f;N , germs of Cr di�eomorphisms in (IRn; 0), f { and thereforeN - hyperbolic di�eomorphisms at 0, kf(x)�N(x)k = o(kxks), s � 1, determine g, thegerm of a di�eomorphism in (IRn; 0) with Dg(0) = Id, in such a way that g�1�f �g = N .Problem 2.3. Given X ;Y , germs of Cr vector �elds at 0 2 IRn, X (0) = Y(0) = 0, andkX (x)� Y(x)k � o(kxks), with X { and hence Y { hyperbolic ows at 0, determine ga germ of a di�eomorphism Dg(0) = Id such thatg�X = Y :If we have a di�erentiable family f� with f0 = N , f1 = f , it is natural to try to�nd a family g� with g0 = Id satisfying(2:1) g�1� � f� � g� = f0 :Analogously, for a family of vector �elds, X0 = X , X1 = Y (for example, X� =�Y + (1� �)X ) it is natural to try to �nd g� with g0 = Id in such a way that(2:2) g��X� = X0 :9

Since f� is a smooth family of di�eomorphisms, we can �nd a family of vector �eldsF� de�ned by(2:3) dd�f� = F� � f� :By the uniqueness theorem of di�erential equations, if f0;F� are C1, there is oneand only one family f� with the given f0 and F�. We also note that, if Dif�(0) =Dif0(0); i = 1; � � � ; k, then, DiF�(0) = 0 ; i = 1; � � � ; k.If we assume that g� can also be written in the form (2.3), taking derivatives withrespect to � in (2.1), we obtaindd� (g�1� � f� � g�) = (g�1� )�(F� � G� + f��G�) � (g�1� � f�1� � g�) :So that, if, given F�, we can �nd a C1 family of vector �elds G� satisfying(2:4) F� � G� + (f�)�G� = 0 ;the family of di�eomorphisms g� determined by g0 = Id; dd�g� = G� � g� | whoseexistence is guaranteed by the fundamental existence theorem of O.D.E.'s | will satisfy:dd�g�1� � f� � g� = 0 ; g�10 � f0 � g0 = f0 ;hence, g�1� � f� � g� = f0.Analogously, denoting _X� = dd�X�, we have:dd� (g�)�X� = (g�)�(LG�X� + _X�):Where, we recall, we have LG�X� = [G�;X�] = �LX�G� : Therefore, �nding C1 solutionsof(2:5) �LX�G� + _X� = 0;solves the original problem.Notice that both (2.4) and (2.5) are linear in the unknowns | the original problems(2.1) and (2.2) are not | and are geometrically natural. The fact that the equations10

are linear, is to be expected since the vector �elds can be considered as in�nitesimaldi�eomorphisms and the equations between in�nitesimal quantities are typically linear.We emphasize that both (2.4) and (2.5) were shown to be equivalent to the originalproblems independently of the fact that there were geometric structures present. Ournext task is to introduce geometric structures and to show that the deformation methodprovides a natural calculus to study them.Notice that all the calculations we have done so far only require that one cantake derivatives and that the solutions of O.D.E's are unique. If we consider higherregularities we �nd it convenient to de�ne precisely.De�nition 2.4. We say that f� is a Cr family if f�(x) is Cr jointly in � and x and,moreover supx;� sup`+j�j�r ������ @@��` @�@x� f�(x)����� � kf�kCr <1 :Remark. Notice that, with this de�nition when the domain is IRn, we can have in�nitelydi�erentiable functions such as x2 which are not C1 in the sense above since we requireuniform control of the derivatives over all of the space. Also, it could have been naturalto require a di�erent number of derivatives with respect to the parameters and withrespect to the variables.Remark. We recall that it is a result in harmonic analysis (see [Kr]) that if r + 1derivatives with respect to the parameter exist and are continuous and r+1 derivativeswith respect to the variables exist and are continuous, then the function is jointly Cr.Unfortunately, (see again [Kr]) it is not true that the function is jointly Cr+1. Thesecomplications in the regularity are due to the fact that the Cr spaces r 2 IN are notwell behaved under many of the operations that appear naturally in harmonic analysis.Nevertheless, they are quite well behaved under composition and, for the purposes ofthis paper, this is quite important.Remark. We also recall that the theorem of existence and regularity on initial condi-tions for O.D.E's shows that if F�(x) is Cr in � and C` in x, then the solutions f� ofdd�f�(x) = F�(f�(x)); f0(x) = x exist and are unique when ` � 1, r � 0. Moreoverthey are Cr+1 in �, C` in x. (See [Ha], Chapter 5.)11

3. Volume and symplectic geometryA k-form on an n-dimensional manifoldM is said to be nondegenerate if the mappingX ! i(X) sending a vector �eld X to the (k� 1)-form i(X) is an isomorphism: thismeans that for all x 2M , X(x) ! i(X(x))[ (x)] is an isomorphism of the tangent spaceTxM with the vector space �k�1(T �xM) of alternating (k � 1)-linear forms on TxM .This can happen only if n = dim(TxM) = dim(�k�1(T �xM)) = � nk�1�, i.e., if k = 2 or n.Let be a nondegenerate k-form; then any (k � 1)-form ! de�nes a unique vector�eld X such that i(X) = !.A (k � 2)-form F is called a hamiltonian of the unique vector �eld XF such thati(XF ) = dF :Clearly, XF = XF 0 if and only if F � F 0 is a closed form. The vector �eld XF is calleda globally hamiltonian vector �eld with hamiltonian F .The vector �eld X! de�ned by a closed (k� 1)-form ! according to i(X!) = ! issaid to be a locally hamiltonian vector �eld. Indeed if U is any contractible open subsetof M , the Poincar�e lemma says that !jU = dF , for some (k � 2)-form, F , on U . Henceon U , X! = XF .Recall that on an n-dimensional manifold only 2-forms or n-forms may be nonde-generate. A nondegenerate n-form is called a volume form. This form is automaticallyclosed. For 2-forms, if some critical calculations are to go through, we must require thisas an independent hypothesis and get the notion of symplectic form: i.e., a nondegen-erate closed 2-form. From now on, stands for a nondegenerate closed k-form (whichis a symplectic form, or a volume form); to emphasize the distinction, we reserve theletter ! for symplectic forms and � for volume forms.Note that even if a hamiltonian determines a unique vector �eld, the hamiltonianof a vector �eld is only determined up to forms with zero exterior derivative. When is a 2-form, the hamiltonians are 0-forms, that is functions and the the only functionswith zero di�erential are constants. Nevertheless, if is a k form with k � 3, there aremuch more complicated (k � 2)-forms with zero exterior derivative.12

Let be a C1 nondegenerate k-form and consider the equation:i(X) = dFIf X is Cr, so is i(X) = dF , hence, using the same construction of integrating alonglines, used in the standard proof of Poincar�e lemma (see [Sp] among others), we obtainthat the hamiltonian F is Cr and the Cr-norm of F can be estimated by a constanttimes the Cr-norm of X. In the symplectic case, it is possible to do better. If F is afunction, the fact that dF is Cr allows us to conclude that F is Cr+1 and that the Cr+1norm of F is bounded by the Cr norm of dF . Conversely if the hamiltonian F is Cr,the vector �eld is Cr�1 and the Cr�1 norm of X is estimated by the Cr-norm of thehamiltonian times a constant (depending on ).Using the standard formulas relating the interior product i(�), the exterior derivatived, and the Lie derivative,i�[X;Y ]�� = LX�i(Y )��� i(Y )(LX�)LX� = i(X) d�+ d i(X)�for all k-forms �, we get the well known:Lemma 3.1. Let be a nondegenerate closed k-form.(i) If X;Y are locally hamiltonian vector �elds, then [X;Y ] is a globally hamiltonianvector �eld with hamiltonian i(X)(i(Y ) ).(ii) If X has G as hamiltonian and Y is locally hamiltonian, then LYG is a hamiltonianfor [Y;X].(iii) If Y has F as a hamiltonian, then �LXF is a hamiltonian of [Y;X].Remark. In the Hamiltonian case, i(X)(i(Y )!) = !(Y;X)Proof. Observe �rst that a locally hamiltonian vector �eld X satis�es:LX = 0 :Indeed LX = d i(X) + i(X) d = 013

since d = 0 and i(X) is closed.To prove (i), compute:i�[X;Y ]� = LX�i(Y ) �� i(Y )(LX )= LXi(Y ) = di�iX iY �+ iXd�i(Y ) �= d�i(X)i(Y ) � :This proves (i).For (ii) we know that i(X) = dG, sod(LYG) = LY (dG) + LY �i(X) �= i�[Y;X]� + i�X�(LY �= i�[Y;X]� :

For (iii), i(Y ) = dF , so:d(LXF ) = LX dF = LXi�Y � = i�[X;Y ]� + i�Y ��LX � = i�[X;Y ]� :A di�erentiable family f� of di�eomorphisms preserves if and only if the associatedfamily F� of vector �elds is locally hamiltonian: indeed, if f�� = , taking derivativeswith respect to �, we get: 0 = dd� (f�� ) = f�� (LF� )and since d = 0, we conclude: LF� = d i(F�) = 0 :In this paper we are interested in what happens near a critical point. Hence, in acontractible neighborhood of the point, F� is globally hamiltonian and we denote by F�its hamiltonian: i(F�) = dF�The hamiltonian F� determines F� which in turn determines f� given f0.14

If f� is a family of -preserving local di�eomorphisms with f0 = N , f1 = f it isnatural to try to �nd -preserving g� such that g�1� � f� � g� = f0, by �rst determiningits hamiltonian G�.We recall the following property of the interior product whose proof is a straight-forward computation:Proposition 3.2. If f is a di�eomorphism, a p-form, and X a vector �eld then,i�f�X� = (f�1)� �i�X��f� �� :If we apply Proposition 3.2 to simplify the result of taking the interior product ofall the terms in (2.4) we obtain:(3:1) F� �G� + (f�1� )�G� = 0 :This equation is equivalent to (2.4) and hence to the solution of the main problem. Inthe symplectic case, G� is a function and (3.1) can be written as:(3:2) F� �G� +G� � f�1� = 0 :An analogous calculation shows that if one is given a family X� of -preservingvector �elds with hamiltonian X�, it is equivalent to �nd a family g� of -preservingmaps solving (2.2) and to solve:(3:3) d��LX�G� + _X�� = 0(Where by _X� we mean @@�X�.) and then determine g� bydg�d� = G� � g� ; g0 = identity.Clearly, a G� solving(3:4) �LX�G� + _X� = 0is a solution of (3.3) and, therefore, we will concentrate our e�orts in solving (3.4). Thisdoes not incurr any loss of generality for the original problem because hamiltonians are15

really de�ned up to exact forms. If �LX�G�+ _X� = R� with dR� = 0, then X�+R �0 Rsdsgenerates the same vector �eld that X�.The previous discusion shows that we can reduce the problem of conjugacy offamilies, all of whose elements preserve a geometric structure to the study of cohomologyequations. To study Problem 2.2 and Problem 2.3, where we are not given families butjust two di�eomorphisms or vector �elds, we need only to show that given such problems,we can �nd families whose initial and �nal points are the given di�eomorphisms or vector�elds and such that all the intermediate vector di�eomorphisms and vector �elds in thefamily preserve the same geometric structure.For vector �elds we observe that the space of locally or globally hamiltonian vector�elds is a subspace of the vector space of all vector �elds. Given X and Y locally orglobally hamiltonian vector �elds, then:X� = �Y + (1� �)Xis a smooth path of locally or globally hamiltonian vector �elds such that X0 = X andX1 = Y.For di�eomorphisms, we need to interpolate f;N : IRn ! IRn with the properties:f(0) = N(0) = 0 (Dif)(0) = (DiN)(0) ; i = 0; : : : ; k < r :In addition we may require that the interpolating maps preserve the form when is either the standard volume form dx1 ^ � � � ^ dxn or the standard symplectic formdx1 ^ dx2 + � � �+ dx2p�1 ^ dx2p (n = 2p).Simply consider f = f �N�1 and f� : IRn ! IRn:(3:5) f�(x) = ( ��1f(�x) if � 6= 0 .x if � = 0 .It is easy to see that f� is a smooth family of di�eomorphisms interpolating betweenthe identity and f [Mi] and preserving the volume or symplectic form . The requiredfamily f� is de�ned by: f� = f� �NClearly f1 = f ; f0 = N ; f�(0) = N(0) = 016

and (Dif�)(0) = (DiN)(0) ; i = 0; 1; : : : ; k < r :Moreover if f and N were -preserving, so are f� for all �. If f and N are Cr, r � 1, f�is a Cr�1 family.4. Contact geometryA contact form on a (2n+ 1)-dimensional manifold is a 1-form � such that � ^ (d�)n isa volume form. The codimension 1 distribution D� whose sections are vector �elds Xwith i(X)� = 0, is the contact structure de�ned by �. A di�eomorphism f preservesthe contact structure D� if and only if f�� = ��, for some nowhere zero function �.If we have a family of di�eomorphisms f� preserving the contact structure D�, thenwe have: f�� � = ��� and taking derivatives with respect to �:LF�� = (f�1� )� �d��d� � �� :Since (f�1� )�� = (�� � f�1� )�1�, we have(4:1) LF�� = ���with �� = � 1�� � d��d� � � f�1� :Since the factor �� and �� obviously depend on the family f�, when there is risk ofconfusion, we will denote them by �f�� ; �f�� if there is any danger of confusion.A vector �eld F satisfying (4.1) is called a contact vector �eld. Again, we willdenote the dependence of the factor �� on the vector �eld by �F��We recall the following fundamental result in contact geometry.Lemma 4.1. Let (M; �) be a contact manifold.Given a function F there is one and only one contact vector �eld F satisfyingF = �(F) :17

Such an F is called the generating function of the vector �eld F .Proof. On the distribution D�, d� is a symplectic structure. Hence it de�nes an iso-morphism fd� between sections of D� (called horizontal vector �elds) and sections of itsdual D�� , which can be identi�ed to 1-forms � vanishing on the orthogonal complementof D�. These forms are called semi-basic forms. The orthogonal complement of D� isspanned by the Reeb �eld Z: i.e., the unique vector �eld Z such that i(Z)� = 1 andi(Z) d� = 0.Given F a smooth function then �F = (i(Z) dF )��dF is a semi-basic form. Hence(fd�)�1[�F ] is a well de�ned horizontal vector �eld. One veri�es thatF = FZ + (d~�)�1h�i(Z) dF �� � dFiis a contact vector �eld with �(F) = F .For a 1-form � and a vector �eld X we write i(X)� or �(X) or h�;Xi interchange-ably. If f� and G� are families of contact di�eomorphisms and vector �elds respectively,we have: �(f��G�)(x) = h(f�� �)(f�1� (x);G�(f�1� (x)i= ��(f�1� (x)) : (�(G�))(f�1� (x))= ��(f�1� (x)) G�(f�1� (x))= [(��G�) � f�1� ](x)Therefore taking generating functions (2.4) becomes:(4:2) F� �G� + (�f�G�) � f�1� = 0 :Analogously, for vector �elds, we observe that the generating function of LFG = [F ;G]is i([F ;G])� = LF i(G)� � i(G)LF�= LFG� i(G)[�F�]= LFG� �FG :Hence (2.5) becomes, for contact �elds:(4:3) �LX�G+ �X� + _X� = 0 :18

To connect a contact di�eomorphism f : IR2n+1 ! IR2n+1 with f(0) = 0, andDf(0) = Id to the identity, the simple formula (3.5) that worked for symplectic andvolume preserving di�eomorphisms, does not work since f� is not a contact di�eomor-phism.Instead we use a chart on a neighborhood U of the set of contact di�eomorphismswith the set J 1M of 1-jets of functions on M . This chart [Ly] is the equivalent of theWeinstein chart for symplectic di�eomorphisms and it is built exactly the same way,that we now describe.Let (M; �) be a contact manifold. Consider M = M �M � IR and denote by �1,�2, t, the projection of M onto the �rst, second, and third factors. On M we have thecontact form � = t � (��1�) � ��2�. Let f be a contact di�eomorphism of (M; �), i.e.,f�� = ��. We de�ne the graph �f of f as the mapping:�f :M ! M : x 7! (x; f(x); �(x)) :Clearly, �f is a Legendre embedding, i.e.,��f � = 0Let L = �f (M) � M and L0 = �id(M) � M . Both L;L0 are Legendre submanifolds.We identify L0 with M as the zero section of the 1-jet bundle J1M over M . Thereexists a di�eomorphism � of a neighborhood M1 of L0 in M with a neighborhood M2of M = L0 in J1M such that �jM0 = identity and ���M = �, where �M is the canonicalcontact form of J1M . Then if f is C2 close enough to the identity, L � M1 and �(L)is a Legendre submanifold of J1M , which is C1-close to L0, hence �(L) = j1(�f ) forsome function on M . The desired contact isotopy f� will have as graph:L� = ��1[j1(��id + (1� �)�f )]where �(L) = j1(�f ) and �(L0) = j(�id) :Observe that for points x 2M such thatf(x0) = x0 and (Df)(x0) = Identity, (f��)(x0) = �(x0)(i.e., �(x0) = 1), hence �id(x0) = �f (x0). Since � is a di�eomorphism the correspondingfunctions have the same jet at the point x0. Therefore the deformation L� is �xed atx0, i.e., L�(x0) = �id(x0) = �f (x0) for all �. Therefore D`f�(x0) = D`f(x0), for all` = 0; � � � ; k. If f and N are Cr, r � 2, then f� will be a Cr�1 family.19

5. Regularity results for cohomology equationsIn this section we study the solution of the cohomology equations. We emphasize thatthe cohomology equations are linear and relate functions taking values in linear spaces.The geometric properties derived for the objects we are interested in are consequencesof the choices of spaces. In particular, it is not necessary to produce solutions using onlygeometrically natural methods. From the point of view of analysis, it is useful to takecoordinates, add elements or apply smoothing operators. Although those operations arenot geometrically natural, they can be used in the present context.We observe that the cohomology equations we have encountered ( (2.4), (3.1), (3.2),(4.2) ) can be written in the form(5:1) '�(x)�M�(x)'� � f�(x) = ��(x)where �� : IRn ! IRm, M : IRn !Mm�m (We denote by Mm�m the space of m �mreal valued matrices.) and f� : IRn ! IRn are given and we have to determine '�.Moreover, for the applications we are considering, we have that f�(0) = 0 and that�� vanishes up to some given order.More generally, in the following theorems, we will give conditions under which wecan solve (5.1) assuming only that for some Banach space E, with the approximationproperty mentioned in the introduction, �� : IRn 7! E, M : IRn 7! L(E;E), f� : IRn 7!IRn. (Here L(E;E) is the set of bounded linear operators from E to itself.) Thisadditional generality will be of use later in the construction.The cohomology equations for vector �elds can, likewise be written(5:2) L�'�(x) +M�(x)'�(x) = ��(x)where L =PiAi(x) @@xi where A(x) 2 Mm�m.We will start discussing both equations in the contractive case (kDfk < 1). Thiswill have direct applications for the general deformation method and for the contactcases. It will also be an intermediate step in the symplectic and volume preservingcases.Theorem 5.1. Assume that in (5.1) we have:20

i) f�; ��;M� are Cr and f�(0) = 0 for all �, Dif�(0) = Dif0(0), i � k,ii) Di��(0) = 0, i � k < r,iii) kDf�(0)k � ~� < 1,iv) kM�(0)k � ~�,v) ~�k+1 ~� < 1.Then there exists one and only one continuous solution '� of (5.1) de�ned in asmall enough neighborhood of zero which satis�esa) k'�(x)k � Kjjxjjk+1. Moreoverb) '� is a Cr family.Remark. Since it is always possible to choose norms in IRn, IRm in such a way thatthe norms of matrices are as close as desired to the spectral radius, conditions iii), iv)are really conditions about the spectrum of Df�(0) and M�(0).Proof. Valuable intuition about the proof can be obtained by considering �rst theparticular case in which M�(x) � M does not depend on x or on � and, likewise,f�(x) � Ax is just a linear map.In that case, we claim that(5:3) '� = 1Xn=0Mn��(Anx)is a solution of (5.1). Since kAnxk � kAkn kxk, we have k��(Anx)k � K kAkn(k+1) kxkso that the series converges uniformly whenever kMk kAkk+1 < 1. The uniform conver-gence justi�es the rearrangements needed to show it is indeed a true solution.The norm of `th derivative with respect to the variables of the n term in (5.3)kMnD`��(Anx) (A`)n k can be bounded by kMkn kAkn(k+1�`)Kkxkk+1�`kAk`n if ` �k + 1 and by kMknKkAk`n if ` � k + 1. Hence, the series obtained by di�erentiatingterm by term the series (5.3) converges uniformly and, hence the series de�nes a Cr21

function. The proof in the general case will follow the same pattern, even if some of theestimates are considerably more involved.We can choose a ball B� of of radius � � 1 centered at 0 2 IRn in such a way that(5:4) supjxj�� kDf�(x)k � � < 1 ;supjxj�� kM�(x)k � �; ��k+1 < 1:Since, by applying (5.1) N times we have'�(x) = NXi=0 24i�1Yj=0M��f j� (x)�35 ���f i�(x)�+ 24 NYj=0M��f j� (x)�35'��fN+1� (x)�:By (5.4) we have ������������ NYj=0M��f j� (x)������������� � �N+1;and if '� is to satisfy a) j'�(fN+1� (x))j � Kj�j(N+1)(k+1). The only possible solution ofof (5.1) satisfying a) is(5:5) '�(x) = 1Xi=0 24i�1Yj=0M��f j� (x)�35 ���f i�(x)� :The proof of Theorem 5.1 will be complete once we show that the sum in (5.5) iswell de�ned and that it satis�es a) and b). In the course of this proof we will denoteby K constants that are independent of i and x. The actual value of the constants maychange from line to line.We start by observing that the series (5.5) converges uniformly on B�. Sincekf i(x)k � ��i, hypothesis ii) implies that we have k��(f i�(x))k � K(��i)k+1. Onthe other hand, kQi�1j=0M�(f j� (x))k � �i, so that the supremum of the general term canbe estimated by K�k+1(�k+1�)i, which, by (5.4) is a geometric series of ratio smallerthan 1. By the Weierstrass M -test, the series converges uniformly.Once we have established uniform convergence, it is easy to show that it satis�es(5.1) by substituting it in the equation and rearranging the order of the terms.22

Moreover, k'�(x)k �Xi�0 �iK�kxk�i�k+1 � kxkk+1K :To establish existence and continuity of higher derivatives, we take derivatives termby term in (5.5) and show that the resulting series converge uniformly.We observe that by the product rule for derivatives,(5:6)D(24i�1Yj=0M��f j� (x)�35���f i�(x)�) =i�1Xj0=024j0�1Yj=0 M��f j� (x)�35DM�(f j0� (x))Df�(f j0�1� (x)) � � �Df�(x)� 24 i�1Yj=j0+1M�f j� (x)�35 ���f i�(x)�+ i�1Yj=0M�f j� (x)�D��(f i�(x))Df�(f i�1� (x)) � � �Df�(x);and that the supremum over B� of each of the terms in the sum can be bounded by�i�1K(��i)k+1. The last term can be bounded by �iK(��i)k�i. Hence, the derivativecan be bounded by (��k+1)i[i�k+1K��1 + ~K�k]which converges absolutely if summed over i.We also observe that kD'(x)k � Kkxkk.For �xed � we have established that the solution is C1.To prove that the solution is Cr as claimed in the theorem, we can use the \tangentfunctor trick." We observe that by taking derivatives of (5.1) we obtain:(5:7) D'�(x)�M�(x)D'��f�(x)�Df�(x) = D��(x) +DM�(x)'��f�(x)� :Note that the equation for D'� is of the form of cohomology equations considered inthe theorem. It satis�es conditions analogous to those satis�ed by '�, except that � is re-placed by � �� (since the linear operator is replaced by D'� !M�(x)D'�(f�(x))Df�(x))23

and condition (ii) is replaced byDiR�(0) = 0, i � (k�1)+ < r, where R�(x) refers to theright hand side of (5.7) and (k�1)+ = max(k�1;�1). This in turn means that condition(v) is replaced by �(k�1)++1(��) < 1. Clearly if �k+1� < 1 then �(k�1)++1(��) < 1, sowe can repeat the same construction as outlined above to construct D'�. Uniquenessimplies that the solution of (5.7) is indeed the derivative of '�(x). Then a calculationlike (5.6) implies '� is C2.If r � 2, we can repeat this procedure, di�erentiating (5.7) a second time withrespect to x. In this case � is replaced by ��2, while in condition (ii), k is replacedby (k � 2)+. Once again if ��k+1 < 1, then (�2�)�(k�2)++1 < 1, so condition (v) issatis�ed and we can construct D2'� and using a calculation like (5.6) conclude that '�is C3.Similarly, we will show that we can bootstrap the number of derivatives with respectto �.We compute the derivative with respect to � of one of the terms in the sum (5.5)(5:8)

@@� 24i�1Yj=0M��f i�(x)�35 ���f i�(x)� == i�1Xj1=0 j1�1Yj=0 M��f j� (x)�"� @@�M���f j1� (x)�+DM��f j1� (x)� X0�j2�j1�1Df j1�j2�1� (f j2+1� (x))� @@�f���f j2� (x)�#i�1Yj=j1+1M��f j� (x)����f i�(x)�++ 24i�1Yj=0M��f j� (x)�35"@��@� �f i�(x)�+D���f i�(x)� i�1Xj1=0Df i�j1�1� (f j1+1� (x))� @@�f���f j1� (x)�#:

Observe that since Di��(0) = 0 i = 0; � � � ; k for all values of �, we have @`@�`Di��(0) = 0,Di @`@�` ��(0) = 0. Hence, � @`@�` ����f j(x)� � K��jkxk�k+1 :24

Similarly, � @`@�` f���f j(x)� � K��jkxk�k+1 :Hence, it is not di�cult to establish again that the norm of the derivative in (5.8)is less than Kkxkk+1(�k+1�)ii2 :Now we observe that the derivative of ' with respect to parameters satis�es(5:9) _'�(x)�M�(x) _'��f�(x)� = _M�(x)'��f�(x)�+M�(x)D'��f�(x)� _f�(x) + _��(x):The �rst term in the right hand side vanishes at zero to order k and, using that _f�vanishes to order k, we see that the second one vanishes up to order 2k � 1 � k andthat it is is Cr�1. Since we established that _'� vanishes up to order k, we conclude that_'� is Cr�1 in the x variables.Applying the result just established to (5.9) we will conclude that _'� admits onederivative with respect to �.Hence, we can proceed by induction. If Da�Dbx'� exist and are continuous andvanish at the origin, we conclude that Da+1� Dbx'� and Da�Db+1x '� exist. The inductionstops only when the corresponding derivatives of �� cease to exist.Remark. It is also possible to prove that the function is Cr by taking formal derivativesterm by term and showing that the resulting series converges uniformly. The estimatesneeded are developed in [BLW].Remark. Notice that the size of the ball on which the solution is de�ned is independentof �. It depends only on f and M . This is, of course, to be expected since �� entersonly linearly. 25

To study the solution of (5.1) in the general hyperbolic case, we will �nd it con-venient to perform several initial reductions that will make it possible to simplify thenotation in the proof, but which do not incurr a loss of generality.We observe that, if (5.1) holds and h� is a di�eomorphism, we have(5:10) '� � h(x)�M� � h�(x)'� � h�(h�1� � f� � h�(x)� = �� � h�(x)so that ~'� = '� � h� satis�es (5.1) with data ~M� = M� � h�(x), ~f� = h�1� � f� � h�,~�� = �� � h�.In particular, if we take h�(x) = ax for some small positive number, a, we haveD ~f�(0) = Df�(0) and Di ~f�(x) = ai�1Dif�(ax),� @@��`Di ~f�(x) = ai�1 � @@��`Dif�(ax) ;so that, for i > 1, i+ ` � r sup�2[0;1]kxk<1 � @@��`Di ~f�(x) ;can be made as small as we please by taking a su�ciently small.Furthermore,sup�2[0;1]kxk<1 � @@��` �D ~f�(x)�D ~f�(0)� � sup�2[0;1]kxk�a � @@��` �Df�(x)�Df�(0)� ;which also can be made arbitrarily small.If �(x) is a C1 function with �(x) = 1 when kxk � 1, �(x) = 0, when kxk � 3=2,we see that de�ning,(5:11) �f �(x) = �(x) ~f�(x) + �1� �(x)�Df�(0)x ;�M �(x) = �(x) ~M�(x) + �1� �(x)� ~M�(0) ;���(x) = �(x)~��(x) + �1� �(x)�~��(0) ;26

�nding a solution of (5.1) de�ned in a su�ciently small neighborhood of the origincan be accomplished by �nding global solutions of (5.1) under the assumption thatkf�(x)�Df�(0)xkCr(IRn�[0;1]) is su�ciently small and thatM�; �� are in Cr(IRn� [0; 1]).We emphasize that the solutions we obtain in this case may depend on the cut-o�function used. There are easy examples that show that indeed di�erent cut-o� functionslead to di�erent solutions. Hence, the solutions produced by Theorem 5.1 are highlynon-unique.There are some further reductions that we will use.We recall the stable/unstable manifold theorem for di�eomorphisms.Theorem 5.2. Let A : IRs� IRu be a linear map and f� a Cr family of di�eomorphismsf�(0) = 0 in such a way thati) The splitting IRs � IRu is invariant under A,ii) kAjIRsk < 1, kA�1jIRuk < 1.iii) kf� � AkCr(IRs�IRu�[0;1]) � � where � > 0 is a number that can be computedexplicitley depending only on A.Then, there exist unique Cr families W s� : IRs ! IRu, Wu� : IRu ! IRs such thata) The graphs of W s� , Wu� are invariant under f�,b) kW s� kCr(IRs�[0;1]) � K�,kWu� kCr(IRu�[0;1]) � K�.where K, can be chosen independently of �.Proofs of invariant manifold theorems with dependence on parameters can be foundin [La] using the graph transform method and in [LW] using Irwin's method.We note that a corollary of the estimates b) in Theorem 5.2 is that if Dif�(0) = 0for i = 1; � � � ; k then, DiW s� (0) = 0; DiWu� (0) = 0; i = 1; � � � ; k. In e�ect, if we consider27

~f�(x) = a�1f�(ax) we see that jj ~f� � AjjCr � Kak�1. On the other hand, the function~W s� (y) = a�1W s� (ay) has a graph invariant under ~f� and, by the uniqueness in theconclusions of Theorem 5.2, we obtain that it should satisfy estimates b). We concludethat jja�1W s� (ay)jjCr � Kak�1 which, can only happen if the derivatives ofW s� of orderless or equal than k vanish at zero. Of course, an analogous argument works for Wu� .The same result can be obtained by examining the graph transform equations in [La].By repeated di�erentiation we can obtain equations satis�ed by the derivatives of W s�and we can check by inspection that if the derivatives at the origin of f� vanish so dothe derivatives of W s� .This theorem allows us to choose the change of variables (5.10) in such a way thatthe invariant manifolds are the coordinate axes.That is, IRn = IRs � IRu and f�(x; 0) = (fs� (x); 0), f�(0; y) = (0; fu� (y)) 8x 2 IRs,y 2 IRu.Therefore, we have established:Lemma 5.3. Let f� be a Cr family of di�eomorphisms de�ned in a neighborhood ofthe origin in IRn such thati) f�(0) = 0, Df�(0) = A, Dif�(0) = Dif0(0), 2 � i � k,ii) A is hyperbolic.Then, if s and u are the dimensions of the stable and unstable subspaces for A,IRn = IRs � IRu, for every � > 0 we can �nd a Cr family h� and a Cr family ~f� de�nedon the whole of IRn such that:a) h�(0) = 0, Dh�(0) = Id, Dih�(0) = 0, 2 � i � k,b) h�1� � f� � h� = ~f� on a neighborhood of the origin,c) k ~f� � AkCr � � ;d) ~f�(IRs � f0g) = IRs � f0g,~f�1� (f0g � IRu) = f0g � IRu : 28

Since for our purposes it is enough to have solutions of (5.1) on a neighborhoodof the origin, we can use cut-o� functions and study (5.1) de�ned in the whole IRn,but nevertheless assume that f� satis�es the conclusions of Lemma 5.3 and that �� hascompact support around the origin.Since the stable and unstable directions will play an important role, we will in-troduce the notation x = (xs; xu), with x 2 IRn, xs 2 IRs, xu 2 IRu. We will alsowrite A(xs; 0) = (Asxs; 0); A(0; xu) = (0; Auxu) :Inspired by the method used in [St] we start by solving the equation on the neigh-borhood of IRs up to high orders in xu.Lemma 5.4. Let f�, �� be Cr families as above. Assume that:i) kAsk � �+ < 1,kAuk � �+ > 1,Di��(0) = 0; i = 0; : : : ; k,kM�(0)k � �+.Let ` be such that:ii) �k+1+ �+�+ < 1 , i.e. ` < � ln �+ln �+ + (k + 1) j ln�+jln�+ :Then, we can �nd a C` family ~'� of compactly supported functions such thata) @j@xju ( ~'�(x)�M�(x) ~'� � f�(x)� ��(x))jxu=0 = 0; j = 0; � � � ; ` :Proof. Since we have good control of the problem on the stable manifold, our �rst goalis to rewrite the problem in an expansion in powers of xu. We will expand the originalequation in powers of xu and try to match the coe�cients of corresponding powers.29

Using a partial Taylor expansion��(x) = Xjij=0 �[i]� (xs)xiu + �[>]� (x):Here �[i]� (xs) 2 L((IRu)i; Rm) is given by Taylor's formula, �[i]� (xs) = 1i! @i@xui ��(xs; 0).Note further that the Taylor remainder satis�es:@i@xiu �[>]� (xs; 0) = 0 i = 0; : : : ; ` ;and that �[i]� is an r� i family with k�[i]� kCr�i � k��kCr . We introduce similar notationsfor M�(x) and its Taylor expansion. We will prove that one can solve recursively for thecoe�cients in an analogous expansion of ~'�. Once we succeed in doing this, we can cuto� the Taylor polynomial by a function tangent to the identity to in�nite order. Sincethe conclusion a) depends only on the Taylor expansions up to �nite order, it will alsobe satis�ed.If ~'i�(x) = ~'[i]� (xs)xiu is the monomial of degree i in the Taylor expansion, then~'i��f�(x)� = ~'[i]� �fs� (xs; xu)�fu� (xs; xu)i= ~'[i]� �fs� (xs; 0)�� @@xu fu� (xs; 0)�ixiu + R[i]� (x) ;where � @@xu�j R[i]� (xs; 0) = 0 j < i+ 1 :The operator ~'[i]� �fs� (xs; 0)�� @@xu fu� (xs; 0)�i belongs to L((IRu)i; IRm) and mapsxiu to ~'[i]� �fs� (xs; 0)�� @@xu fu� (xs; 0)xu�i If ~'i�(x) is a C` family, then R�(x) is a C`�1family, and if � @@xs�j ~'[i]� (0) = 0 j � k ;then � @@xs�j R[i]� (0; 0) = 0 j � k :The derivatives of order ` of R[i]� with respect to @@xu can also be evaluated in termsof tensor products of derivatives of f up to order ` and '[i]� (xs).30

Therefore, if(5:12) '[�]� (x) = X'[i]� (xs)xiu ;then, '[�]� �f�(x)� = Xi=0 T [i]� (xs)xiu + R[�]� (x)where, R[�]� is of high order andT [i]� (xs) = '[i]� �fs� (xs; 0)�� @@xu fu� (xs; 0)�i + C [i]� (xs) ;where C [i] is an expression that involves derivatives of f and '[j]� for j < i.If we substitute the expression (5.12) for '[�]� into (5.1) and equate similar powerswe obtain for i � ` �xed,(5:13) �[i]� (xs) = '[i]� (xs)�M [0]� (xs)'[i]� �fs� (xs; 0)�� @@xu fu� (xs; 0)�i� i�1Xj=0M [i�j]� (xs)C [j]� (xs) :Remark. The ~C [i]� 's in (5.13) are rede�ned with respect to the C [i]� 's in the previousequation by absorbing additional terms of lower order in the derivatives of f and '[j]with j < i.We observe that (5.13) is of the form of the cohomology equations we considered inTheorem 5.1. The data, C [i], �[i]� and the unknowns, '[i] are de�ned on the stable man-ifold and fs� (xs; 0) maps the stable manifold into itself and is a contraction. Therefore,we want to verify the quantitative hypothesis of Theorem 5.1.Note that the linear operator M� that appears in Theorem 5.1 is replaced by theoperator '[i](�) 7!M [0]� (xs)'[i](�)� @@xu fu� (xs; 0)�i The norm of this operator is boundedin a neighborhood of the origin by supremum of the norm of M [0]� and by the supremumof the norm of @@xu fu� (xs; 0) raised to the power i. Hence, by choosing the neighborhoodsu�ciently small, we can bound the operator norm by ��i where � and � are as close as31

one likes to �+ and �+. Similarly, we can assume that jj @@xs fs� (xs; 0)jj � �, where � canbe made arbitrily close to �+ by considering a su�ciently small neighborhood of theorigin. Hypothesis (v) of Theorem 5.1 becomes �k+1��i < 1, which is satis�ed becauseof ii) of the hypothesis of the Lemma, provided that we consider only a su�cientlysmall neighborhood of the origin.Once we have candidates for what the derivatives should be, the rest of the proofof Lemma 5.4 is just verifying that we can extend them using a cuto� function.If : IRu ! IR is a C1 function taking the value 1 in a neighborhood of zero andwith compact support, then ~'�(x) = '[�]� (x) (xs) ;where '[�]� is as in (5.12) and the coe�cients are obtained solving (5.13) is a solutionof the problem.We observe that, if '�(x)�M�(x)'��f�(x)� = ��(x)and ��(x)�M�(x)���f�(x)� = ��(x) ;then(5:14) ('� � ��)(x)�M�(x)('� � ��)�f�(x)� = ��(x)� ��(x) :Hence, to �nd solutions of (5.1), by considering '�� ~'� where '� is produced usingLemma 5.4 we can consider only the case where ( @@xu )i��(x) vanishes on the stabledirection for jij � `.Now, we try to solve the equation (2.4) for functions which vanish to a high orderin the unstable direction.Lemma 5.5. Let f�, M� be as before. Assume that �� is a Cr family with support ina neighborhood of the origin and that, in addition to the hypothesis of Lemma 5.4 wehave: 32

i) kA�1s k � �� > 1kA�1u k � �� < 1ii) Di��(xs; xu) � Kjxujk�i+1iii) kM�(0)�1k � ��Let ` 2 IN be such thativ) �����k�`+1� < 1 ; that is; ` < (k+1)j ln��j�ln ��(ln��+ln��) :Then, there exists a C` family '� solving (5.1) on a neighborhood of the origin.Remark. The conditions imposed in iv) are too conservative. The natural conditionsfor the method of proof presented here seem to be:iv0) �����k+1� < 1 ; ` < � ln ��ln�� + (k + 1) j ln��jln�� :At the end of the proof we will sketch how to improve the argument to get thisresult.Proof. The proof is very similar to Theorem 5.1.Again, considerable intuition can be obtained considering the case where M isconstant and, likewise, f� � A.We just check that(5:15) '�(x) = 1Xi=1M�n��(A�nx)is a solution.In e�ect, since k(A�nx)uk � �n�kxuk, we have that ��(A�nx) � K�n(k+1)� kxuk. Sothat the series (5.15) converges provided that �k+1� �� < 1.As before, if we take a derivatives with respect to x and b derivatives with respectto � in the general term, we obtain a general term:M�nDaxDb���(A�nx)�Aa��n33

whose norm can be bounded by ����k+1�a� �a��nKkxuk if a � k + 1.To prove the general case, observe that, by cutting o� the function M and f� aswe did in (5.11) we can assume that the bounds assumed for the derivative at zero arevalid globally, with slightly worse constants. We can arrange that these new constantsalso satisfy assumption iv).We also observe that (5.1) can also be written(5:16) '�(x)�M�1� �f�1� (x)�'�(f�1� (x)) = �M�1� �f�1� (x)����f�1� (x)� :We claim that a solution of (5.16) can be obtained by setting(5:17) '�(x) = 1Xi=1 24 iYj=1M�1� � f�j� (x)35 �� � f�i� (x) :The estimates to show that the sum in (5.17) converges are very similar to thoseused in the study of (5.5). Hence, it will be important to �nd out how fast f�i� (x)approaches the set where �� vanishes.Proposition 5.6. Under the conditions of Lemma 5.5 denote f�n� (xs; xu) = (xsn; xun).Provided that f� is C1-close enough to A, we can �nd � as close as desired to jA�1jEu jand K > 1 so that jxunj � K�nkxu0k.Proof. We denote:(5:18) f�1� (xs; xu) = (A�1s xs +Ns(xs; xu) ; A�1u xu +Nu(xs; xu)) :We note that Nu(xs; 0) = 0 due to the invariance of the unstable manifold andthat the C1 norm of Ns and Nu can be assumed to be as small as desired.Hence, we can estimatejA�1u xu +Nu(xs; xu)j � jA�1u xu +Nu(xs; xu)�Nu(xs; 0)j� (jjA�1u jj+ jjNujjC1)jjxujj :34

To show that the R.H.S. of (5.17) converges, we just observe that iYj=1M�1� � f�j� (x) � (�� + �)iwhere � can be made arbitrarily small by making the neighborhood under considerationsu�ciently small. Moreover, by Proposition 5.6 and the assumptions on � we havek�� � f�i� (x)k � K(�� + �)i(k+1) :By assumption iv) the general term of (5.17) is bounded by a geometric series of ratioless than 1 if � is su�ciently small.The rest of the argument is very similar to that in the proof of Theorem 5.1. Weobserve that (5.17) has the same form as (5.5) with M� replaced by M�1� , f� replacedby f�1� so that, if we take derivatives term by term we obtain analogues of (5.6), (5.8).Consider, for example, the analogue of (5.6). We have:(5:19)

D0@ iYj=1M�1� � f�(x)1A � � f�i� (x) == i�1Xj0=024j0�1Yj=0 M�1� (f�j� (x))35DM�1� (f�j0� (x))��Df�1� (f�(j0�1)� (x) � � �Df�1� (x)) 24 i�1Yj=j0+1M�1� (f�j� (x))35 ��(f�1� (x))++ i�1Yj=0M�1� (f�j� (x))D��(f�i� (x))Df�1� (f�i+1� (x)) � � �Df�1� (f�1� (x))Most factors in this expression have been estimated before. The only additional com-ment we need to make is that, in any neighborhood of the origin, we can boundkDf�1� k � (�� + �) and � can be chosen arbitrarily small by taking the neighborhoodof the origin small. Thus, the norm of (5.19) can be bounded by(5:20) i�1Xj0=0K1(�� + �)i�1(�� + �)j0(�� + �)i(k+1) +K2(�� + �)i(�� + �)i(�� + �)ik35

Note that, to estimate D��(f�i� ) we have used hypothesis ii) and Proposition 5.6.We can bound (5.20) by K(�� + �)i(�� + �)i(�� + �)ik and, if � is su�cientlysmall, hypothesis iv) will ensure that the sum over i converges. Higher derivatives andderivatives with respect to � are handled in a similar fashion.As remarked before, the conditions imposed in Lemma 5.5 are too conservative.In the following paragraphs, we will give a proof only in the case where f is linearand just sketch the modi�cations needed in the general situation. Even if these im-porvements lead to better values of A, we did not think it was worth to write a wholeproof. Of course, these remarks are only meant for the very motivated reader.For the case that M�(x) �M and f�(x) = Ax, we observe that, if we denote by Dsand Du derivatives along the stable and unstable directions respectively, the results inLemma 5.4, allow us to assume:ii0) kDu��(x)k � Kkxukk+1�`;kDs��(x)k � Kkxukk+1; rather than the more conservative ii) in Lemma 5.5.As before, the solution of (2.4) is given byP1i=1M�n��(A�nx). If we estimate theterms that we obtain when we apply Ds and Du to the general term, we obtain:kM�nDs��(A�nx)(A`s )�njj � (���k+1� ��)nKkxukk+1kM�nDu��(A�nx)(A`u )�njj � (���k+1�`� ��)nKkxukk+1�`Hence, under the hypothesis iv0), we know that ` derivatives along complementarydirections exist and are uniformly bounded. It is well known (see e.g. [Kr]) that, ifa function has ` derivatives along complementary directions and those are uniformlybounded, the function is C`��. Of course, if we consider ` =2 IN with the usual meaningof H�older regularity for the derivative of order [`], then this lemma allows us to removethe � in the conclusions.In the case that f� is not a constant linear map, the argument can be generalized.We remark that, since f� is uniformly close to A in the whole space, then, one can prove36

stable and unstable foliation theorems completely similar to the ones usually statedfor compact manifolds. These foliations, have smooth leaves but are not very smoothin transverse directions. Nevertheless, it is shown in [LMM] that one can considerdi�erential operators along the leaves. Moreover, one has regularity results analogousto those for coordinate foliations, namely that regularity along both the stable andunstable foliations implies global regularity { with a loss of � in the integer case and noloss in the non-integer case { (See [LMM], [Jo].) The existence of derivatives along thestable and unstable directions can be used by estimating the derivatives term by term.The details of these estimates are in [LMM]. Moreover, one can use regularity lemmasthat show that when a function is di�erentiable along the stable and unstable leaves,then it is di�erentiable. This completes the sketch of the proof that (iv) in Lemma 5.5can be replaced by (iv0).As a consequence of Lemma 5.4 and Lemma 5.5 we have the main result of thesection.Theorem 5.7. Let f�, M�, �� be Cr families as before. Assume that Df�(0) = A ishyperbolic, and thati) kAsk � �+ ; kA�1s k � �� :kAuk � �+ ; kA�1u k � �� ;kM�(0)k � �+ ; kM�(0)�1k � �� ;where �+; �� < 1, ��; �+ > 1.ii) Di��(0) = 0 ; i � k < r:Let ` 2 IN be such that:` < (k + 1) j ln�+jj ln��jln�+(ln�� + j ln��j) � ln �+j ln��jln�+(ln�� + j ln��j) � ln ��(ln�� + j ln��j) :Then, it is possible to �nd a C` family '� satisfying (5.1) on a neighborhood of theorigin.Proof. Using Lemma 5.4 we can construct a C ~ family ~�� that solves (5.1) whenrestricted to the stable manifold and that satis�es a) of Lemma 5.4 vanishing up to37

order ~, where ~ is any integer satisfying:(5:21) ~< � ln �+ln�+ + (k + 1) j ln�+jln�+ :Hence, we can choose ~ to be:(5:22) ~� � ln �+ln�+ + (k + 1) j ln�+jln�+ � 1:If we now consider the family �� � ~��, we see that it will satisfy the same equation(5.1) but with a right hand side that satis�es the hypothesis of Lemma 5.5.We can now obtain a solution which is C` for any ` such that:(5:23) ` � � ln ��(ln�� + ln��j) + (~+ 1) ln��(ln�� + j ln��j) :If we now take a ~satisfying (5.22), we see that we can take ` as claimed in the statementof Theorem 5.7.Remark. The assumption that ` 2 IN does not enter in the study of the cohomologyequation. It is quite possible to develop a theory of cohomology equations when thecoe�cients are in the usual H�older spaces. To use them in the deformation method, wewould need a theory of solutions of ordinary di�erential equations when the coe�cientsare H�older.>From Theorem 5.7 we can deduce the results claimed in the main theorems if weobserve that in all cases, if we have a Cr family of mappings, the generators of the ow are a Cr�1 family and, hence, the hamiltonians are also a Cr�1 family. (This isthe reason why the condition k < r � 1 appears in Theorem 1.1 rather than k < r inTheorem 5.7.) Furthermore, if our family of di�eomorphisms f� is tangent to order kat the origin, the vector �eld F will vanish also to order k. In the symplectic case, thisimplies that the hamiltonian F� vanishes up to order k + 1. In the volume preserving38

case, since we need to integrate, we can only conclude that the hamiltonian vanishesup to order (k� 1) at the origin. In the contact case, since the hamiltonian is obtainedjust taking interior products, it vanishes up to order k.Now we turn to compute the values for A, B for the di�erent geometric structuresthat we claimed in the remarks after Theorem 1.1 To do that, we just need to relate thebounds on M to the bounds of the derivatives in the original problem { the operatorsM are constructed out of the derivatives of the original problem { and to study therelation of the regularity of the hamiltonian to the regularity of the original problem.If we look at the way that the cohomology equations were derived, we see that in thecase that no structure is preserved, the equation (2.4) can be written in components asF�(x)�G�(x)+(Df��f�1� (x))G�(f�1� (x)) = 0. In order to compare it more easily with theequation (5.1) discussed in Theorem 5.7, we write it as F�(f�(x))�Df�1� (x)G�(f�(x))+G�(x) = 0 so that ��; �� have the same meaning in Theorem 5.7 and in Theorem 1.1.In that case, M = Df�1 and, hence we can take �+ = �� and �� = �+. Substitutingthis into the claim of Theorem 5.7 leads to the value in the remarks after Theorem 1.1.Similarly, in the case that the mapping is symplectic, we take M� = 1 and hence�+ = �� = 1. The order of tangency in the cohomology equation is one more than thatappearing in the hypothesis of Theorem 1.1.In the case that the ow is volume preserving, we take M� = �Df�1� �^(n�2) {notice that we are acting on antisymmetric forms { hence, we can take �+ = �(n�2)� ,�� = �(n�2)+ . Notice that the previous bounds are very conservative and that we couldtake as �+ the product of the (n�2) { maybe after changing the norm to an equivalentone { largest absolute values of points in the spectrum. An analogous statement holdsfor ��. This is the improvement alluded to in the remarks at the end of the statementof Theorem 1.1. The order of tangency in the cohomology equation is one less than theorder of tangency of the di�eomorphisms. So that the k appearing in Theorem 5.7 inthis case is one less than the k appearing in the hypothesis of Theorem 1.1.In the contact case, we takeM to be the factor by which the push forward multipliesa form. hence, we can take �+ = ��, �� = �+Note that in the symplectic, volume-preserving, and contact cases, to construct the ow from the solution of the cohomology equation we have to take one derivative. Thus,39

if we know that solution of the cohomology equation is C`, we can conclude that theconjugating di�eomorphism is C`�1.For the case of ows, it is quite possible to give a very similar treatment which weonly sketch.We observe that if L�, '�, M�, �� are as in (5.2), if we call f t� the time t map of the ow and introduce the matrix valued function �t� by:ddt�t�(x) = �t�(x)M�(f t� (x))�0� (x) = Id 9=; :then (5.2) can be written asddt�t�(x)'�(f t� (x)) = �t�(x)M�(f t� (x))'�(f t� (x))+ �t�(x)[L�'�] � f t� (x) == �t�(x) ��(f t�x) :Hence '�(x) = � Z s0 �s�(x)��(fs� (x)) ds+ �t�(x)'�(f t� (x)) :As in the case of the discrete equation this suggests that, if f t� (x) is a contradiction wecan write a solution as(5:24) '�(x) = � Z 10 �s�(x)��(fs� (x)) ds :Notice that this equation is quite similar to (5.5) and an analysis analogous to theone we performed there can establish that the improper integral above converges andis di�erentiable with respect to x and to parameters.For the case of ows, it is also possible to prove an invariant manifold theorem andthe same argument we gave can be adapted without di�culty. We leave the details tothe reader.Theorem 5.8. Let L�, '�, M�, �� be Cr families of vector �elds, vector valued asbefore. Assume that DL�(0) = A is hyperbolic in the sense of ows and that if �(A)denotes the spectrum of the operator A, we have:40

i) ��� � Re(�(As)) � �+,��� � Re(�(Au)) � �+,kM�(0)k � �+,kM�(0)�1k � �,ii) Di��(0) = 0 i � k < r.Let ` < (k + 1) j�+jj��j(�+)(�� + j��j) � �+j��j(�+)(�� + j��j) � ��(�� + j��j) :Then, it is possible to �nd a C` family '� satisfying (2.5) on a neighborhood of theorigin.>From this, we can derive the values of A and B in the remarks after Theorem 1.2in a way analogous to the way we derived the values of in Theorem 1.1 from those inTheorem 5.7.6. References[B] A. Belitskii: Equivalence and Normal Forms of Germs of Smooth Mappings. Russ.Math. Surv. 33, 107-177 (1978).[BLW] A Banyaga, R. de la Llave, C.E. Wayne: Cohomology equations and commutatorsof germs of contact di�eomorphisms. Trans. A.M.S. 312, 755{778[Ch] M. Chaperon: G�eom�etrie di�er�entielle et singularit�es des syst�emes dynamiques.Ast�erisque 138{139 (1986).[Che] Chen, K. T.: Equivalence and decomposition of vector �elds about an elementarycritical point. Amer. J. Math. 85, 693{722 (1963).[CM] P. Cherno�, J. Marsden: Properties of in�nite dimensional hamiltonian systems.Springer, Lecture Notes in Math. 425, (1974).41

[DL] A. Delshams, R. de la Llave: Some algorithms for computations of normal formsand their object oriented implementation. Manuscript[Gr] J. Grey: Some global properties of contact structures. Ann. of Math. 69, 421{450(1959).[Har] P. Hartman: \Ordinary Di�erential Equations", Birkhauser, Boston (1982).[I] Y. Ilyashenko and S. Yakovenko: Finitely smooth normal forms of local families ofdi�eomorphisms and vector �elds. Russ. Mat. Surv. 46, 1{43 (1991).[Jo] J{L. Journ�e: A regularity lemma for functions of several variables. Rev. Mat. Iber.4, 187-193 (1988).[Kr] S. Krantz: Lipschitz spaces, smoothnes of functions and approximation theory.Expo. Mat. 3, 193{260 (1983).[La] O. Lanford: Bifurcation of periodic solutions into invariant tori: The work of Ruelleand Takens. Springer, New York, Lecture Notes in Math. 322, (1972).[Le] H.I. Levine: Singularities of di�erentiable mappings, notes of a course by R. Thom.In \Singularities at Liverpool I, C. T. Wall ed.". Springer Verlag, Berlin, Lec.Notes in Math. 192, (1971).[Li] P. Liberman: Sur les automorphismes in�nitesimaux des structures symplectiqueset des structure de contact. Louveim, Colloque de Geometrie di��erentielle globaleBruxelles 1958 (1959).[Li2] P. Liberman and Ch-M. Marle: Symplectic geometry and analytical mechanics. D.Reidel, Dordrecht (1987).[LW] R. de la Llave, C.E. Wayne: On Irwin's proof of the pseudostable manifold theorem.Math. Zeit., In Press[LMM] R. de la Llave, J.M. Marco, R. Moriy�on: Canonical perturbation theories of Anosovdi�eomorphisms and regularity results for the Livsic cohomology equation. Ann.of Math. 123, 537{611 (1986).[Ly] V.V. Lychiagin: On su�cient orbits of a group of contact di�eomorphisms. Math.42

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[St3] S. Sternberg: The structure of local homeomorphism III. Amer. J. of Math. 81,578{604 (1959).[Ta] F. Takens: Singularities of vector �elds. Pub. Mat. IHES 43, 47-100 (1974).

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