del all ave marco mori yon

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Canonical Perturbation Theory of Anosov Systems and Regularity Results for the Livsic Cohomology Equation Author(s): R. de la Llave, J. M. Marco and R. Moriyon Source: Annals of Mathematics, Second Series, Vol. 123, No. 3 (May, 1986), pp. 537-611Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1971334Accessed: 10-09-2015 09:41 UTC

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Annals of Mathematics, 123 (1986), 537-611

Canonical perturbation theory of Anosov systems and regularity results for the

Livsic cohomology equation By R. DE LA LLAVE, J. M. MARco, R. MORIYON

Introduction

The goal of perturbation theories is to use knowledge of a " well understood" system to obtain information about similar ones. In dynamics, one of the most effective ways of doing that is looking for changes of variables that reduce, at least in an approximate way, the perturbed system to the well understood one. The main advantage of this procedure is that we obtain information about the global orbit structure.

This strategy can be implemented, in principle, for a general dynamical system or flow [BM] but historically its main use has been canonical perturbation theory around integrable systems, the reason being that the use of generating functions allows one to transform the equation for the change of variables-an equation between diffeomorphisms-into an equation between generating functions, much simpler objects.

However, it can be remarked [CEG] that there are hamiltonian systems which are well understood, even if they are non-integrable, and one could hope to apply canonical perturbation theory to them. Indeed, the authors referred to above construct canonical perturbation theories for geodesic flows on surfaces of constant negative curvature. (Related ideas appear in [GK1], [GK2], [GK3].)

The goal of this paper is to extend their results to all Anosov symplectic diffeomorphisms and to all hamiltonian flows Anosov on each energy surface.

It turns out that generating functions, being geometrically unnatural objects, are not suitable tools for this task (they cannot even be defined in some manifolds) and our first objective is to formulate canonical perturbation theory using only geometric objects. As a reflection of this natural geometric structure, the equations we will have to deal with are extremely simple so that we hope this formalism will be useful in other problems of perturbation theory.

In this formalism we are led to consider cohomology equations. For general Anosov systems Livsic [Lii] found the conditions of solvability in the class of CA functions 0 < a < 1 and, in some particular cases using harmonic analysis, he

537


538 R. DE LA LLAVE, J. M. MARCO, R. MORIYON

and others showed them to have Coo or CW solutions ([Li2], [GK2], [GK3], [CEG]).

In this paper we prove a C' version of Livsic's theorem (thus solving a question raised in [GK3]) and Coo dependence on the parameters involved.

For the problem of conjugation this allows us to discuss convergence without any appeal to Nash-Moser iteration as in [CEG].

Besides this main result, there are some other spin-offs. For example, we can modify the method to yield CW results in some cases, which include all the ones treated previously by harmonic analysis, and we also show that all Co topological invariant measures are actually C'. We have not explored the consequences, if any, that those results have for other sources of the same problem (e.g. statistical mechanics [B2]).

We also study asymptotic perturbation theories for analytic families of analytic canonical diffeomorphisms and flows and show that, in the cases considered before, Anosov systems, when these theories can be carried formally to all orders, there is a Coo family of Coo canonical transformations that reduce the perturbation to the well understood system. Even if in some cases it is possible to obtain CW rather than C', we do not know how to prove it in general and the methods are different since they involve Nash-Moser techniques.

The first section of the paper contains an introduction to the formalism of globally hamiltonian isotopies, as well as all the results concerning conjugation of canonical mappings and flows. These include the construction of complete sets of invariants.

For some manifolds, these invariants of conjugation for families can be reformulated in a more suggestive way as the preservation along the family of some invariants that can be computed for the system (map or flow) itself, independently of the family in which it is embedded. In the case of flows, the invariants are the actions along periodic orbits. This was worked out in [CEG], by another formalism, for the surfaces of constant negative curvature; here we do it for diffeomorphisms of tori. A more general construction working for diffeomorphisms of symplectic manifolds admitting a prequantization will be published by one of us elsewhere [Ma].

The second section is devoted to the study of the regularity of solutions of cohomology equations for Anosov diffeomorphisms and flows, including dependence with respect to parameters. We also apply these results to study the regularity of invariant measures.

In Appendix A we give a proof of the Anosov structural stability theorem for flows based on the implicit function theorem in manifolds of maps along the lines of [Mol] (see also [Mat] for the case of maps). This proof, which we could not


CANONICAL PERTURBATION THEORY 539

find in the literature, shows that the change of variables and the time change depend in a C' fashion on the perturbation; the standard proof in [A] would yield Lipschitz dependence and this improvement turns out to be crucial for the regularity results of Section 2.

In Appendices B and C we discuss generating functions and Lie techniques for perturbation theories.

Perhaps the greatest shortcoming of our approach is that it uses the assumption that the perturbed system is embedded in a family of perturbations and all the conditions are expressed in terms of the family. We would like to say something about small perturbations not necessarily embedded in families, especially because some of our theorems have the form that two systems are conjugate when they are connected through a family keeping the invariants of conjugation constant. It is very tempting to conjecture that the same result would be true when the two systems have the same invariants and are sufficiently close (see [CEG]).

1. Globally hamiltonian isotopies and conjugacy problems

The goal of this section is to introduce a formalism to deal with conjugacy equations between families of canonical maps.

The classical method [Po] was to write the conjugacy for diffeomorphisms in terms of their generating functions and then to study the resulting equation. The conjugacy problem was thus reduced to solving an equation among functions rather than an equation among diffeomorphisms and was, hence, susceptible to the tools of analysis. For our purposes, the main inconvenience of this strategy is that generating functions are not geometrically natural objects and cannot even be defined on some manifolds (see Appendix B).

In this section we find another geometrically natural way of reducing conjugacy equations to equations between functions. As a consequence of this naturalness, the resulting equations are very simple.

Immediately after introducing the definitions we can state our theorems about conjugation and prove them by applying the results of the section on cohomology equations.

It turns out that, in the cases where generating functions can be defined, the formalism based on them is equivalent to the one of this section if we consider only problems involving families of diffeomorphisms rather than diffeomorphisms on their own.

We refer the reader to [Ar], [AM] and [Th] for concepts and results of symplectic geometry which we use without explanation.



Let M be a symplectic manifold with symplectic form w.

Definition. We say that a Coo family (g,) of C' diffeomorphisms of M is a globally hamiltonian isotopy (GHI) when

d

where the generator Se is a globally hamiltonian vector field of hamiltonian GE (for all e in an interval I c R).

GHI's can also be defined on a domain smaller than M. In that case, each gE will be a diffeomorphism onto its image, which can depend on e.

We analogously define locally hamiltonian isotopies (LHI); as is well-known all Coo families of Coo canonical diffeomorphisms are LHI.

We use a small italic letter to denote an LHI, a script letter to denote its generator, and, if it is a GHI, an upper case italic letter to denote its hamiltonian.

Clearly, a GHI g, is determined by its initial value go and GE. Conversely, the GE are determined up to an additive function of e. In order to determine GE completely, we will assume throughout this paper that M has finite volume (dV = X A f) and normalize the hamiltonians to have zero average. In general, it could also happen that the flow of Se is not complete and, hence, some GE will not give rise to a GHI. However, in the applications here, we will make compactness assumptions that guarantee completeness. Hence, we can associate GHI's and their hamiltonians.

The GE will play here a role similar to that of generating functions: We rewrite conjugacy equations in terms of them.

We will start by discussing diffeomorphisms rather than flows since the results are simpler. Theorems 1.1 and 1.2 demonstrate the naturalness of the study of GHI rather than the more general LHI.

The problem we consider is the following. Let fe: M -- M be an LHI. We want to find another LHI such that

(1.1) e o g. = g ? fo, go = Id. Taking derivatives with respect to e, and after some manipulations, we

obtain an equivalent equation for the generators

(1.2) F

The symplectic form allows us to identify naturally locally hamiltonian vector fields and closed 1-forms. Using the canonicity of Le, we have

i(E )w = -(E)w f*0(E)W



Since f0& = fe4: H1(M) -- H1(M) because fo and fg are isotopic, taking cohomologies we obtain

(1.3) [(( )] (Id f)[i()]

In particular:

PROPOSITION 1.1. In order that there exist an LHI (gE) verifying (1.1) it is necessary that [i(E)w] E Range(Id - f04).

Remark. Since for a globally hamiltonian ,e [i(


Remark. One could worry that, since there are so many ways of reducing an LHI (f) to a GHI, the possibility of conjugating the resulting GHI to a constant through GHI's might depend on the way this reduction is made.

However, if Id - f0& is a bijection, either any GHI conjugated to ft can be conjugated to fo or no such GHI can be conjugated to it. This is a consequence of Theorem 1.1. In particular, given an LHI of Anosov diffeomorphisms ?e, and assuming the conjecture in the last remark holds, in order to check the solvability of the conjugation problem (1.1) it suffices to find a GHI of the form h71 o J. o hE and to study the corresponding problem for this GHI.

In view of the previous theorems, from now on we will only consider the problem of conjugating GHI's to their initial value under another GHI.

For convenience of the reader we collect here some useful identities between hamiltonians of GHI that are the basis of a calculus. Notice that the corresponding identities between generating functions (when they exist) are much more involved.

PROPOSITION 1.2. Let (fe), (g.) and (hE) be GHI's. (a) AE = FE + G. o f1 is the hamiltonian of a. =fEo g E. (b) BE = -HE o hE is the hamiltonian of bE = hE1 (c) If k is a symplectic transformation, then C, = H. o k is the hamiltonian

of CE = k - o hEo k. (d) If dE = o f o ? E? then we can determine the hamiltonian DE for d E

by the identity

DE =F g - G. ? gE + G. ? ? gE. All the above identities are proved by taking derivatives with respect to e

and translating the identities among vector fields into identities among hamiltonians; we leave the details to the reader since they are fairly simple.

Notice that the implication works both ways: If we have hamiltonians satisfying the relations above we can conclude that their isotopies satisfy the corresponding identity.

Using these properties we obtain that, for GHI's (1.1) is equivalent to

(1.4) GE GE Eof- =FE.

Remark. This equation corresponds in our formalism to the Hamilton- Jacobi equation for generating functions. Notice it is linear and besides it has some "group structure". Hence, if we know how to solve it for fo in place of fE and the solution satisfies "tame" estimates, we could apply Nash-Moser techniques. (The exact algorithm used is outlined in more detail later.)



We can immediately deduce some necessary conditions:

PROPOSITION 1.3. If there is a GE satisfying (1.4), then FE satisfies that, given a periodic point x, = 0,

1 N-1 (1.5) N Fi(0,E(x = 0.

In some cases, the previous necessary condition can be reformulated in a more suggestive way as the preservation of some invariants.

The invariants are constructed out of periodic orbits. Their construction would be easy if there were a symplectic potential 0 (that is dO = - c), but by Stokes' theorem this can never happen on a compact manifold. This main idea can, however, be readily fixed to yield non-empty results for the torus; we notice that in the universal cover there is a symplectic potential and perform there all the steps requiring it: the structure of the torus allows us to jump from the universal cover to the manifold very easily. These invariants, whose construction for the torus is given in detail below, can be generalized by different methods to all symplectic manifolds admitting a prequantization; the details of the general construction will appear elsewhere [Ma].

We need to assume that there is an interval Eo < E < El where the periodic orbits are nondegenerate, that is, for any periodic point xE of L. of period N. (LE)t( x ) does not have 1 as an eigenvalue. Under this assumption, the periodic points xE are persistent for Eo < E < El, and they have C' dependence on E. The previous condition is automatically satisfied if the periodic orbits are hyperbolic, and in particular if each L. is Anosov.

Suppose that M = T2d, the 2dimensional torus, with periodic coordinates (qi, pi), and X = E dqi A dpi. Then 0 = Ypi dqi is not a form on T2d, but on its covering space R2d. Let L.: T2d , T2d be a GHI and J.: R2d R2d a lifting of L. Since in R2d all closed forms are exact we have

E- 9 = dSE.

Denote by (xE, 0) < j < N - 1, a closed orbit of J., and set XE j = XE(0), 0 ? j ? N. where x ,O is a lifting of (xE0 ) to R2d. We have x N = + (k, 1) for some k, 1 e Zd. In this case we have:

PROPOSITION 1.4. In the situation described above, conditions (1.5) are equivalent to

(1.6) d E S (x - I ] = - K()



The constants K(Ec) will vanish if we add suitable constants in the definition of the actions SE. In this case, (1.6) means that the term between brackets (the action along the periodic orbit) remains constant along the deformation.

Proof Since

-0) =fELf =Lf*[di(JV ) + i(5 dH] = *(O )5

once SO is chosen we can set

SE =So + lf7* ( 0 ir-51n -5 dq . 0 If we compute the derivative on the left-hand side of (1.6) we obtain

Now, taking derivatives of tEx ,) = ?E j?L we obtain

This can be rewritten as

1 N-1 A~~d1 N-

FE E(j + (d + - de) - - Z FfxE)

N j =_ _ _ _ _ _ _= N j

since we assume that PE, N = PEO ? 1, qE, N = qfO ? k. Even if all the previous necessary conditions are not sufficient in general, it

is remarkable that for Anosov systems they are:

THEOREM 1.3. Let M be a compact symplectic manifold and (fE) a CHI consisting of Anosov diffeomorphisms. Then the folloting facts are equivalent:

(a) There exists a CHI (g E) such that

A ogE =g o0, g0 EId JW gE x EE joX I

(b) If x is a periodic point of period N for fas then 1 N-i -EZ FE(fE (XE)) = o?

Moreover, if Id - feo: Ho1(M) ncsHa1(M) is a bijection, then these facts are equivalent to



(c) There exists an LHI (g,) as in (a). In this case (g,) is a GHI. When M = T2d, the previous conditions are equivalent to

(d) The action invariants are constant.

Proof. The only thing to prove is that the conditions in (b) are not only necessary but also sufficient for the existence of smooth families, which solve (1.4). This is a consequence of Theorems 2.1 and 2.2 in the next section.

Remark. It is amusing to notice that, for the cohomology equation in the case of Anosov diffeomorphisms and when FE is the hamiltonian of f, (as in 1.4) we can prove differentiability of G. with respect to E very simply, without appealing to Theorem 2.2.

As remarked in Appendix A, the proof of the structural stability theorem based on the implicit function theorem automatically yields Coo dependence on parameters. Hence, there should be a Cx family of homeomorphisms g, satisfying fo = E E Jo.

On the other hand, the G. solving (1.4) should be a Co family of C' functions as can be seen by a closed graph argument: If FE varies in a C' compact set of the form { FEJ IIFII < rk, FE satisfies (1.5)} then G, also varies in a Cx compact set {GJ IIGEIICk < Sk}; but (1.4) is closed under C' limits. (See the proof of Lemma 2.11 where this argument is used in more detail.)

If G. is a C0 family of Cx functions, the hamiltonian vector field is C0 in E and C' in the manifold coordinates. Since g, is obtained integrating this vector field, it follows from standard O.D.E. theory that it is a C1 family of C' diffeomorphisms. As we discussed before, these g, solve the conjugacy equation (1.1). But the solutions of (1.1) close enough to the identity are unique, so that g, should coincide with - for all values of E (divide the interval I into pieces so small that uniqueness applies). Therefore g,(x) should be differentiable infinitely many times with respect to E and also with respect to x. It follows that it is a Coo function of (E, x) jointly. (The arguments to show the existence of mixed derivatives will be discussed at great length in Section 2; but for this case, they are relatively easy.)

Remark. Notice that in the above theorem there is no smallness assumption on E. The only smallness condition is that the family is always Anosov.

COROLLARY 1.1. Under the assumptions of Theorem 1.3, if there exists a GHI of C' diffeomorphisms g, with C' dependence on E satisfying

fe g = g E Jof, go =Id,

then each g, is of class Cx and depends in a Cx fashion on E.



Proof. The argument leading to (1.5) goes through under the regularity assumed in this corollary; so by Theorem 1.3 there exists a smooth GHI solving the conjugacy equation, and the corollary is a consequence of the uniqueness of conjugating GHI's by the Anosov structural stability theorem.

Remark. This corollary can be extended in several directions; for example, if the conjecture on (id - f04) is true, it would also apply for LHI's. It can be also adapted for flows, but then the statement is more messy because of the non-uniqueness of the conjugacy. The tedious (albeit simple) details are left to the reader.

THEOREM 1.4 (analytic version of Theorem 1.3). Let M be a compact, analytic, symplectic manifold and ft a C' GHI consisting of CA' diffeomorphisms. Then, if fo is Anosov, its stable and unstable foliations are analytic and iffe is sufficiently close to fo, the following facts are equivalent.

a) There exists a Co GHI (go) of C'a diffeomnorphisms such that

fo g0 g= &ofo, go = Id.

b) If x? is a periodic point of period N for ft we have

1 N-1

E F,(f,?(X,)) = 0. i=O

The proof of this theorem proceeds along very different lines from that of Theorem 1.3. We use a Nash-Moser iteration scheme. The advantage of this strategy is that we only have to assume C' regularity, with tame estimates, of the solutions of the Livsic equation at fo, even though we have to assume smallness conditions. For the Cm case, where regularity of solutions in an interval of parameters is automatic, this is irrelevant, but this strategy is useful for the cases where the Livsic equation is solved through the use of harmonic analysis or the Cw Anosov case where, (with the present technology) it requires analyticity of the foliations (see Section 2).

The method we use is an adaptation of that of [CEG] but, in our version, we do not use the fact that the invariants are also variational principles. However, we have not been able to do without the use of parametric families.

Proof of Theorem 1.4. We will just sketch the inductive step and show it can be iterated indefinitely. Given the tame estimates we prove in Section 2, showing convergence in a smaller domain of analyticity, under suitable smallness conditions, is standard Nash-Moser lore. Particularly useful in this context are versions in [Br, ?4], [Gal] or [CEG].



We will introduce the following notation. If FE is an analytic family of functions M R, we will set FE = X>FF, FN] = nNJnFn and analogously for < > >

LEMMA 1.1 (inductive step). Let fE be as in the assumptions of Theorem 1.4 and moreover F. = Q(cN). Then the equation F I


can only get estimates in a domain fo(U), which is a factor (independent of k) thinner and this leads to disaster. We will check that, arranging the estimates properly, the mismatches are controlled by II F,(kII and this goes to zero extremely fast. I I I denotes the sup of the modulus in a compact neighborhood of the manifold. At each step, it will have to decrease.

The second one is more delicate but we will say much less since it is almost the same as in many other treatments.

We rewrite

(k+ 1 (F(k[ 2k - _ (k + G (k o 1 )og(k + Fk[ 2k] g k

+(Gk o fl 0fo( (k - GE fk ) T gEk

Then, the first parenthesis is zero because of the way G(k is constructed. The norm of the second term can be bounded by using analyticity estimates in the E variable:

sup

||(Fe)

|| < sup

PF'

11( ) (1

- (-))

To control the norm of the parenthesis in the third term we observe it can be bounded by

IIV(G(kJ7L) II

d(f0(L(k


For hamiltonian flows we have similar results, namely necessary conditions for conjugation under GHI, which are sufficient when the flow is Anosov on each energy level.

Let d? , C I, be a C' family of C' hamiltonian vector fields on M without critical points, the parameter - varying over a real interval I, 0 c I. We fix an interval of energies [E1, E2] c Ho(M), and we consider the following problem:

Problem A. Find a new interval J, 0 E J c I, a GHI with variable range, g,: Ho 1(El, E2) E=-M, E, and a C' family of C? functions 4D: (El, E2) R, c E J, such that go = Id, I0 = Id, and

(1.7) g? E = Xo on Ho1(E1, E2),

where H ( = o H. We can look at g, in Problem A as a canonical transformation sending the

flow with hamiltonian He into the flow with hamiltonian Ho, up to reparametrizations.

A remarkable variant of Problem A is the following:

Problem B. Find a new interval J, 0 E J c I, and a GHI with variable range, g,: H-1(El, E M C= J, such that g0 = Id, and

(1.8) *-X' =o on H1(El, E2).

We assume in both cases that the hypersurfaces of constant energy of A0? are compact for each E. This implies that the hamiltonian vector fields are complete, and also that M is fibrated by the hypersurfaces for each E, since one can follow a suitable multiple of the gradient of H, in order to pass from one level set to another.

We can assume that the families of vector fields and their corresponding families of hamiltonians are defined on domains D E C M that depend continuously on E E I.

We now study Problem A. Problem B will be shown to be equivalent to it. We derive a cohomology equation equivalent to (1.7). First of all, (1.7) is

equivalent to

(1.9) (Do He og? = Ho

for a suitable choice of 4D. In order to get an equation for the hamiltonian G, of the isotopy we differentiate both sides of (1.9) with respect to E, and the equation we get is equivalent to (1.9) because the initial conditions for E = 0 are fixed and we can integrate back. Identifying hamiltonians and composing with



gE 1 on both sides we obtain

oHe + ?e He d G

(where { , } means Poisson bracket). (1.9) and the nonexistence of critical points for AE imply that me when it

exists cannot have critical points. Hence we can transform the previous equation into

(1.10) ()= d - [H.. (V)']oH.

Notice that as a consequence of the fact that (D 0, (D is actually a diffeomorphism from (E1, E2) onto (E(fE1), (E2)). If we denote its inverse by I,, the term between brackets on the right-hand side of (1.10) is just -(dI/dc)o(,

The cohomology equation (1.10) plays a role similar to that of (1.4) in the discussion of diffeomorphisms. The independent term in (1.10) looks more complicated, and this can be understood as a consequence of the fact that we have not made a definite choice of the hamiltonians involved, as we did for diffeomorphisms.

The first consequence of (1.10) is that DE can be determined immediately from the hamiltonians HE: since their domains of definition are fibrated by the energy level hypersurfaces E E = H- 1(E), there are natural measures tL E induced on each E so that

coA n =dH AIE onY2~E E E, E E, The measures P,, E are invariant under E, since both X A n and H. are invariant. If we integrate both sides of (1.10) along E E with respect to

P,, E' the term on

the left-hand side cancels according to Green's formula, and we have

where dH

E(E) =

IE,E( E,E) dd E, E

The function A is determined by HE, and (1.11) is an ordinary differential equation for 4, that is satisfied with initial conditions p0 = Id. This determines 4E and hence D =.

Let us now make a more geometrical description of the previous discussion. Let HE = o H,; then (1.9) is equivalent to

H H E0gE = Ho



Differentiating with respect to E and composing with g 9 on the right we have

EdH (1.12) Xe G = d

and integrating on E E = H- (E) with respect to the measure , E determined by

An = dH A EE we have

dHEl

E

Equality (1.13) is equivalent to

(1.14) [dHe H (, e J, E?1 < A < B < E2. This can be proved applying Stokes' Theorem to the (2n + l)-dimensional manifold with boundary

N = {(E, x) E J X M: a < e < , A < 1I(x) < B},

and the (2n)-form Hf~o. This gives

dH YIEP B |H Hc Anf An Ja[J~kke E Ej T'(AB) 'c(A, B)a

An ~~~An +B o fl-AJ( H- l(B) n dN H-'(A) n N

But on H'-(E) n 8N= {(e, x) e dN: H (x) = E}, since dH = 0,

An =dX EE=d j EP.=dHid co d E A E, E (H - e de )AE E d~ 1 E, A de

so that

| [| ( - de A dE| de H--wn. fa[LB~fde dE 'E(Af B) + |s|

- A i Ed Al dA d e Ak A d~ r,, dj



Differentiating both sides with respect to 13 at /3 = a, we see that

'HE (A B) An] fA[E(J1 d -diEE)1 dE and the equivalence of (1.13) and (1.14) follows from this. Notice that the previous calculation proves that

d [( A]=dHF

de H[L-'(A, B) J H-(A, B) de Remark. The previous calculations can be reinterpreted in the following

way: The natural data for Problem A are not families of hamiltonians but rather families of hamiltonians " up to reparametrization." (Two hamiltonians are equivalent from our point of view if we can reparametrize them so that they coincide.)

Then, we can think of (1.14) as a normalization condition that picks a reparametrization of our data. This normalization is very natural and useful as witnessed by the fact that, for hamiltonians satisfying it, Problem A becomes Problem B. Since this normalization can-as we have shown-always be achieved, it suffices to discuss Problem B.

Notice the analogy of this with what we did for diffeomorphisms normaliz- ing the hamiltonian to have zero average. Since this is a useful property we will record it in a definition.

Definition. A C' family of C' hamiltonians H., E E J, is normal for E1 < HE < E2 if (1.14) holds.

A C' family of C' hamiltonian vector fields HOE' 6 E J. is normal from E1 < HO < E2 if it admits a family of hamiltonians HE, normal for E1 < HE < E2 with HR = HO on HO -(E, E2).

Given a C?? family of C' hamiltonian vector fields XE, its normalized family of hamiltonian vector fields from E1 < HO < E2 is the family HOE determined by HE = oD ? HE according to (1.11), if we assume (E1, E2) is in the range of HE for E e J. The hamiltonians HE are the normalized hamiltonians.

If we integrate both sides of (1.12) along periodic points, we get necessary conditions for the solvability of Problem A:

PROPOSITION 1.5. A necessary condition in order to solve Problem A is that for any E J and any periodic orbit -y of .O'E with energy H,(y) e (El, E2),

(1.15) f E dt= O. d hr

where HE denotes the normalized hamiltonians from El < HO < E 2



Remark. Condition (1.15) can be expressed as a condition on the original hamiltonians HE as follows:

(1.16) - dt = 11E( Et EE) dHede E'

where T is the period of y with respect to eE , and E = HE(y). When the symplectic form co on M has a symplectic potential, that is

X - dO, we can define action invariants for Problem A:

PROPOSITION 1.7. Under the above assumptions, suppose that the periodic orbits are nondegenerate on each energy level; then a necessary condition in order to solve Problem A is that for any Coo family YE of periodic orbits of YE with fixed energy HE(Y.) = E E (El, E2),

d (1 .17) de 0 0 J.Ye

Remark. Nondegeneracy of the periodic orbits holds automatically if they are hyperbolic and in particular if the flows are Anosov on each energy level.

Proof Given a family YE of periodic orbits as above, as a consequence of their nondegeneracy, gE(yO) = yE, and (1.17) follows from the fact that the g, are action-preserving, according to Proposition 1.4, a). In fact, conditions (1.15) and (1.17) are equivalent, independent of their relation to Problem A:

PROPOSITION 1.7. Suppose there is a symplectic potential, w = - dO, and Ye is a C' family of periodic orbits of XE with fixed energy HE(-YE) = E. Then (1.15) with HE instead of HE is equivalent to (1.17).

Proof. Consider the map

Y: J X S' M

(E) s) YE(sTE)

where TE is the period of yE. We have

d d d'~~(YE y - = TE dHE e

d dH

TE [- (HE(yE)) - E(YE)

dH =-T E(Y)



so that we can express ye*a as

dH yX TE d(YE) d E A ds.

Next we apply Stokes' theorem in the cylinder a' < E < s E Si, and we get the equality

l l 16 ldH ffj- fdIdtdA Ye' e E YESd

so that d dH

d-f Jd dt de Yde which proves our result.

A different set of invariants can be formulated in terms of the periods. As we will see this set of invariants turns out to be weaker than (1.15) or (1.17).

PROPOSITION 1.8. Suppose that the periodic orbits are non-degenerate on each energy level; then a necessary condition for solving Problem A is that for any CU family of periodic orbits yE of XE with fixed energy HE(Y>)= E E (E1, E2), and period TE

(1.18) dTE -

Proof. It is an immediate consequence of the fact that g (-yO) = yE and

PROPOSITION 1.9. Assume that YE E is a C?? family of periodic orbits of J# with energy H(YE. E) = E E (E1, E2). If

= 0 |E d

for any E E J. E E (E1, E2), then dTEE = 0

de for any E E J. E E (E1, E2).

Proof: We consider the map

y: J X (E1, E2) X S' M



Just as in the proof of Proposition 1.8,

a * i(- ) =- TE E E ( Y )

Moreover,

W* ( - ) = C(TEOEYE EYEE) -E ( d E) = TEaEE = TE,

so that y * w can be expressed as

dH Y O TE E ds A dE + Te.E d (YE, E) deds + AdE A dc

for a certain function A. Since yea is closed, it vanishes when integrated along the boundary of the

region E' < E < E", E' < E < E", s E S', and we have, taking into account the hypothesis we made in the proposition, that

Te E) dE = 0 for any E', E (E1, E2), E, EJ.

The proposition follows immediately.

Remark. If the periodic orbits are nondegenerate on each energy level there are 2-parameter families Ye, E of periodic orbits of -Y, with energy E.

Then, according to (1.11) and (1.16) the parametrization (DE giving a normal family He = 4E o HE is determined uniquely by what happens in one such family of periodic orbits. Indeed, if there was an IE it should satisfy

d PE (()) = 1X dHE d~~ I TE fl Edd

for all E E J and all E E (E1, E2) where T, E is the period of Ye, E. This ordinary differential equation and the initial condition %0 = Id determine bDE uniquely.

The compatibility condition tells that we would obtain the same function (DE no matter which family of periodic orbits we use.

In general, it is not true that this (DE would also satisfy (1.11) but, for hamiltonian systems Anosov on each energy surface, the measure dpl EE Eis in the weak closure of the convex combination of the invariant measures [BR], [Si] on each periodic orbit and so, in this case, (E should also satisfy (1.11). Therefore, for hamiltonian flows Anosov on each energy surface, we can rephrase the



compatibility conditions, as

1 dH Ed _yfjEdt

is only a function of E and the energy of the closed orbit y. As in the case of diffeomorphisms, these necessary conditions are also sufficient under hyperbolic- ity assumptions. We have:

THEOREM 1.5. Assume that he generates an Anosov flow on each energy hypersurface. Assume (1.15) (or (1.16) or the version in the previous remark) holds for any periodic orbit y of dE with reparametrized energy Hf(Y) E (E1, E2), for all - e J. Then, Problem A has a solution.

In case there is a symplectic potential, condition (1.15) can be replaced by condition (1.17), if we assume it is satisfied by all C?? families of periodic orbits y, with fixed energy H,(-y) = F E (E F2), E E J.

Proof Assume (1.15). We have to prove the existence of a C?? family of Co real valued functions G. satisfying (1.12) on H?-'(El, E2), E J. This will be a consequence of the results of the Livsic cohomology equation in Section 2, once we prove that ? is hyperbolic for every c (for E small this is a consequence of Anosov structural stability results). This is proved using the following lemma:

LEMMA 1.2. If X is a C' vector field on a compact manifold with an Anosov flow, and f is a C' real, positive function, then X = fX generates an Anosov flow.

The proof will be given at the end of the proof of the theorem. Once we know that 7? is hyperbolic, by Theorems 2.1 and 2.2 in the next section we know that the conditions (1.15) are not only necessary for the solvability of (1.12), but also sufficient. The regularity with respect to variations of the energy is a consequence of the regularity with respect to parameters (Theorem 2.2) and the fact that HR -(E,, E2) is fibrated by He, as was explained before. The second part of the theorem is now a consequence of Proposition 1.8.

Proof of Lemma 1.2. We construct explicitly the stable and unstable bundles Es, u for the flow of X, as follows:

jES = {Z + a(Z)X: Z E Es),

where Es is the stable bundle of the flow of X, and a E I(ES*) will be specified in a moment. Some calculations show that Es is invariant under the flow 4t of X



if and only if

Lx(f 'a) = f 2df; so we can determine a by

a(vS) = - f(x)f f(4t(x))* df(t * vx) dt,

VE EEs. The above integral converges uniformly by the contractivity of the flow on stable directions, and so a is continuous.

In order to show that ot is contractive on ES, we note that it is given by

ft(X ) =T 0(X)( X),

where, d d t(X = fTt(4X)(X)) ) "(X) = 0;

so there exist c > 0, C > 0, such that

Ct < Tt(X) < Ct. t 2 O.

Then, for any i5s = vs + a(vs)Xx E E', with vs Es, and any t> 0

I4t(V-)I= 0T(X)VSx + a(Af(X)VX) XX ?< Ckli where X = XV < 1. The unstable bundle is constructed similarly.

Remark. The hypotheses of the previous theorem are satisfied by the most natural hyperbolic hamiltonian flows, namely geodesic flows on compact manifolds of negative curvature, and suspensions of canonical Anosov diffeomorphisms, as follows:

If (N, o) is a compact symplectic manifold and f is a CX canonical Anosov diffeomorphism on N. we consider the manifold M = (N x R2)/ where the equivalence relation is defined by

(x, s, a) (x, s, fi)


turn uniqueness of the solutions of the conjugacy equation (1.1), in accordance with Anosov's structural stability theorem.

In the case of flows, solutions of the cohomology equation (1.10) are determined by their value at a curve transversal to the energy hypersurfaces once be is fixed, and by the ergodicity of Anosov flows the solutions are in this case unique up to additive functions of the hamiltonian He. This means that the generator We of the conjugating equation (1.7) is determined up to the addition of reparametrizations of ,E and this implies some nonuniqueness of solutions for both Problems A and B. This can also be explained by the symmetry of the systems E , since any time t flow of any hamiltonian reparametrization of Xe is a canonical diffeomorphism that conjugates -e to itself.

From the above discussion we deduce immediately the following results concerning Problem B.

PROPOSITION 1.10. If Problem B can be solved, then oe must be a normal family of hamiltonians vector fields from E1 < Ho < E2, and for any 0 E J. and any periodic orbit y of Xe with energy He(y) E (E15 E2), (1.15) must hold, where He denotes the normal family of hamiltonian.

In case there is a symplectic potential, co = - dO, (1.17) must hold for any family of periodic orbits ye of -e with constant energy He(-y) = E E (E1, E2).

Remark. The normality of Xe can be expressed as a condition on H., namely

d_ dHE 1 dE ~~~d~ dl F]=

for all (e, E) in the appropriate range. In this way, the expression that is differentiated with respect to E in the previous formula defines a function 4(Ec) for E E J. and the normal hamiltonian is given by

He= He - (a) da.

As a consequence of this, (1.15) is equivalent to

1 dH1 IC

Th do |X=J (a)da (see (1.16)),

and H can be determined from any family (e of periodic orbits of Xe by

1 dHF (1.19) HR= He T dId



THEOREM 1.6. Assume HOE generates an Anosov flow on each energy level, consider any C' family of periodic orbits (E of H'O, and define H. by (1.19). If (1. 15) holds for any periodic orbit y of XE7 with HE(y) E (E1, E2), then Problem B can be solved.

In case there is a symplectic potential, X = - dO, condition (1.15) can be replaced by (1.17), with the assumption that it holds for any family of periodic orbits yE of H'O with constant energy HE(y,) = E E (E1, E2).

Remark. From the previous considerations, we see that the above conditions are necessary and sufficient in order to solve Problem B. Also the normality of the given vector fields is a consequence of (1.15), (1.16) or (1.17) if HE is defined by (1.19) in the case of Anosov flows. This can be explained as a consequence of the existence of many periodic orbits, so that a function with average equal to zero along all of them has the analogous property along energy shells. However, the construction of the normal hamiltonians HE by solving (1.11) has still some relevance in this case, since it gives directly the maximal domains where the previous conditions must be checked.

Remark. In case M is the cotangent bundle T*V of a compact manifold V, the HE are energy functions of some metrics and we choose for the potential 0 the canonical 1-form, the normality of J-7 is equivalent to the fact that all the metrics have the same total volume. Conditions (1.15), (1.16) and (1.17) are equivalent to the fact that the lengths of the closed geodesics are invariant under the perturbation. This is a consequence of the fact that O(O) = 2HE. If the closed geodesics are nondegenerate, a deep theorem of Duistermaat and Guillemin, [DG], says that the above conditions hold if the spectrum of the laplacian remains constant under the perturbation.

If the metrics are negatively curved, the assumptions of the theorem of Duistermaat and Guillemin and Theorem 1.5 are verified. In case the spectrum of the laplacian is constant, if we could insure that the conjugation g. we obtain is linear along the fibers of M, it would induce an isometry in V, and we would have spectral rigidity theorems. In this direction see [GK 1,2,3], where those results are proved for two-dimensional manifolds and in higher dimensions with a pinching condition on the curvature.

We turn now to the study of conjugation problems for families of locally hamiltonian vector fields. We assume that M is a compact symplectic manifold, and J-, 5 E I, is a C' family of C? locally hamiltonian vector fields on M without critical points, the parameter e varying over a real interval I, 0 E I. In a first step we consider the simplest conjugation problem:



Problem B'. Find a new interval J, 0 c J C I, and an LHI, g, e e J, such that g0 = Id, and (1.20) g ,=e X0 on M for E E J.

We start looking for necessary conditions in order to solve this problem. Taking derivatives with respect to E on both sides of (1.20), we see that it is equivalent to

(1.21)= de

If we set ae = i(e)W /3e = E)E A'M, taking into account that L',' = 0, (1.21) is equivalent to

da (1.22) L'e= d=

From this we get immediately the following condition on dyE

PROPOSITION 1.11. A necessary condition for solving Problem B' is that for any E cJ,

(1.23) [a.] = [a0] inH'(M). Moreover, if a C' family of C' functions De is fixed with

da E de =dDE, eel,

then for each E E J there exists CE C R such that whenever the periodic orbits Hy of JY with period Tie, 1 < i < N. satisfy

N A Xi [yiE] = 0 in H1(M, R) i=1

for some X1,. .., XN e R. then

(1.24) ( xi De) XiTi ce Proof If PE = dG on U C M, then

LAf;3=d( ?'G) = d f E3* ) on U.

By (1.22), it follows that da~/dE is exact on M, and (1.23) follows by integration on e.

Now (1.24) is a consequence of the fact that for any compact interval -y of an orbit of Ye' contained in U as above,

fI3e=f1'G = f(DE + CE).



If J0? has hyperbolic behaviour, the conditions described above are sufficient in order to solve Problem B'. We shall make the following assumptions on the vector fields: For each e, the (codimension 1) foliation associated to a. has compact leaves, and ye defines an Anosov flow on each leaf.

Examples satisfying the above hypothesis appear naturally as suspensions of Anosov diffeomorphisms in the following way: If (N, w) is a compact symplectic manifold and f: N -- N is a canonical Anosov diffeomorphism, we consider on N x R2 the relation

(x) , ,) 9, (xi)


codimension one foliation on a compact manifold with compact leaves that is globally defined by a 1-form has a quotient manifold.

LEMMA 1.3. In the above situation, there exists a C?? family of C' curves (? e E J. transversal to the corresponding foliation, and intersecting every leaf exactly once.

Proof. Let us construct 40. We fix a riemannian metric on M so that a0 has length 1, and we consider the vector field X that corresponds to a0 by the metric. Given x0 E M, let us use the symbol xt for the points in the integral curve of X passing by x0. By the compactness of M, there exist t1, t2 E R, t 1 = t2, such that x t and xt2 are in the same leaf. We can assume 0 = t1 < t2. By the compactness of the leaves, { t: x0 and xt are in the same leaf) is discrete. We denote by to its smallest strictly positive number. Then, xS and xt are in different leaves for 0 < s < t < to, since otherwise x0 and x t-s would be in the same leaf. Moreover, the union of the leaves passing through xt, 0 < t < t0, is M, since it is an open and closed set. Consider now a simple closed curve qo that follows xt between x0 and xto and then goes back to x0 along its leaf. The desired curve 4o can be constructed by a small Co perturbation of mo. Finally, the family (e can be constructed from 4o by a straightforward perturbation argument. This finishes the proof of Lemma 1.2.

We come back to the proof of Theorem 1.7. By Theorems 1.1 and 1.2, there exists a C?? family of C' functions (De on M such that

'e)E= AE and he =0?

A simple computation shows that da

LX,~ (q, + d 0,)e= ?1 + d Ae dDe d '7 ~~~~~~~de Thus, equation (1.22) has a solution and Problem B' can be solved.

Remark. Equation (1.22) has many solutions, as happens in the case of globally hamiltonian flows (see the second remark after Theorem 1.4). In fact, if f3e is a solution, and Le is a C' real function on M satisfying dLe A a, = 0, then 3e, + fea, is again a solution of (1.22). This reflects the fact that any time t flow of any locally hamiltonian reparametrization of He is a canonical diffeomorphism that conjugates e to itself. However, there is the possibility that the systems of A', present a bigger symmetry, and equation (1.22) has more solutions. This is the case if J?0 is the locally hamiltonian suspension of a canonical Anosov diffeomorphism described above, since then the vector field a/au commutes with 9o, and the corresponding 1-form, ds, satisfies that



g3o + Cds is a solution of (1.22) for e = 0 whenever go is another solution. This reflects the invariance of X0 under translations in the variable u. This extra symmetry that can appear for locally hamiltonian flows is reflected in the solvability conditions (1.23) and (1.24) for the conjugation problem by the fact that there are some cases where the constants CE in (1.24) are not uniquely determined, since both the numerator and the denominator on the left-hand side vanish.

In the globally hamiltonian case the suspended flows are also invariant under translations in the variable u, but the corresponding vector field is not globally hamiltonian, and no GHI can be constructed out of it. In that case, the dependence on arbitrary constants CE is not possible for the solutions of the cohomology equations, since the hamiltonians must be normalized.

Remark. A similar theorem can be proved if it is assumed that the locally hamiltonian vector fields 9e define Anosov flows on M. However, it is not clear whether there are any systems satisfying these hypotheses.

Finally, we consider the extension of Problem A to families of locally hamiltonian vector fields:

Problem A'. Given a C' family Dw, e E I, of C? locally hamiltonian vector fields on M without critical points, find a new interval J. 0 E J C I, a CX family of Cx functions DE1: M - R e C J. and an LHI, g, e e J. such that go = Id, (Do = Id, and

(1.25) e *e dy' e 0

We assume, just as we did in Problem B', that the foliations associated to the vector fields have compact leaves. Then we can consider, by Lemma 1.2, a C' family of C' curves her transversal to the corresponding foliation, and intersecting every leaf exactly once. We assume each (e is parametrized by t E [0, 1]. If we set

CE =|a

where we keep the notation of Problem B', then a Cx family of Cx functions He: M -* R/Z = S', constant over each leaf of the foliation corresponding to ,h',, is determined by

HE(JE(t)) = cc-, aE FelO. t]

where we write s instead of s + Z E S'. These functions can be considered as



hamiltonians of C7 4'9 in the sense that

ae = C dHe.

In this situation, if (1.25) holds, since (I . Ai% is a locally hamiltonian vector field, it is clear that there exists fe: S' -* R such that

OF f o Ho.H

and (1.25) is equivalent to

(1.26) C7-' d(H, o g,) = CQ'(f o) dHO,

so that

fe (Ho) dMO = CO/CE

and a family of diffeomorphisms 4,: S' -- S' is defined by

OE(HO) C JH'f(s) ds. Co

Then (1.26) is equivalent to

(1.27) H' o g,= ? Ho + K, for some constant KE. Differentiating with respect to e we get

dH_ d4 -l - ? f + (CH,) ? g, = c d Ho + K

Composing with g-', and using (1.27), we obtain

dHe (1.28) e + We H = Me 1 He + F

where SF is the generator of the family of diffeomorphisms, fe, determined by

d c

Conversely, (1.28) implies (1.25) by the uniqueness of solutions of ordinary differential equations.

If we differentiate (1.28) along M we obtain

(dHF (1.29) L43E =d ) + d(Q o H.) and this implies, just as in Problem B', the following necessary conditions for the solvability of Problem A':



PROPOSITION 1.12. If Problem A' has a solution, then there exists a C?? family of C?? functions RE: S' -1 R, such that whenever the periodic orbits Y-y of A?' 1 < i < N. with fixed energy HE(-y[) = H, and periods Te, satisfy

N

E X [yie] = 0 in H'(M, R) i=1

for some X1..., XN eR, then

(1-30) ( L AiJ E XiTi MJ~eHE). Proof Integrate (1.29) along LXi[yi] taking into account that L,43B=

d(f3 *e).

In the case of Anosov flows on each compact hypersurface, the above conditions suffice to solve Problem A':

THEOREM 1.8. If the locally hamiltonian vector fields dy satisfy (1.30), then Problem A' can be solved.

The proof of Theorem 1.6 can be adapted to this case. We leave the details to the reader.

Remark. The discussion above shows that in the Anosov case, condition (1.23) can be replaced by the apparently weaker condition CE = C0, E E J. and this, together with (1.30) with ME(HE) = KE, still suffices in solving Problem B'.

Remark. Theorem 1.4 has an analogue for flows; namely, if we can solve the Livsic equation (with tame estimates) for one particular flow embedded in a family satisfying the compatibility conditions, then there are a smooth GHI and a smooth reparametrization reducing the family of small perturbations to the initial flow. (The proof is a straightforward adaptation of that of Theorem 1.4.)

Again, this theorem is more interesting when applied to Cw cases. Unfor- tunately the Cw theory for Livsic equations for flows is less general than the Co. Solving the Livsic equation for a Hamiltonian flow entails solving it on each energy surface and showing good dependence (we would need tame) on the parameters. The first part goes through under the assumption of analytic foliations but we have not been able to show analytic dependence in general. Nevertheless, analytic dependence on the energy is quite easy to show for geodesic flows because changing the energy is reparametrizing the flow by a constant factor. This generalizes the results about conjugacy of [CEG] to geodesic flows on homogeneous, negatively curved manifolds (so that the foliations are analytic) of any number of dimensions.



2. Smoothness results for the Livsic cohomology equation

The goal of this section is to prove existence of smooth solutions for cohomology equations for transitive Anosov flows and diffeomorphisms as well as smooth dependence with respect to parameters.

A source of motivation was discussed in Section 1, but there are others; notably, such equations appear also in statistical mechanics of one dimensional systems. See [B2].

The main theorems we prove here are the following:

THEOREM 2.1. Let m be a compact manifold and kt be a transitive Cx Anosov flow on M (resp. fa Cx diffeomorphism of M), and 'q: M R be a C' function. Then, the following are equivalent:

a) There exists A: M - R of class C' satisfying

d (2.1) dtA ? (Ptt

dt t=O

(resp. 4'o f -\=7) b) For any periodic orbit yt = 4t(yo) of period T.

(2.2) f 71 Ot(yo) dt = 0

(resp. for any periodic point x of period N.

N-1

E 7q(fix) = o). j=O

If At (resp. f) is analytic, as well as the stable and unstable bundles and the function 'q, then 4 is analytic.

Remark. Theorem 2.1 naturally breaks up in two statements: One that there exists a C0 4, and second that a C0 4 which satisfies (2.1) with a C' RHS is of class Cx. The first part is the Livsic theorem and it is the only place where transitivity is needed; the second part holds for all Anosov systems regardless of transitivity (and the same happens in Cw when the foliations are analytic).

COROLLARY 2. 1. Assume kt is an Anosov flow (resp. f is an Anosov diffeomorphism) of class C' on a compact manifold M, and It is a positive continuous measure, invariant under (t (resp. f). Then pL is of class Cx. If 4t



(resp. f) is analytic, as well as the stable and unstable bundles, then IL is analytic.

Proof Assume IL = p dv; then

d(log p o ) if L) (d) A

dt 0 fL(d)'d .t=O

and the corollary is a trivial consequence of Theorem 2.1 in the case of a flow. In the case of a diffeomorphism it is simpler, since then

log p of - log p = det f'.

Notice that for this corollary we do not require transitivity of the system since only the regularity part of Theorem 2.1 is invoked.

THEOREM 2.2. Assume that for - ranging in a certain interval 4?t is a C' family of transitive Cx Anosov flows (resp. fT is a Cx family of transitive Cx Anosov diffeomorphisms) and that ? is a C' family of Cx functions satisfying the condition b) of the previous theorem for 4, t (resp. fE). If the additive constants of the 4? are determined in such a way that +?(x?) is a C' function of E for some C' curve x?, then 4, is a C' family of C' functions.

Remark. A careful reading of our proof will show that the operator assign- ing A to - is a tame operator satisfying

(2.3) II 1 II Ck < Kk(JX'q||Ck?v + ||4'IICO) where v is a number independent of the dynamical system (v is any number > 0) and k> 0.

It is now apparent that there are also versions with only a finite number of derivatives in the flows (or diffeomorphisms) and aq.

Remark. A C' (O < a < 1) version of Theorem 2.1 was proved by Livsic [Li 1]. Smooth versions like the one here were also proved in particular cases: Livsic [Li 2] proved it for automorphism of the torus (he also proved analytic versions) and V. Guillemin and J. Kazhdan [GK 1] did the same for geodesic flows in negatively curved surfaces. P. Collet, H. Epstein and G. Gallavotti [CEG] proved a Co version for geodesic flows of surfaces of constant negative curvature.

All the Coo, Co results quoted above were obtained through the use of some kind of generalized harmonic analysis and, hence, can only work for manifolds



with some symmetry. The argument we use is based on some geometric ideas and it is worthwhile to give it first for the case of automorphisms of the torus.

Proof of Theorem 2.1 for automorphisms of the torus. As remarked before, the Livsic theorem reduces our task to proving that 4 is C00 once we know it exists and is Co (of course, the other implication in the statement is trivial). Repeatedly using (2.1) we get, when fN(x) = y, A(x) = N- 1kN-q(fk(X)) ? A(y) and, when x is in the stable orbit of the origin, A(x) =- ' S oq(fk(x)) + A(0). The Ca Livsic theorem guarantees that we can extend the function continuously to the whole manifold even if the series is only defined on the stable manifold and does not converge uniformly.

If D, is a derivative along a stable eigendirection, we can differentiate formally to get

00

(2.4) Ds84(x) =k- E XsDsq(fk(x)) k=o

where Xs is the eigenvalue along the direction. (We assume for simplicity of notation that f is diagonalizable, but the argument would go through also when it is not. Instead of having Ds-q( fkx), we would have other derivatives, that would be the same from a certain k on and we would have to include a Ck" in front of Xk.)

Now observe that the RHS of (2.4) is a uniformly convergent series. Hence, it is a bona fide derivative for all the points in this stable manifold. We also observe that, rather unexpectedly, it also converges for all points in the torus.

For all x, y, in the stable manifold of the fixed point and separated by a vector in the direction of Ds we have

A(x) - A(y) = f'Ds8(x + ty) dt

where Ds4 stands for the RHS of (2.4). Since all the functions involved are continuous and the stable manifold of the fixed point is dense, this formula is true everywhere and, since Ds4 is continuous, is the derivative of 4.

Proceeding in the same way for f 1 we also can get existence of derivatives along unstable eigendirections.

Renark. The argument as it stands now is enough to get that 4 is C' whenever - is C' and it works without change for any Anosov diffeomorphism, not only diffeomorphisms of the torus.

In the CW case, the series (2.4) converges uniformly in a neighbourhood of x in the complex extension of the stable manifold associated to the torus, and this proves that 4 is real analytic along stable directions with uniform radius of convergence of its power series. Since it is also real analytic along the unstable



directions, a simple computation using its Fourier series shows the estimates for derivatives of 4 needed in order to prove its global analyticity on the torus.

In order to get higher derivatives we do the following: We observe that the same argument to justify formal differentiation would work for repeated derivatives along stable eigendirections (considering f-' we can also get the high order derivatives along expanding eigendirections). The problem now is to recover the mixed derivatives.

For automorphisms of the torus, there are simpler methods that would get by (see the proof of Theorem 2.3), but the following would generalize better: We can form [(-D'- -D )N+(-D21 - - Du),fDu and we have proved it is a continuous function. But we can consider this as the expression that an elliptic operator applied to 4 gives a continuous function. We can, then invoke standard elliptic operator theory to prove that 4 is in the Sobolev space HN. But N is arbitrary, so that 4 is C'. C1

The proof of Theorem 2.1 for C' functions in general proceeds with the same strategy, but, it is considerably more delicate.

A (not very important) difference is that we are going to prove the theorem only for flows; the standard suspension trick, performed with due care, shows that we can always reduce to that case.

The first obvious difficulty is that, when the diffeomorphisms are not linear, it is not clear how we can take derivatives of order higher than one. We will need to control all the extra terms coming from higher derivatives of f (this will be done by choosing a coordinate system well adapted to the problem), and to do that, we will need rather precise information about the smoothness of the stable and unstable foliations.

The second difficulty comes from the fact that the elliptic operators that we can form have very irregular coefficients and one cannot make sense of their value when acting on functions that are not regular, in any suitable generalized sense. We observe the possibility of using symmetric operators, that can be defined on L2. However, it is not clear that there are selfadjoint operators constructed out of derivatives along stable directions. The standard method of proving selfadjointness is integrating by parts, and this becomes problematic on general grounds for operators as irregular as ours. Nevertheless, we observe that the possibility of integrating by parts is related to the smoothness of the jacobian of translations along the field. Such a problem was considered by Anosov ([A], Thin. 10), where he proved it is continuous. By repeating his proof more carefully, we get the extra regularity we need.

Another proof of absolute continuity of the foliations can be found in [PS]. Even if our version here is based on the proof of Anosov, the same results can be reached improving the proof in [PS].



Now, we start to collect systematically all the information necessary for the strategy outlined before.

Reduction to flows. If f: M -* M is Anosov, constnuct the manifold M = I x M/ - where - is defined by (x, 0) - (y, 1) - y = f(x). Then the flow kt of generator a/at is Anosov.

Let p(t) be a C' function supported in [1/4,3/4] such that fJlp(t) dt = 1. A solution of (d/dt)4(4,)I,-o = p(t)'q(x) has the form 4(x, t) = A(x) +

x (x)f0op(s) ds where 4 is a solution of the homology equation for f. Hence, we can study the problem for flows only.

Notation. By definition of Anosov flow, there are a continuous splitting of the tangent bundle TM = Es ? Eu ? R4o (where 40(x) = (d/dt)4,(x)jIo) and a metric such that

(2.5) 14;v811 < ?XtIvSII, Vs E EsE t > 0,

IkkQ vUII < ?tIvuII vu E Eu t > 0

for some X < 1. We call Es = Es?R and Eu = EU ? Roo and we denote by Es, Eu,

Es, EU, the corresponding bundles of balls of radius 8 centered at the origin. Wxs (resp. Wxu) will be the stable (resp. unstable) manifold of the point x and Wxs = -tktWxs (resp. Wxu = UtktWxu) the stable (resp. unstable) manifold of the orbit of x. We will sometimes call the stable manifolds of the orbit of x the center stable manifold and will refer to WS as the center stable foliation, similarly, of course, for the unstable.

LEMMA 2.1. There exist 8 > 0 and a continuous map ws: Es -* M such that, for each x E M, Wx = Ws I Es is a C' embedding onto a neighbourhood of x in Wxs with ws(O) = x. Since the wxs have the Coo topology, they depend continuously on x. Moreover, the ws also depend continuously on ft when it is given the Coo topology of mappings M x R -* M.

Analogous statements can be made for the unstable, center stable and center unstable foliations. Q

This lemma can be summarized as saying that the four foliations have C' leaves with continuous oo-jet which depend continuously on the point and the flow.

Proof. Since i!5xs(vs + t4o(x)) = 4t(ws(vs)) (and analogously for the unstable) it suffices to prove the lemma for the stable manifold.



Except for the statements about dependence on x and ft this lemma is a consequence of Theorem 6.1 in [HPS].

The continuity of w' is immediate if one organizes the proof of this theorem as an application of the implicit function theorem. For diffeomorphisms, this is done in detail in [Sh], p. 94, and indeed, continuous dependence with k, is mentioned explicitly there. For flows, the case of interest here, such proof can be obtained from [I]. Combining the proof of Theorem 6.1 with the remarks at the end of Chapter 6, one obtains the statement of this lemma for what in [I] are called a-stable manifolds. However, the uniqueness statement of Theorem 6.1 in [HPS] shows that the a-stable manifolds are the stable manifolds.

Remark. The properties mentioned above do not determine the w uniquely; there is still some arbitrariness that can be used for example to obtain a nice expression for the adapted metric. A natural choice that we will use is to take wX as the exponential mapping of Wxs with the adapted metric. (This choice is used in the proof of [Sh] alluded to above.)

If we not only take coordinates along the stable direction but also on the base point we are naturally led to the following.

Definition 2.1. We say that AS: U X V -* M (U C Rs open, V C Ru+l open) is a local parametrization of the stable foliation when it is a homeomor- phism from U x V to an open set of M and for each y E V, As Y: U-* M given by AS Y(x) = A8s(x, y) is a C' immersion whose image is an open set of a leaf of the stable foliation and, moreover, for any a, daA/dxa is continuous on U x V. There is an analogous definition for the unstable foliation.

As a consequence of Lemma 2.1 those parametrizations exist around each point.

In view of the sketch of the proof of Theorem 2.1 we gave, the following definitions are natural.

Definition 2.2. We will say that a function A: M -* R is of class Cji, o < j < so when its restriction to each Wx5 is of class Ci and the j-jets of these restrictions depend continuously on x (analogously C1i). This is equivalent, to saying that h o ws are of class Ci for each x and that its i-th order derivatives (all i < j) define a continuous section of the bundle of i-linear forms on Es. When i < so, we give these spaces a natural norm topology analogous to that of Ci spaces (using the sups of the derivatives along the appropriate directions). For j = so there is also a natural topology analogous to C'.

Remark. If we take a local parametrization of the stable manifold, locally, it is the same to say that fE Cs(M) as to say fo As: U X V -* R has derivatives



of all orders with respect to the arguments in U and that they are continuous. We will denote such functions by C7?(U X V).

Definition 2.3. We denote by OJl the space of first order differential operators on M satisfying:

a) They are tangent to Ws at any x E M. b) They can be written in the form EaeDe where ae E C700 and De are

regular differential operators with Coo coefficients.

Remark. The generator of 4t is an operator in Os' l Oul, Ci c Cs, n Ot, C1 = Cs' n C,],.

Remark. The operators in O, map CQ+ into C] continuously.

Remark. By the previous remark, we can compose operators in Os' to obtain higher order operators. They are still differential operators with continuous coefficients. In this way we generate an algebra that we will denote by Os (similarly for Ou).

LEMMA 2.2. Assume 4 E Co and (d/dt)(l4 o ?t)t=0 = r E Csu. Then

Proof. Assume x is a periodic point of period T > 0 and y E Wxs. We have

(2.6) A(Y) = A(X)- lim fNI,(40t(y)) dt (N an integer).

The above sequence of integrals converges locally (in the Wxs topology, not in the M topology) uniformly.

Now, we can take formal derivatives along a vector field Ds tangent to the stable manifold,

00 (2.7) {'(y)Dys = q|'(Ot(y))Ot'(y)Dvs dt.

By (2.5) this is a uniformly convergent integral, and hence, a true derivative in the stable manifold of a periodic point. However, as we argued before, the formula for the increments can be extended by continuity and thus, (2.7) is an expression for a genuine derivative in the closure of the stable manifold of periodic points. But this should be the whole manifold because in [Sm] (Corollary II.5.3) it is shown that the stable manifolds of points in the non-wandering set is dense and also, the periodic points are dense in the non-wandering set.

To compute higher order stable derivatives of 4 we will need to control higher derivatives of the stable manifold and our task would be greatly simplified by an adequate choice of local coordinates, which is our next task.



For a given x in M we call x, = 4,(x) and set

(2.8) IDt (w 1 otWxw 4X ,t (WXt) ?ot?

Then, (Ds is a C' mapping from Es to Es If the ws are chosen as the exponentials of Wxs under the adapted metric,

we have

(2.9) |((Ds V o)| Ai xl s (= Es. t > 0, = (may hold for some other X < 1 than the one in (2.5)). Moreover,

(2.10) ?4?X(VX + It+O(X)) = (I )'(vs) + ?L)J(t) The group property of the flow induces the following property for (D.

(2.11) (D r = X(s,t ? (sr x E M; r, t E R.

In this new notation, where the coordinates are explicitly inserted, (2.7) becomes

(2.12) (4'o ix)'(i5x)(Dxs + Ifo(x))

= ix ) ? WX(Vx ) - |0 [( ?ix WX) Jit(x)] X(a(VX) Xd

=~~~~~~~~~~( ( [(4 s-)sX - ( ~t ~(Xl[ t)f(vs)] Dxs dt

where vs is the projection of -xs over the Es and x E M, t E R, Dxs E- Es V-s E Es are arbitrary. The remarkable property of this formula is that the only x x arguments of a function, the -xs, vs, belong to linear spaces and we can think of the xt as parameters. The higher order derivatives along the stable direction are obtained by taking derivatives with respect to v. We can reduce now to matrix-valued functions.

Since 7 0 ws is a C' function, for the previous argument to justify the higher derivatives, it will suffice that the derivatives of Is are bounded uniformly in x, by a function of t integrable in [0, x]. Using the semigroup property (2.11) repeatedly we obtain

X,t XN, t-N XN-1, 1 X, 1

(where N is the integer which satisfies N < t < N + 1). If we take derivatives, we get

(2.13) (kxt)' =(RXNt-N) XNI RXN,1) 0XN-1I [(?x,1).] (high derivatives of (ks 6t) are multilinear forms and their product means



contraction in the appropriate indices). In this case, by (2.9), it is easy to show that 11(k, )'ll < Xt-N.X X < t.

In general, we will proceed analogously: We will take derivatives (since the product of multilinear form is multilinear, the usual rules for the derivatives of a product hold); we will estimate all the first derivatives using (2.5) and the higher derivatives by their supremum. (Notice that, sup II(( D)(K 1) is finite).

xcM TE[O, 1]

We compute the n-th derivative in the following way: apply the rule for the product, the chain rule and the first derivatives generated by it, and substitute by (2.13). We arrive at an expression with A(N, n) terms each of which has at most B(N, n) factors.

The following properties are true: a) A(N, n + 1) < A(N, n)B(N, n), b) B(N, n + 1) < B(N, n) + (N + 1). c) The derivatives of higher than first order only involve T - e [0, 1]. d) In each factor, the 's of the first derivatives add up at least to

N - n - 1. a) is true because by the rule of the derivative each term splits into as many

terms as it has factors. b) is true because the only way that the number of factors in each term grows is by the new 1" appearing from the chain rule and its subsequent expression via (2.13). c) is true because, by use of (2.13), all the first derivatives on which to take higher derivatives only have time parameters less than 1. d) is true because each term contains at least N factors and only n of them can be derivatives of higher than the first order. But all the terms with first derivatives except one have time parameters greater than 1.

The recursion relation for A, B can be easily solved and the initial values can be read off from (2.13); so we have

((D, t)(n|? < (n + 1)!(1 + Cn)tn+lXt-n- where Cn= sup Jj(1Ix t)(eII.

T[O, 1] xcM e~n

From these bounds and the fact that we can also obtain uniform bounds for o ?w (notice that, by compactness, we can cover the whole manifold by a finite

set of coordinate patches), we obtain that the integrand in (2.7) is absolutely integrable. This concludes the proof of Lemma 2.2.

The proof of Theorem 2.1 in the COO case will be finished when we prove LEMMA 2.3. Cs, n C,?? = C?.

The proof of this lemma will be based on the fact that a function 4 in C,00 n CT satisfies linear differential equations CL 4 = q with CL in Os + Ou and

cE q + Cu' C C0. These equations are satisfied in the sense of the remarks



following Definition 2.2. The operators El can be chosen to be elliptic, but the coefficients will be only Holder continuous, and we will need a regularity theory for them. The differential operator El: H2m(M) -* H0(M) can be considered as a densely defined closed operator on H(M) that can be extended to a larger domain of definition by the remarks following Definition 2.2. The natural way to prove regularity of 4 in this situation is by proving that Hm is a maximal domain of definition for E1. We do not know how to prove this in general, but we can handle the case of self-adjoint operators, which turns out to suffice for our purposes. We are going to prove Lemma 2.3 when the foliations W' and W' come from an Anosov flow, but not in general. The following lemma will allow us to construct systematically self-adjoint elliptic operators. It says that the algebras OS and O, are closed under the operation of taking adjoints.

LEMMA 2.4. Fix any Coo measure dv equivalent to Lebesgue measure. Then, for any P E Os', there is an mqs E Cs7? such that

JM(Pf)gA = ff( tg) dv M ~M

for all f, g E Cs' where Ptf = - Pf + mqsf (and analogously for 021).

We will prove Lemma 2.3 assuming Lemma 2.4 and then show that Lemma 2.4 is true for Anosov systems.

Proof of Lemma 2.3. The lemma has a local character. Using C' cut-off functions and local systems of coordinates we can reduce it to a problem in Rn or the torus.

Let PI,..., Pa generate Es and Q1, ..., Qb generate E't in the neighbourhood of some point x0 E M. They can be taken in such a way that they define first order operators Pi EO' for 1 < i < a, and Qj E 02tl I., for 1 < j < b. We will use the same symbols for the differential operators they define in a neighbourhood of the origin in Rn via a system of coordinates.

Given any m > 0 we introduce the operator

a ~~b Ma El,= (- ? E (Pit) pim + L Qt Qm +E a (x) i~l j=l J II?2m dxa

where p is a number whose choice will be specified later, and the adjoints are taken with respect to a volume form that induces the measure dxl A ... A dxn in a neighbourhood of x0. By Lemma 2.4, ,1 is a differential operator of order 2m with continuous coefficients.



We also consider the operator a

z 0= Z a(?(O) Jal C(I + j R2)m, ( n

for some C > 0. As a first step, we are interested in choosing AL, . and 4 in such a way that

El defines an isomorphism between H2m(TU) and H0(T') = L2(Tn). This can be done as a consequence of the theory of elliptic equations (elliptic estimates), developed in the early 60's mainly as a tool for the study of nonlinear partial differential equations (see [ADN], Cor. to Thin. 12.8 for the case of Holder spaces and references given there). Those techniques allow us to show that any elliptic differential operator on a compact manifold with continuous coefficients has discrete spectrum. In the next paragraph we give a schematic proof of the facts just mentioned that is well suited to our situation. We do this since most expositions are developed in a different context, and the reader might find our exposition useful.

By (2.14), L0 is an isomorphism between H2U1 and Ho, whose inverse will be denoted by E0. Then

LiEo = I + 42(ol - 0O)EO + (4)[L1, 4)]Eo + A [L0, A]]Eo + [EO). If supp 4 is small enough, I + 42(Ei1 - DLO)EO is an automorphism on

L2(Tn), due to the continuity of the coefficients of L1. Since the commutators are operators of order (2m - 1) with continuous coefficients, the last term is a compact operator. Hence, LEo is a Fredholm operator.

We want to show that DLEO is injective on Ho, because then, applying the Fredholm alternative, it would be invertible and, hence El would be an isomor-



phism. We just have to prove that LI is injective. This can be seen as a consequence of Lemma 2.4 as follows: If u is a function in C? fn Cu? (in particular, if it is in Cx), we can repeatedly apply Lemma (2.4) to obtain:

f[(Pk)"Pm u] uidv = L A dv (and analogously for Qk). Hence we have

a b (2.15) Ref (Lu)idv ? Z I IPimouI2dv + E | A

Tn i=1 T j= n

+ ? f2U2 d + CZ ( + IkI 2) I1(4u)k 12, Tn k

where we denote by Vk the Fourier coefficients of the function v, v(x)= L ke 2iikx

By density of C' in H2n, this inequality is also true in H2n' and so, injectivity is also true.

Now, we want to show that, if u E Cs? fn C?? then u is the function v in H2m' that satisfies Llu = [lv (notice that we use the same symbols; the operators LI on both sides act on different spaces).

As before, we remark that it suffices to show this in a sufficiently small neighbourhood of an arbitrary point of M. So, we keep the notation of the argument before. Moreover, multiplying u by a C? function supported where

= 1, we can assume that u is a function on the torus. We can apply inequality (2.15) to the functions u - Zkexp(ik - k2/r)vk,

r > 0. We would like to apply (2.15) to u - v, but we do not know that v is in Cf n C?; so we have to regularize first. Letting r go to infinity, we obtain

a

f(Du - Lv)(U - V f) I I Piu - Pi "1kuI2 Tri T i= 1

b

+ Z IoQ7Mu - Q7mOv12 + Ifr2iU - V12 j=1

+cE(1 + Ik12) + (U - V))k 12 k

(Notice that, even if the two Li operators are understood in different senses, they allow the manipulations and the right-hand side is unambiguous.)

In particular, we can deduce that 42(u - v)2 = A2(U - v)2 = 0. Since 02 + p2 = 1 the result is established.

Remark. In the above proof we have used the self-adjointness of Li in order to prove its injectivity, and hence that it has nonempty resolvent. A proof



independent of self-adjointness can be given using the elliptic estimate (see [ADN]). The only point where we have used the self-adjointness of MI in an essential way is in considering its adjoint as an operator Lt: Ho -* H-2', and proving that this extension of Ol is also injective, so that H1 is the natural domain of definition of LI as an unbounded operator on L2.

Even if for present purposes our result is enough, from the point of view of P.D.E. theory it would be interesting to prove Lemma 2.3 without assuming self-adjointness, hence without using Lemma 2.4. The problem is closely connected to the characterization of maximal domains of definition of elliptic operators, as stated before.

Remark. Theorem 2.1 can be stated in terms of functions and flows of diffeomorphisms of class Ck. In that case, if -q is of class Ck, 4 is of class Ck-N for a suitable N that can be computed by checking carefully all the steps of the proof. In particular, when applying Lemma 2.3 one has to use Sobolev's embedding theorem, proving that C22m n Cu2m c H2m C C2m-N as far as N > n/2, n being the dimension of M. A better result can be obtained if one uses the LP theory, p > n, instead of p = 2 as before, so that Cs2m n Cu2m c Wp2m71c C 2m-. The only difference in this context is that in order to prove that L0 is an isomorphism between W2, and W? = LP, one has to use the theory of Calderon-Zygmund singular integral operators in its periodic version, namely the theorem of Marcinkiewicz and Zygmund (see [Z]).

Remark. Comparing the geometric proofs of the Livsic theorem [Li 1] or proofs using harmonic analysis, one is lead to very intriguing equivalences (see [CEG] for the case of SL(2, R)). For hyperbolic automorphisms of the torus one can prove the following:

Let 'q be a C function on the torus and f a hyperbolic automorphism. Then, the following are equivalent:

a) 1/N~I~j.N- 1 fk(X) = 0 for all x such that fN(x) = x, b) EkEZ% k1 = 0 for all 1 C Zd,

where B = ft and, as usual, ml denotes Fourier coefficients. It would be very interesting to give a direct proof of this-or the similar

equivalences in more complicated cases. If such a proof could lead to bounds of the left-hand side of b) in terms of those of a) when they are not zero, it would be a major step towards eliminating the use of parametric families in Theorem 2.1.

We prove now the real analytic version of Theorem 2.1. The proof follows essentially the same pattern of the Coo case, but there are important differences in the proof of each lemma, that we show now.



First of all, the reduction to the case of flows we made in the C' case is not valid in the CW case; so two different proofs have to be provided (one for maps and another for flows). We prove now the result for flows, and use the same notation of the Coo case.

Since M is real analytic, it can be extended to a (noncompact) complex analytic manifold M. By our hypothesis about the analyticity of the invariant subbundles, we can assume they are extended to stable and unstable holomorphic subbundles Es and Eu of TM, transversal at every point. Since Es and Eu are integrable, we see that Es and Eu are also integrable. So they define stable and unstable holomorphic foliations WS and Wu in M. The vector field X: M -> TM that generates the flow ot on M can be extended to an analytic vector field X: M -> TM with a flow kt(z) that is not going to be complete any more. However, Es and Eu are semi-invariant under the flow 4 in the sense that ft*EPs(t) ~,= E8' U if t is in the domain of definition of 4 at z.

The usual adapted metric can be taken to be analytic, and by shrinking M we can assume that it extends to a metric in M, and estimate (2.5) can be extended to M assuming all the terms are well-defined. In particular, M is a compact hyperbolic subset of M. We shrink M further, substituting for it by

Ma = {z E M/d(z, M) < 8}, with s small enough. Then, for any x E M, W~n is actually the stable manifold of 4 through x, in the sen

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