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IMS and Department of Mathematics, SUNY at Stony Brook,

arXiv:math/9210228v1 [math.DS] 10 Oct 1992OPTICAL HAMILTONIANS ANDSYMPLECTIC TWIST MAPSCHRISTOPHE GOL´EIMS and Department of Mathematics, SUNY at Stony Brook,Stony Brook NY 11794-36511Abstract: This paper concentrates on optical Hamiltonian systems of T ∗Tn,i.e. those for which Hpp is a positive definite matrix, and their relationship withsymplectic twist maps.

We present theorems of decomposition by symplectic twistmaps and existence of periodic orbits for these systems. The novelty of these resultsresides in the fact that no explicit asymptotic condition is imposed on the system.We also present a theorem of suspension by Hamiltonian systems for the class ofsymplectic twist map that emerges in our study.

Finally, we extend our results tomanifolds of negative curvature.1. IntroductionIn a previous paper [G91b], the author explained how symplectic twist mapscould be used to decompose Hamiltonian systems on the cotangent bundleof a compact manifold M n, thus deriving a discrete variational approach tothe search of periodic orbits for such systems.

This method can be seen as ageneralization of the so called “method of broken geodesics” in differentialgeometry. A similar method was introduced by Marc Chaperon for Hamil-tonian systems [Ch84].

Although our method is very similar to his, it is infact even more akin to the original method (see e.g. [Mi69].

)In [G91b], we put a boundary condition on the Hamiltonian, however: ithad to equal the metric Hamiltonian H0(q, p) = 12 ∥p∥2 on a fixed level set{H0 = K}. It could be anything inside {H0 < K}, including time depen-dent.

The result (see [G91b,92b]) was then the existence of at least cl(M) (orsb(M) if all nondegenerate) contractible periodic orbits inside {H0 < K}for such systems. Other results for non contractible orbits were obtained1 Partially supported by an NSF Postdoctoral Fellowship DMS 91-07950.1

if M supports a metric with negative curvature, or for Tn (comparable toTheorem 7.4 in this paper. )Here we swap the boundary condition (and the compactness of {H0 ≤K}) for a convexity condition which gives a Hamiltonian system with apriori no compact invariant set: the systems we study here are optical, inthe sense that the second derivative in the fiber direction, Hpp is positivedefinite .The main result of this paper is:Theorem 5.2 Let H(q, p, t) = Ht(z) be a twice differentiable function onT ∗Tn × R satisfying the following:(1) sup∇2Ht < K(2) The matrices Hpp(z, t) are positive definite and C < ∥Hpp∥< C−1 forsome C.Then the time 1 map of the associated Hamiltonian flow has at leastn + 1 distinct periodic orbits of type m, d, for each prime m, d ∈Zn × Z,and at least 2n in the generic case when they are all non degenerate.

(An m, d–orbit is one for which the dth iterate of each point of the orbitis a translation by (m, 0) of this point, in the covering space R2n of T ∗Tn. )Earlier results on the existence and multiplicity of periodic orbits can befound for Hamiltonian systems or symplectic twist maps of T ∗Tn in [BK87],[Che92], [CZ83],[Fe89], [G91a], [J91].

The two first are perturbative results,i.e. for systems close to integrable ones.

The four latter works are global inthat sense, and do not require the system to be optical, but instead requiresome asymptotic condition on the first derivative of H (or of the time 1map F). Only the two last works consider homotopically nontrivial orbits.Note also that, via the Legendre transformation, Theorem 5.2 appliesto Lagrangian systems whose Lagrangian function satisfies the same con-ditions as H in our theorem (it is not hard to see that these conditionstranslate under the Legendre transformation.) Hence Theorem 5.2 extendssome existing theorems for such systems (see, e.g., [MW89], Theorem 9.3.

)We start (Sections 2 and 3) with some background on symplectic twistmaps. In Section 4, we give the proof of a theorem of existence and multi-plicity of periodic orbits for compositions of symplectic twist maps with aconvexity condition (Theorem 4.3.) A proof of such a theorem was givenin [KM89].

Unfortunately, the multiplicity part of their proof is wrong. Wereproduce here their proof of existence of a minimum, and present a newproof of the multiplicity.In Section 5, we prove Theorem 5.2.

This results derives from the formertheorem on symplectic twist maps, and a decomposition technique. Theresulting discrete variational method is interpreted as a method of brokengeodesics.2

In Section 6 we show that a symplectic twist map with the convexitycondition can be suspended by a Hamiltonian (our proof does not forcethe Hamiltonian to be optical, unfortunately ), extending a result of Moser[Mo86] and a remark about it by Bialy and Polterovitch [BP92].In section 7, we indicate how to extend Theorems 4.3 and 5.2 to thecotangent bundle of a manifold of negative curvature.It is very likely that, with a little care, these techniques could extend tooptical Hamiltonians on the cotangent bundle of any compact manifolds.The author would like to thank J.D. Meiss and F. Tangerman for use-ful conversations, and D. McDuffand J.D.

Meiss for the corrections theysuggested.2. Symplectic Twist Maps of Tn × RnLet Tn = Rn/Zn be the n–dimensional torus.

Its cotangent bundle T ∗Tnπ→Tn is trivial: T ∗Tn = Tn × Rn, the cartesian product of n cylinders. Wegive it the coordinates (q, p) in which the symplectic structure isΩ= dq ∧dp =nXk=1dqk ∧dpk.As in any cotangent bundle, Ωis exact: Ω= −dλ, where λ = pdq.It is useful to work in the covering space R2n = ˜Tn × Rn of T ∗Tn, withprojection pr : R2n →Tn × Rn.Of course, pr is an exact symplectic map (see Definition 2.1) , as wehave pr∗pdq −pdq = 0.The group Zn of deck or covering transformations is the set of integervector translation in R2n of the form:τm.

(q, p) = (q + m, p),m ∈Zn.A lift of a map F : T ∗Tn →T ∗Tn is a map ˜F : R2n →R2n such that pr◦˜F =F ◦pr. Since pr is a local, symplectic diffeomorphism, ˜F is symplectic if andonly if F is.

On the other hand, ˜F will always be exact symplectic when it issymplectic, which is not the case for F, as the example (q, p) →(q, p + p0)shows.We will fix the lift of a map F once and for all, remembering that twolifts only differ by a composition by some τm.Definition 2.1A map F of T ∗Tn is called a symplectic twist map if(1) F is homotopic to Id. (2) F is exact symplectic: F ∗pdq −pdq = dh for some h : T ∗Tn →R.

(3) (Twist Condition) If ˜F(q, p) = (Q, P ) is a lift of F then the mapp →Q(q0, p) is a diffeomorphism of Rn for all q0, and thus the map3

ψ : (q, p) →(q, Q)is a diffeomorphism (change of coordinates) of R2n.Comments 2.21 .The twist condition (3) implies the more familiar looking:det ∂Q/∂p ̸= 0.It also implies that :(2.1)˜F ∗pdq −pdq = dS(q, Q)where S is the lift of h written in the (q, Q) coordinates : S = ˜h ◦ψ−1,with ˜h = h ◦pr. Equivalently, we can write:p = −∂1S(q, Q)P =∂2S(q, Q).S(q, Q) is called a generating function for ˜F.2 .

Condition (1) is equivalent to the fact that on any lift ˜F of F:(2.2)˜F ◦τm = τm ◦˜F, i.e.˜F(q + m, p) = ˜F(q, p) + (m, 0).Example 2.3The family of mapsFs(q, p) = (q + A (p −∇Vs(q)) , p −∇Vs(q))where A is a nondegenerate symmetric matrix, Vs is a C2 function on Tn iscalled the standard family. Usually, V0 ≡0.

The generating function forFs is given by:Ss(q, Q) = S0(q, Q) + Vs(q).WhereS0(q, Q) = 12⟨A−1(Q −q), (Q −q)⟩is the generating function of the completely integrable map:F0 : (q, p) →(q + Ap, p),At = A, det A ̸= 0. ( The term “completely integrable” comes from the fact that F0 conserveseach torus p = p0, on which it acts as a rigid “translation”.

)This general standard family includes the classical standard family ofmonotone twist maps of the annulus where A = 1 andVs(q) =s4π2 cos(2πq)and also the Froeschl´e family on T2 × R2 with A = Id and4

Vs(q1, q2) =1(2π)2 {K1cos(2πq1) + K2cos(2πq2) + λcos2π(q1 + q2)}.In this case the parameter s = (K1, K2, λ) ∈R3. (See e.g.

[KM89].) Thestandard map can be interpolated by an optical Hamiltonian (see Section7).As this paper will suggest, many examples of symplectic twist mapscan be derived from Hamiltonian systems, and used to understand thesesystems.The following results are also helpful to construct symplectic twist maps.Their proofs can be found in [G93] (see also [H89] for Corollary 2.7.

)Proposition 2.4There is a homeomorphism between the set of lifts ˜F ofC1 symplectic twist maps of T ∗Tn and the set of C2 real valued functionsS on R2n satisfying the following:(a) S(q + m, Q + m) = S(q, Q),∀m ∈Zn(b) The maps: q →∂2S(q, Q0) and Q →∂1S(q0, Q) are diffeomorphismsof Rn for any Q0 and q0 respectively. (c) S(0, 0) = 0.This correspondence is given by:(2.3)˜F(q, p) = (Q, P ) ⇔p =−∂1S(q, Q)P =∂2S(q, Q).Lemma 2.5Let f : RN →RN be a local diffeomorphism at each point,such that:supx∈RN(Dfx)−1 = K < ∞.Then f is a diffeomorphism of RN.Corollary 2.6Let S :R2n →R be a C2 function satisfying:(2.4)det ∂12S ̸= 0sup(q,Q)∈R2n(∂12S(q, Q))−1 = K < ∞.Then the maps: q →∂2S(q, Q0) and Q →∂1S(q0, Q) are diffeomorphismsof Rn for any Q0 and q0 respectively, and thus S generates an exact sym-plectic map of R2n.Thus , if S satisfies (2.4) , as well as the periodicity condition (a) ofProposition 2.4 , it generates a symplectic twist map.Corollary 2.7Let F be an exact symplectic map of T ∗Tn, homotopic toId.

Let ˜F(q, p) = (Q, P ) be a lift of F. Suppose that5

(2.5)supz∈R2n∂Q∂p−1z < ∞.Then ˜F is a symplectic twist map.3. The Variational SettingAs in the classical case of twist map (n = 1), the generating function of asymplectic twist map is the key to the variational setting that these mapsinduce.Proposition 3.1 (Critical Action Principle) Let F1, .

. ., FN be symplectictwist maps of T ∗Tn, and let ˜Fk be a lift of Fk, with generating functionSk.

The sequence {(qk, pk)}k∈Z is an orbit under the successive ˜Fk’s (i.e. {(qk+1, pk+1) = ˜Fk(qk, pk)}k∈Z , with ˜Fk+N = ˜Fk, Sk+N = Sk) if and onlyif the sequence {qk}k∈Z in (Rn)Z satisfies:(3.1)∂1Sk(qk, qk+1) + ∂2Sk−1(qk−1, qk) = 0,∀k ∈Z.The correspondence is given by: pk = −∂1Sk(qk, qk+1).Equation (3.1) can be interpreted formally as:∇W(q) = 0withW(q) =∞X−∞Sk(qk, qk+1).This interpretation makes mathematical sense when one is concernedwith periodic orbits of a symplectic twist map F:Definition 3.2A point (q, p) ∈R2n is called a m, d–point for the lift ˜Fof F if ˜F d(q, p) = (q + m, p), where m ∈Zn and d ∈Z.Let ˜F = ˜FN ◦.

. .

◦˜F1. The appropriate space of sequences in which tolook for critical points corresponding to m, d–points of ˜F is:X∗= {q ∈(Rn)Z | qk+dN = qk + m}which is isomorphic to (Rn)dN: the terms (q1, .

. .

, qdN) determine a wholesequence in X∗.To find a sequence satisfying (3.1) in X∗is equivalent to finding q =(q1, . .

., qdN) which is a critical point for the function:6

W(q) =dNXk=1Sk(qk, qk+1),in which we set qdN+1 = q1 + m. To see this, write :pk = −∂1Sk(qk, qk+1),P k = ∂2Sk(qk, qk+1).Then Fk(qk, pk) = (Qk, pk) and with this notation, the proof of Proposition3.1 (for m, d–points) reduces to the suggestive:∇W(q) =dNXk=1(P k−1 −pk)dqk.A little more care must be taken in order to let the topology of Tn play arole. Note that because of the periodicity of S ((a) in Proposition 2.4), Wis invariant under the Zn action on X∗:τm(q1, .

. .

, qdN) = (q1 + m, . .

., qdN + m).Moreover, if we want our variational approach to count m, d–orbits, andnot the individual m, d–points in each orbit, we should use the fact that Wis also invariant under the N–shift map:σ{qk} = {qk+N}.Let :X = X∗/σ, τbe the quotient of X∗by these two actions. We continue to call W thefunction induced by W on the quotient X.One can show ([BK87], Proposition 1 or [G91a]) that X is the totalspace of a fiber bundle over Tn, and that the projection map X∗→X is acovering map.

(One makes the change of variables:v = 1ddNX1qktk = qk −qk−1 −m/din which v is the base coordinate, t the fiber.) In particular, each criticalpoint of W on X corresponds to an infinite lattice of critical points of W onX∗.

Whereas the original variational problem ∇W = 0 on X∗would pickup the (infinitely many) m, d–points of the lift ˜F of F, when we restrict itto X it exactly gives m, d–orbits of F.7

4. Periodic Orbits and the Convexity ConditionLet ˜F(q, p) = (Q, P ) be the lift of a symplectic twist map of T ∗Tn, andS(q, Q) its generating function.

In this section, we impose the:4.1 Convexity Condition There is a positive a such that:⟨∂12S(q, Q).v, v⟩≤−a ∥v∥2 .uniformly in (q, Q).Remark 4.2 . Note that:∂Q∂p (q, p) = −(∂12S(q, Q))−1 ,as can easily be derived by implicit differentiation of p = −∂1S(q, Q).

Theconvexity condition 4.1 thus translates to:(4.1)*∂Q∂p−1v, v+≥a ∥v∥2 ,∀v ∈Rn.uniformly in (q, p). This means that F has bounded twist.

MacKay, Meissand Stark [MMS89] imposed this condition on their definition of symplectictwist maps, a terminology that we have taken from them.Theorem 4.3Let F = FN ◦. .

. ◦F1 be a finite composition of symplectictwist maps Fk of T ∗Tn each satisfying the convexity condition.

Then, foreach prime (m, d) ∈Zn × Z, F has at least n + 1 distinct periodic orbits oftype m, d. It has at least 2n of them when they are all nondegenerate.By a prime pair m, d we mean that at least one of the components mkof m is prime with d.Remark 4.4 . One can show ([G91a]) that an m, d–point of F is nondegen-erate if and only if the sequence q of X∗it corresponds to is a nondegen-erate critical point for W. The hypothesis that F has only nondegeneratem, d–points is thus equivalent to the one that W is a Morse function.

Fur-thermore, this is a generic condition on the space of symplectic twist maps[G92a]. Note also that an m, d point is also an km, kd–point for all k ∈N.The reason for restricting ourselves to prime m, d is that if we were to lookfor km, kd orbits, we would also find the prescribed number of them, butwith no guarantee that they would be any different from the m, d orbitsalready found.8

P roof.The first part of the proof, due to Kook and Meiss, [KM89] con-sists in proving that the function W is proper, and hence has a minimum.The following lemma and corollary were proven in [MMS89], and [KM89].Lemma 4.5Let S be the generating function of a symplectic twist mapsatisfying the convexity condition 4.1 . Then there is an α and positive βand γ such that:(4.2)S(q, Q) ≥α −β ∥q −Q∥+ γ ∥q −Q∥2 .P roof.We can write:S(q, Q) = S(q, q) +Z 10∂2S(q, Qs).

(Q −q)ds,where Qs = (1 −s)q + sQ. Applying the same process to ∂2S, we get:S(q, Q) = S(q, q) +Z 10∂2S(Qs, Qs).

(Q −q)ds−Z 10dsZ 10⟨∂12S(Qr, Qs). (Q −q), (Q −q)⟩dr≥α −β ∥Q −q∥+ γ ∥Q −q∥2 ,where α = minT n S(q, q), β = maxT n ∥∂2S(q, q)∥and γ = a2.⊓⊔Corollary 4.6For F as in Theorem 4.4 , there is a minimum for W (andhence an m, d–point for F.)P roof.Equation (4.2) as applied to each Sk implies that Sk has a lowerbound, thus W does as well.

We have to prove that this lower bound is notattained at infinity, i.e., that W is a proper map.The set {(q, Q) ∈(Rn × Rn)/Zn | S(q, Q) ≤C} is compact since (4.2)implies that S ≤C corresponds to bounded ∥q −Q∥. Likewise the setS = {q ∈X | W(q) ≤C}is compact.

Hence W must have a minimum in the interior of S, for C bigenough. This point is a critical point.⊓⊔Remark 4.7 .

We have thus found at least one m, d–orbit corresponding toa minimum of W. The reader should be aware that, unlike the 1 degree offreedom case, this does not imply that the orbit is a minimum in the senseof Aubry (see [H89]. )We now turn to the proof of existence of at least n + 1 distinct orbits oftype m, d, and 2n when they are all nondegenerate.9

Remember that X is a bundle over Tn . Let Σ ∼= Tn be its zero section.Let K = supΣ W(q) .

Trivially, we have:Σ ⊂W K def= {q ∈X | W ≤K}( since W is proper, for almost every K, W K is a compact manifold withboundary, by Sard’s Theorem.) From this we get the commutative diagram:(4.3)H∗(Σ)k∗−−−−−−→H∗(X)i∗ցր j∗H∗(W K)where i, j, k are all inclusion maps.

But k∗= Id since Σ and X have thesame homotopy type. Hence i∗must be injective.If all the m, d–points are nondegenerate, W is a Morse function (ageneric situation ) and by [Mi69], §3, W K has the homotopy type of afinite CW complex, with one cell of dimension k for each critical point ofindex k in W K. In particular, we have the following Morse inequalities:#{critical points of index k} ≥bkwhere bk is the kth Betti number of W K, bk >nkin our case sinceH∗(Tn) ֒→H∗(W K).

Hence there are at least 2n critical points in thisnondegenerate case.If W is not a Morse function, rewrite the diagram (4.3) , but in Co-homology, reversing the arrows. Since k∗= Id, j∗must be injective thistime.

We know that the cup length cl(X) = cl(Tn) = n + 1. This exactlymeans that there are n cohomology classes α1, .

. ., αn in H1(X) such thatα1 ∪.

. .

∪αn ̸= 0. Since j∗is injective, j∗α1 ∪.

. .

∪j∗αn ̸= 0 and thuscl(W K) ≥n + 1. W K being compact, and invariant under the gradientflow, Lusternik-Schnirelman theory implies that W has at least n + 1 criti-cal points in W K (The proof of Theor`eme 1 in CH.2 §19 of [DNF87], whichis for compact manifolds without boundaries can easily be adapted to thiscase.)⊓⊔5.

Periodic Orbits for Optical Hamiltonian SystemsAssumption 5.1H(q, p, t) = Ht(z) is a twice differentiable function onT ∗Tn × R (or T ∗M × R, where ˜M = Rn) and satisfies the following:(1) sup∇2Ht < K(2) The matrices Hpp(z, t) are positive definite and C < ∥Hpp∥< C−1.Theorem 5.2 Let H(q, p, t) be a Hamiltonian function on T ∗Tn×R satisfy-ing Assumption 5.1 . Then the time 1 map h1 of the associated Hamiltonian10

flow has at least n + 1 distinct periodic orbits of type m, d, for each primem, d, and 2n in the generic case when they are all non degenerate.P roof.we can decompose the time 1 map:h1 = hNN ◦(hN−1N )−1 ◦. .

. ◦hkN ◦(hk−1N )−1 ◦.

. .

◦h1N ◦Id.and each of the maps hkN ◦(hk−1N )−1 is the time1N of the (extended) flow,starting at time k−1N , or in other words, the time 1/N of the HamiltonianKt = Ht+ k−1N . Proposition 5.4 shows that, for N big enough, such maps aresymplectic twist and satisfy the convexity condition 4.1 .

The result followsfrom Theorem 4.3.⊓⊔Remark 5.3 . Remember that Hamiltonian maps on cotangent bundles areexact symplectic.

More precisely, the time t map ht of a Hamiltonian systemon T ∗M satisfies:(5.1)(ht)∗pdq −pdq = dStwhereSt(q, p) =Z ht(q,p)(q,p)pdq −Hds,and the path of integration is the trajectory (hs(q, p), s) of the (extended)flow. Obviously ht is isotopic to Id.

The twist condition is what remainsto be checked – it is clearly not always satisfied. The following propositionshows that it is, for small t, under Assumption 5.1 .Before that, let us remark that the method of proof that we are using inthis section is analogous to the so called method of broken geodesics [Mi69]:by (5.1) , the function W that we appeal to above in our use of Theorem4.3 can be interpreted as:W(q) =XkZγkpdq −Hdswhere γk is the orbit of ht starting from (qk, pk) at timekN , and ending at(qk+1, P k) at time k+1N .

The broken curve whose pieces are the γk projects,via the diffeomorphisms ψk (see definition 2.1) to a continuous, but onlypiecewise differentiable curve of Tn. In the case where H is the Hamiltoniancorresponding to a metric, this curve is a piecewise, or broken geodesic andψk is the exponential map.

Proposition 3.1 can then be interpreted as sayingthat, among broken geodesics, the smooth ones are exactly the ones thatare critical for W (See [G93] for more details. )The following applies without change to Hamiltonians in cotangent bun-dles of Riemannian manifolds of negative curvature.

It is, however, the pointat which our method breaks for the cotangent of arbitrary manifolds: sym-plectic twist maps cannot be defined on all of T ∗S2, for instance.11

Proposition 5.4Let hǫ be the time ǫ of a Hamiltonian flow for a Hamilto-nian function satisfying Assumption 5.1 . Then, for all sufficiently small ǫ,hǫ is a symplectic twist map of T ∗Tn.

Moreover, hǫ satisfies the convexitycondition 4.1 .P roof.We can work in the covering space R2n of T ∗Tn, to which the flow lifts.The differential of ht at a point z = (q, p) is solution of the linear (variation)equation:(5.2)˙U(t) = J∇2H(ht(z))U(t),U(0) = Id,J =0−IdId0We first need a lemma that tells us that U(ǫ) is not too far from Id:Lemma 5.5Consider the linear equation:˙U(t) = A(t)U(t),U(t0) = U0where ∥A(t)∥< K, ∀t. Then :∥U(t) −U0∥< K ∥U0∥|t −t0|eK|t−t0|.P roof.Let V (t) = U(t) −U(t0), so that V (t0) = 0.

We have:˙V (t) = A(t) (U(t) −U0) + A(t)U0= A(t)V (t) + A(t)U0and hence:∥V (t)∥= ∥V (t) −V (0)∥≤Z tt0K ∥V (s)∥ds + |t −t0|K ∥U0∥For all |t −t0| ≤ǫ, we can apply Gronwall’s inequality to get:∥V (t)∥≤ǫK ∥U0∥eK|t−t0|and we get the result by setting ǫ = |t −t0|.⊓⊔We now finish the proof of Proposition 5.4 . By Lemma 5.5we canwrite:U(ǫ) −Id =Z ǫ0J∇2H(hs(z)).

(Id + O1(s))dswhere ∥O1(s)∥< 2Ks, for s ≤ǫ small enough.Let (q(t), p(t)) = ht(q, p) = ht(z). The matrix bǫ(z) = ∂q(ǫ)/∂p, is theupper right n × n matrix of U(ǫ).

It is given by:12

(5.3)bǫ(z) =Z ǫ0Hpp(hs(z))ds +Z ǫ0O2(s)dswhereR ǫ0 O2(s)ds < Kǫ2. From this, and the fact thatC ∥v∥2 < ⟨Hpp(z)v, v⟩< C−1 ∥v∥2 ,we deduce that:(5.4)(ǫC −Kǫ2) ∥v∥2 < ⟨bǫ(z)v, v⟩< (ǫC−1 + Kǫ2) ∥v∥2so that in particular bǫ(z) is nondegenerate for small enough ǫ.

The set ofnonsingular matrices {bǫ(z)}z∈R2n is included in a compact set and thus:(5.5)supz∈R2nb−1ǫ (z) < K′,for some positive K′. We can now apply Corollary 2.7 to show that hǫ is asymplectic twist map with a generating function S defined on all of R2n.Likewise, from (5.3) , and the fact that ⟨H−1pp (z)v, v⟩> C∥v∥2, oneeasily derives that hǫ satisfies the convexity condition 4.1.⊓⊔6.

Suspension of Symplectic Twist Maps byHamiltonian FlowsIn [Mo86], Moser showed how to suspend a monotone twist map of the com-pact annulus into a time 1 map of a (time dependent) optical Hamiltoniansystem. Furthermore, he was careful to construct the Hamiltonian in such away that its flow leaves invariant the compact annulus (when the map does)and also such that it is time periodic.As announced by Bialy and Polterovitch [BP92], Moser’s method can beadapted to suspend a symplectic twist map whose generating functions S issuch that ∂12S(q, Q) is a positive definite symmetric matrix satisfying theconvexity condition 4.1.

In particular their result shows that the StandardMap is the time 1 of an optical Hamiltonian flow, periodic in time.Here we present a suspension theorem for higher dimensional symplectictwist maps, without the assumption that ∂12S is symmetric. Our result ismodest in that we do not obtain a convexity condition on the Hamiltonian,or show that the Hamiltonian we construct can be made time periodic.

Ourmethod is different from Moser’s.Theorem 6.1Let F(q, p) = (Q, P ) be a symplectic twist map of T ∗Tnwhich satisfies the convexity condition 4.1. Then F is the time 1 map of a(time dependent) Hamiltonian H.13

P roof.Let S(q, Q) be the generating function of F. Condition 4.1 canbe rewritten:(6.1)inf(q,Q)∈R2n⟨−∂12S(q, Q)v, v⟩> a ∥v∥2 ,a > 0, ∀v ̸= 0 ∈Rn.The following lemma, whose proof is left to the reader shows that thisinequality implies (2.4). Hence whenever we have a function on R2n whichis suitably periodic and satisfies (6.1) , it is the generating function for somesymplectic twist map.Lemma 6.2Let {Ax}x∈Λ be a family of n × n real matrices satisfying:supx∈Λ⟨Axv, v⟩> a∥v∥2,∀v ̸= 0 ∈Rn.Then :supx∈ΛA−1x < a−1.We construct a differentiable family St of generating functions, withS1 = S, and then show how to make a Hamiltonian vector field out of it,whose time 1 map is F. LetSt(q, Q) =(12af(t)∥Q −q∥2for 0 < t ≤1212af(t)∥Q −q∥2 + (1 −f(t))S(q, Q)for 12 ≤t ≤1.where f is a smooth positive functions, f(1) = f ′(1/2) = 0, f(1/2) =1, limt→0+ f(t) = +∞.

We will ask also that 1/f(t), which can be continuedto 1/f(0) = 0 be differentiable at 0. The choice of f has been made so thatSt is differentiable with respect to t, for t ∈(0, 1].

Furthermore, it is easyto verify that:sup(q,Q)∈R2n⟨−∂12St(q, Q)v, v⟩> a ∥v∥2 ,a > 0, ∀v ̸= 0 ∈Rn, t ∈(0, 1].Hence St generates a smooth family Ft, t ∈(0, 1] of symplectic twist maps,and in fact Ft(q, p) = (q + (af(t))−1p, p),t ≤1/2), so that limt→0+ Ft =Id, in any topology that one desires (on compact sets.) Let us write˜St(q, p) = St ◦ψt(q, p),where ψt is the change of coordinates given by the fact that Ft is twist.

Itis not hard to verify that ψt(q, p) = (q, q −(af(t))−1p),t ≤1/2. so that:˜St(q, p) = 12(af(t))−2∥p∥214

In particular, by our assumption on 1/f(t), ˜St can be differentiably contin-ued for all t ∈[0, 1], with S0 ≡0. Hence, in the q, p coordinates, we canwrite:F ∗t pdq −pdq = d ˜St,t ∈[0, 1].A familly of maps that satisfies this with ˜St differentiable in (q, p, t) iscalled an exact symplectic isotopy.

The proof of the theorem derives fromthe standard:Lemma 6.3Let gt be an exact symplectic isotopy of T ∗Tn (or T ∗M, ingeneral.) Then gt is a Hamiltonian isotopy.P roof.Let gt be an exact symplectic isotopy:g∗t pdq −pdq = dStfor some St differentiable in all of (q, p, t).

We claim that the (time depen-dent ) vector field:Xt(z) = dgtdt (g−1t (z))whose time t is gt, is Hamiltonian. To see this, we compute:ddt(d ˜St) = ddtg∗t pdq = g∗t LXtpdq = g∗t (iXtd(pdq) −d(iXtpdq)) ,from which we getiXtdq ∧dp = dHtwithHt =(g−1t)∗dStdt −iXtpdq,which exactly means that Xt is Hamiltonian.⊓⊔7.2 Cotangent Bundle of Manifolds with NegativeCurvatureWe indicate in this section how some of the previous results can be obtainedin the cotangent bundle T ∗M of a compact manifold M which supports ametric of negative curvature.

Such a manifold is always covered by Rn ( Asbefore we denote by pr : ˜M(= Rn) →M the covering map.) The definitionof symplectic twist map carries through verbatim for the cotangent bundleof such manifolds, as well as Propositions 2.4 and 3.1, Corollaries 2.6 and2.7.

The action by translations of π(Tn) = Zn on R2n is replaced by themore general action of π1(M), the deck transformation group of T ∗˜M. Note15

also that the convexity condition 4.1 still makes sense in this more generalcontext. For more details, see [G91b], [G93].The first resistance we encounter to an extension of our results to suchmanifolds is Definition 3.2 of m, d–orbits.

The clue to define such an orbit inthis new context is Remark 5.4: we saw there that, in the case where the mapF considered is Hamiltonian and decomposed into symplectic twist maps,an m, d–sequence gives rise to a closed, piecewise smooth curve in Tn (a“broken geodesic”). The integer vector m classifies these broken geodesicsup to homotopy with or without fixed base points.

This is because the groupπ1(Tn) = Zn is abelian.In general manifolds, two loops through a base point that representdifferent elements in π1(M) might be homotopic if we allow the homotopyto move the base point: we say then that the curves are free homotopic.Free homotopy classes are in one to one correspondence with the conjugacyclasses in π1(M).Coming back to our broken geodesics, the natural classification for pe-riodic orbits of a Hamiltonian system is that of free homotopy class: eachof these classes represent a connected component in the loop space. Thismotivates:Definition 7.3Let m be a representative of a free homotopy class of loopsin M. A sequence {qk} of points in ˜M = Rn is called a m, d–sequence if, forall k ∈Z, pr(qk) = pr(qk+d) and (any) curve ˜γ of ˜M that joins qk and qk+d,projects to a closed curve of M in the free homotopy class m, independentof k. The orbit {(qk, pk)} of a map of T ∗M is an m, d–periodic orbit ifthe sequence {qk} is an m, d–sequence.We can now state:Theorem 7.4Let M be a compact Riemannian manifold with negativecurvature.

Let F = FN ◦. .

. ◦F1 be a finite composition of symplectic twistmaps Fk of T ∗M satisfying the convexity condition 4.1.

Then, for each freehomotopy class m and period d, F has at least 2 periodic orbits of typem, d. If m = 0, the class of contractible loops, then there are at least cl(M)orbits of type m, d, and sb(M) if they are all nondegenerate.P roof.It is shown in [G91b], Lemma 7.2.2, that the set X of m, dNsequences (modulo π1(M) and shift; X is denoted Om,d/σ in that paper),has a deformation retraction onto the set, that we call Σ, formed by theunique geodesic of class m (remember that M has negative curvature) thatis, X has the homotopy type of S1. The proof of Theorem 4.3 can now berepeated, keeping in mind that sb(S1) = cl(S1) = 2.When m is the trivial class, the set X retracts on the set of constantloops, naturally embedded in it ([G91b], Lemma 6.2).

This set, that we call16

Σ again is homeomorphic to M. A simple adaptation of Lemma 7.2.2 in[G91b] shows that Σ is in fact a deformation retract of X and hence onceagain, we can repeat the proof of Theorem 4.3.⊓⊔Assumption 5.1 and Proposition 5.4 apply without a change to our newcontext and hence we have:Theorem 7.5Let M be a compact Riemannian manifold with negativecurvature. Let H(q, p, t) be a Hamiltonian function on T ∗M satisfying As-sumption 5.1 .

Then the time 1 map of the associated Hamiltonian flow canbe decomposed into a product of symplectic twist maps. It has at least 2 peri-odic orbits of type m, d, for each m, d. When m is the trivial class, there areat least cl(M) orbits of type m, d, and sb(M) if they are all nondegenerate(i.e.

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