Convex Hamiltonian energy surfaces and their periodic trajectories

In this paper we introduce symplectic invariants for convex Hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations. In particular we show that they can be used to prove multiplicity results for the number of periodic trajectories.


LI. Dynamical and Geometrical Formulation of the Problem
Denote by <•, •> the usual inner product on R 2 " and let J be the standard complex structure on R 2 " given by the matrix Associated to <•, •> and J is the symplectic form Ω given by Assume H:R 2iV ->R is a smooth map. The so-called associated Hamiltonian vectorfield is defined by the formula The corresponding differential equations (HS) x = X H (x) is called a Hamiltonian system. If x solves (HS) theñ

X=Ω(X H (XXX H (X)) = 0
so that H is constant on x. Therefore it is natural to ask for periodic solutions of (HS) having a prescribed energy H. Though the problem of finding a periodic solution with a prescribed energy seems to belong to the theory of dynamical systems, it is possible to formulate it in purely geometrical terms. This can be done in great generality (see [W 2]). Here, however, we shall restrict ourselves to the cases we shall in fact study, namely convex smooth hypersurfaces in R 2n . More precisely we say ScR 2n satisfies condition (Jf) if the following holds: 2 " is a compact C^-manifold bounding a convex region. Moreover S has a nonvanishing Gaussian curvature and S encloses 0eR 2π . The collection of all S satisfying (Jf) will be denoted by jf.
The condition that 0 e R 2 " is enclosed by S is only some kind of normalisation and has nothing to do with the results obtained. We defined a 1-form θ on R 2 " by Then dθ = Ω. Denote by λ the restriction of θ to S and put ω = dλ. Then kern(ω) must be nontrivial since dim (S) is odd. In fact, where n(x) is the outward pointing normal vector at x e S and moreover λ(Jn(x)) =4< / since (ffl) holds. Therefore λ A ω n ~1 is a volume on S. Hence (S, ω) is a manifold of contact type in the sense of Weinstein, [W2]. As a consequence of our previous discussion we have the following (1) See [W 2] for the easy proof. Since JS? S C TS we have a one dimensional and therefore integrable distribution on S.
Definition ί. Let SeJtf.A periodic Hamiltonian trajectory on S is a submanifold Γ of S which is diffeomorphic to S 1 , satisfying The collection of all Hamiltonian trajectories will be denoted by 3Γ(S).
If H: IR 2 "->IR is now a Hamiltonian having S e #? as a regular energy surface, say H=i, then the periodic solutions of the corresponding Hamiltonian system with energy 1 on S are just parametrisations of Hamiltonian trajectories Γe &~(S). In fact each x 0 e Γ is the initial data for a periodic solution x lying entirely on Γ.
By results of Weinstein [W 1] and Rabinowitz [R 1] it is known that ^~(S) + 0 for SeJf. Knowing that 3Γ(S) φ 0 for S e J4? one can ask for its cardinality. Let α f > 0, i = 1,..., n, so that the α f 's are independent over ΊL. Define S = S(α ί ,..., α n ) by n I Σ α i( X ? + x l+ n) ί = 1 One easily shows that # $~{S) = n. As far as the cardinality is concerned this is the worst known example. Hence the following conjecture.
Conjecture 1. If SeJt, then A few partial results are known to be true [E-L, E-La, E1, B-L-M-R], see also [A-M, HI].
In this paper we shall associate to SeJ^ its index interval σ(S) which is a compact interval in (0, oo). We show in particular that σ(S) degenerates to a point if #iΓ(S)<oo. To the Hamiltonian trajectories ΓE3^(S) we shall associate two positive numbers y(Γ) and y(Γ) which are independent. They are called the totaland the mean-torsion at Γ. In the main result of this paper we shall prove that σ(S) and the collections {y(Γ)} and {y(Γ)} are not independent and that always certain inequalities and equalities have to hold. The inequalities turn out to be optimal. This new approach gives besides new results for Hamiltonian systems a much deeper insight to the problem of periodic Hamiltonian trajectories than previous results. Several open problems are mentioned. For instance, it is shown that if ^(S) is finite, then where σ{S) = {σ}. Moreover y{Γ)>\ for all Γe#~(S) if rcΞ>2. So in particular the above inequality implies that # 2Γ(S) ^ 2 for n ^ 2, thus improving the results of [E-La], where this was proven for n^3. Further it will be shown that the above inequality is optimal in the sense that there exists an S for which we have equality.

The Index Interval of an Energy Surface and Torsion Indices for Its Hamiltonian Trajectories
We start with a definition Note that A is S 11 -invariant. In the following we write (most of the time)  (B G We define for SeJtf a map α s :(-oo,0)-»N, N={0,l,2 9 ...} by where (//)* : H(B G ) -> ίJ(M d s G ). It requires of course some proof that α s (d) < oo. This will be provided later. For specialists this is clearly the Fadell-Rabinowitz index of M|, see [F-R]. We define a subset σ(S) of IR + = [0, -f oo) called the mdex interval of 5 by ί e σ(S) o lim inf α s (d) |d| ^ ί ^ lim sup a s {d) \d\.
Denote by c β the collection of all compact intervals in (0, + oo) which we equip with the Hausdorff topology and Hausdorff metric. As we shall see later the following holds: Sometimes it is possible to compute σ(S). For example for S = S(α l5 ..., α π ) with α f > 0, we have as we shall see later.
Next we introduce the torsion indices for Γ e SΓ{$\ where SeJ^. Fix SeJtf* and denote by ζ the associated vectorfield defined by ΞO (15) One easily verifies that where H'(x) is the gradient of H = H S in R 2n . The right-hand side of (16) defines a Hamiltonian system on R 2 ". Let x:R->ΓeIR 2 " be a solution of Sometimes V(Γ) is also called the action of Γ.
Note that by (15) and the fact that T X Γ = IR((x), λ\Γ is a nonvanishing 1-form on Γ and defines therefore a volume-element. Linearizing the Hamiltonian system (HS) around x:IR->Γ gives Denote by (R(t))teΈί, R(0) = Id the fundamental solution of (LHS). Then ). Denote by R* the adjoint of R defined by and define Then B is the "unitary part" of R, see [C-Z1]. That is, B(t) commutes with J and \B(t)y\ = \y\ f°r every t eR and j/eR 2 ". J defines a complex multiplication on R 2n by .
Definition 4. Let SeJt? and Γ e $~{S). The total torsion at Γ is the real number The mean torsion at Γ is the number Now we formulate our main result.
So Corollary 1 implies conjecture 1 for the case n = 2. In [E-La] this was claimed too, however due to a faulty argument it was actually unproved (the arguments in [E-La] hold only for n^3.) Under the general hypotheses of Theorem 2 the inequality in (iii) is optimal. Namely let S = S(OL U ...,α n ) with a t >0 independent over Z. Then as we shall see later We mention another conjecture. Denote by τ^ the topology on #? which is induced from the weak Whitney topology on C GO (R 2 "\{0},IR) via #. Then we have Conjecture 2. For a residual subset J-fj of 3tf the following holds: For SeJ^Ί the map ^(S)->R:Γ->7(Γ) is injective.
A simple corollary of this conjecture is that # &~{S) = + oo for S e M u because y:^{S)-^lR. cannot be injective if #5%S)<oo by Corollary 1. Finally we mention the following.
Problem 4. How does y behave on periodic Hamiltonian trajectories close to a generic elliptic one? Is it injective?
There is of course some connection between Conjecture 2 and Problem 4.

Critical Point Theory
Consider the C u ^functional A S = A\M S on M s . As Riemannian metric on M s we take the one induced by ( , ). Then one verifies easily that for de(-oo,0) If ||grad^s(x fc )|| -»0 and A s (x k )-+d<0, then (x k )cM s is precompact in M s .
(PS) d Solving the differential equation x'=on M s in forward time we obtain a continuous map, R + xM s -+M s :{t,x)-^x*t, which is the restriction of a not necessarily globally defined flow. The map t-+A s (x * t) is non-increasing for fixed x e M s . A well-known consequence of (PS) d is the following where we take S 1 =R/Z. θ operates by isometries on E via

((a,k)*x)(t)=lx(kt + a). (4)
k One easily verifies that θ * M S = M S . Moreover if Cr(S) denotes the set of critical points of A s then θ * Cr(S) = Cr(S). But caution, note that A s is noί (!) θ-invariant. In fact Moreover θ induces by restriction to S 1 ^S 1 x {1} the usual ^-action. Denote by " ^ " the smallest equivalence relation containing the relations x->(α,k)*x for all (α,k)eθ and xeE. One verifies easily that S 1 *y has the desired properties. In fact ΦG y =l.
(i) If XECΓ(S) then we have for some number <5φO, where Ψ(x) = J H*( -Jx(t))dt and the prime denotes the gradient in E. Let k = # G x . o Then y defined by y(t) = /cy ί -J is a minimal representative for [x]. One computes \/c/ easily Άy = kδΨ\y).
(iii) Let [x] eCv(S)/~ and z a minimal representative. We have G z = {1} and for some (5 Φ 0, Taking the inner product with z and using that A' and Ψ' are positively 1-homogeneous we infer since Ψ(z)=ί, Using (6) we find that for arbitrary h E E, Since h has mean value zero we find a constant c 1? eR 2 " such that ι = \δ\H*' (-Jz(t)).
By (7) again we conclude from this that with c = \δ\~1c 1 parametrizes an element in «^~(S). Now starting with some Γ and doing the whole procedure backwards we end up with a class [x] eCr(S)/~.
(iv) Using (12) and the definition of z 2 we see that the minimal period Γ of z 2 is | <5| ~1 = |A s (z)| ~x and that We use now the Fadell-Rabinowitz index [F-R], denoted by ind. We have already seen that (formula 1.12) d = a s (d). (13)

In particular if a s is discontinuous at d, then Cr(d)Φ0. Moreover if a s (d)
oc s (d~) ^ 2, then Cr(d) contains infinitely many S ι -orbits. Consequently in this case #iΓ(S)=oo.
Since the proof is essentially contained in [F-R] we can be sketchy.

Proof d-^κx s (d) is non-decreasing by the monotonicity property of ind. To see that oc s (d)< -h oo, decompose E as follows
where xeE ± is given by If de(-oo,0) one easily finds NeN such that xeM s and ,4 s (x)^d implies By a variant of Lemma 4 we find ε > 0 such that By the properties of ind we infer from (14) and (15) Proof. Equation (16) follows from our second assertion. Since k = a s (cί~) +1 we see that (/")V" ι ) = 0.
By exactness of the row in (17) we find using (18)

A Finite Dimensional Reduction
Recall the definition of the Hubert space E. For N e N* we denote by The orthogonal projection E-+E N is denoted by P N . Moreover we put Q N = ld-P N . Define as before ΨEC U \E,TR) by Proof. Define τ(y, z) by Then τ(y, z) > 0 since y + z Φ 0, and moreover it is a C 1 ' 1 -map since this is true for Ψ on £\{0}. Consequently σ is C 1 ' 1 . It is clear that P N σ(y,z)ή=0, so im(σ)cM SjiV .
Clearly the map u-*(y,z) is C 1 * 1 and an inverse to σ. Π Next we need Lemma 9. Given d 0 e(-co,0), there exists iVj^jeN* such that Proof. We find c ί >0 such that Hence if x e M s we infer Now let x e Mf. Then Denote by c>0 a monotonicity constant for Ψ, that is We shall express A = A\M S by "local coordinates" in S N x (EN) 1 , that is we consider the map of class C ίΛ given by τ y (z) = τ(>;,z).
We equip the vectorbundle S N x (E N ) λ ->S N with the metric [ , •] induced from the inner product on E i(y,z),{y,zj\:={z,z).
Now combining (23) and (30) we obtain Therefore for a suitable constant c 12 , ddc-\d\~~ ^c^ \\z-z\\ ^c ί2 \\y-y\\, so for a suitable number N 2 (d 0 ) ^ N ι we find α = <x(d 0 ) > 0 independent of y, z, and where z is a solution of Γy{z) = 0, Γ y (z)^d 0 and similarly for y and z. Π Proof. Since (Ψ(y + z n )) is bounded away from zero the sequence (τ y (z n )) must be bounded. Let us also show that (τ y (z n )) is bounded away from zero. Arguing indirectly and eventually passing to a subsequence we may assume Hence Consequently (34)

So
On the other hand Λ(τ y (z
Since (u n ) is bounded we may assume eventually taking a subsequence that u n -^u weakly in E, Άu n ->Άu strongly in E.
So (39) gives using that Ψ':E~+E is a homeomorphism, is converging strongly. Hence, with σ y (z n ) = u n we find z n -^z strongly for some z and Γ y (z) = 0, Γ y {z) = d. D Therefore we have just proved that Γ y satisfies (PS) d for all de (-oo,d 0 for every y e S N such that the right-hand side in (40) Consider the C 1?1 -map The preimage of Z 1^ consists of all (y, z)eS N x (Ex) 1 such that We solve the parameter dependent differential equation Since 1/ carries as an open subset of S N the induced C 00 -differentiable structure coming from the standard differentiable structure, we can equip t d s°N with a smooth differentiable structure uniquely characterized by the requirement that the map in (41) From this we shall infer since τ and σ are C 1 and y-+z y is Lipschitz continuous, that f' is of class C 0 ' \ i.e., Γ is of class C 1 ' ι . We compute with Γ'(x) defined by (42), Now dividing the above inequality by \\y ί -y o || and taking the lim sup for y x we infer where we use that {Ψ~1{yι+z 0 )-Ψ~1(y 0 + z 0 ))/\\y ι -y 0 \\ can be replaced in the limit by

Ψ'(y o + z o
Similarly one proves that yi-^yo Note that we had in principle to work in local coordinates to establish that Γ is differentiable at y 0 and has Γ'(y 0 ) given in (42) as gradient. However, taking an exponential chart for a suitable small zero neighborhood W, we see that is the identity so that actually (43) and (44) imply the assertion in the approach using local coordinates. So we have till now proved that (42) gives indeed the gradient. Since by construction of Γ we have we infer that gvadA s (σ y (z y )) = 0, if Γ'(y) = 0. On the other hand if grad^l s (x) = 0 with A s (x)<d 0 , then writing x = σ y (z) we see that z is a critical point of Γ y (z\ so that by our previous discussion z = z y . Hence y is a critical point of Γ.
Next we have to prove the assertion concerning the smoothness of the G-action and of A\Σs Ό tN near a critical orbit.

A k = {xeC\S\Έi 2n )nE\x(t) + 0 \fteS 1 }.
One verifies easily that we have the following commutative diagram: where the top arrow is a smooth map. We have to exclude x with x(t) = 0 because H" and H*" do not exist at zero. Here the space Δ k and C\S ι , R 2 ")n£ are of course equipped with the C fe -topology. Define a map by for yeS N and zeΛ k r\(E Ή ) L such that (y + z)'(t)ή=0 VteS 1 . So the map is in particular smooth around pairs (y, z) such that σ y (z) is a critical point of A\M d s°. The partial differential with respect to z at σ y (z y ) is given by where Ψ"(y + z y ) is given by -} (J JH*"{ -J(y(τ) + z y (τ))) (-J/l(τ))dτ ) o\o / By the definition of Λ^ it follows that the ^-extension of the map (48) (E^MiStf) 1 : ft-^'Λ -A( σy (z,))β^y"0; 4z y )h is an isomorphism. Now let ze(E N ) 1 nC /c (S' 1 ,IR 2 "), and pick he (Ex) 1 with By a simple regularity argument it follows that /2eC /c (5 ίl ,R 2 ")n(£ ΛΓ ) 1 . So by the open mapping theorem the map given in (48) as a map of the /z-variable is a topological isomorphism. By the implicit function theorem there exists a smooth map C k -+C k :y-+z y defined for y close to a critical orbit of Γ such that By uniqueness z y = z r Since fc e N* was arbitrary we see that the points in Σ d s°iN close ("close" is independent of k) to a critical orbit belong to C GO (5' 1 ,R 2 ")n£. Moreover the map y-^Γ y (σ y (z y )) is smooth for y close to a critical orbit. So A\Σ d £ N is smooth near critical orbits. S 1 acts smoothly on S N , so it acts smoothly on {σ y (z y )}, provided the y are close to a critical orbit. In fact, close to a critical orbit the map y->σ y (z y ) is smooth and a*σ y (z y ) = σ aφy {z a4ιy ) 9 implying our assertion. • 442 1. Ekeland and H. Hofer Recall that a critical point x of Λ s satisfies x(ί)=t=0, teS 1 . Therefore the following definition makes sense. where h e T X M S .
Clearly Q x has a finite index m~(x) which is the maximal dimension of a linear space in T X M S on which Q x is negative definite, and a finite nullity m°(x), which must be of course bounded by 2n. We call m~(x) and m°(x) the formal index and m°(x) the formal nullity of the critical point x.
We shall show that there is a close relation between m~(x), m°(x) and the index and nullity of x as a critical point of A\Σ d s°N for d 0 sufficiently close to 0, d o <0. More precisely,

Γ(x) = m~(x) and i°(x) = m°(x).
This is quite standard and we will be somewhat sketchy. See also [E 1] for a related result for a different reduction method.
Let y 0 + z yo = x. Then Γ is smooth near y 0 by our previous discussion. Differentiating (51) at y 0 gives for heT yo S N ,

Γ"(y Ό )h = τ yo (z yo )P N lA'(h)-Γ(y o )Ψ"(y o + zj (h + z^)] . (52)
On the other hand by the construction of Γ we have By the proof in Lemma 14 the map y-*z y is smooth in the C k -setting if y is close to y 0 . So we infer differentiating (53)

%Γ"(y Q )K h) = τ yo (z yo )Q x (h + z' yo h).
This implies in particular that index(f "(j/ 0 )) = z'~(x) since f is a local coordinate description of A\Σ d s°N Sm~(x) by (55), Hamiltonian On the other hand assume X is a linear subspace of T X M S with Q x being negative definite on X. Then by the definition of N we infer that P N u + 0 for u e X\{0}. Note that z' yo h is defined by the minimum problem [as a unique solution which follows from the definition of N(d 0 ) in (33)]

ve(E N μ
Let v h = z f yo h. Then defining a subspace X of T^M S by we see that βJX is negative definite and by construction dimX = dimX. So and similarly nullity(f"(j;o))^w o (x). Π (59)

Critical Points with Prescribed Formal Index
Definition 7. The discontinuity sequence (S k ) ke^* for α s , 5 e Jf, denoted by dis(S), is the non-decreasing sequence consisting of all points d<0 at which a s is not continuous. Moreover each point d is repeated according to its multiplicity a s (d) The aim of this section is to prove the following: We don't claim that the x t are mutually different.
If x k is a minimal representative of [x fe ] we may assume for some and the second part of (2) is proved. If now j ^ 1 we can argue as follows. Define Then (2)  Pick b ^j+1 and δ 0 >0 such that all critical points on level d being c> 0 -close to x k , ...,Xi have a Morse index m~ satisfying m'ix^m'^m'ix^-i-wPix^-ί for l = k,...J.
(The -1 comes from the fact that we have a nontrivial S ι -action.) Now according to (5) we can take a new x i + ι corresponding to fr^j+1 and <5 0 as above. If x / + 1 coincides again with some of the x k ,..., x t we find a critical point x t + ί different from the orbits G * x k ,..., G * x t on level cί which is δ 0 close to one of the critical points in {x k , ...,xj. It satisfies by (6) Now combining (4) and (6) gives Since m°(x i+ί )^2n, this yields We take x ί+1 for our new x i+ί and the second part of (2) is proved. Define
(10) For c<d<d 0 the inclusion
By the compactness of Cr(d) there are only a finite number of equivalence classes. The Riemannian metric on S N induced by the inner product on E induces a Riemannian metric for Σ. We denote by R + xI^:(ί,x)^n (13) the restriction of the minus-gradient flow associated to Ά, that is We shall also denote by x * t for t < 0 the image of x in backward time as long as the flow is defined on [£, 0]. Note that Σ d is compact for every d<d 0 . Now fix δ > 0 and denote by [u{\ δ ,..., [u m{δ) '] δ the mutually disjoint equivalence classes of Cτ(ct). Note that every [wj ό is G-invariant, open and closed in Cr(5).
We find ε((5)e(0, ε 0 ) and compact G-neighborhoods K t in Σ of [wj 5 such that The G-action and A are smooth on an invariant neighborhood We define K; =K^d~E {d \ Lemma 16. The inclusion induces an isomorphism in equivariant cohomology. Here \} denotes disjoint union.
Results of this type are well known if the critical orbits are isolated. That they are isolated is however not assumed here.
Proof. We use the strong excision property of Alexander-Spanier cohomology. Since we shall work in the finite-dimensional manifold Σ we can define equivariant cohomology by taking the G-product with E G = S 2k~1 instead of E G for some sufficiently large k. The inclusion (20) induces a bijection (U KMU κn^(Σ"-ε{δ) Moreover if we take the G-product of the data involved in (21) with E G9 k large, we obtain a similar assertion to (21): Moreover the inclusion is a closed map since a closed set in the left-hand space is compact. Recall that the suffix G means product with E k G for k large enough. By the strong excision the inclusion in (20), say j = {jj, induces an isomorphism Here H G (X):=H(X G ) by definition. H G is called an equivariant cohomology theory. This construction is due to Borel, [B]. By condition (18), using the map * :IR + x Σ^Σ, we can easily construct a continuous map such that VxeΣ a " ε(<5) , Using (24) we obtain the following G-homotopy commutative diagrams:

J (/")• (26)
where / ί? / ± are induced by a classifying map (see Lemma 7), everything else is induced by an inclusion. Recall the cohomology class σ exhibited in Lemma 7. We easily infer from (26) Moreover the nontrivial cohomology given in (28) "lives" above or on level <?, namely we have the commutative diagram (de[β, -ε(δ), ct+ε(δ)~]), That the vertical arrow on the right is a isomorphism follows as in the proof  (15), (17), (18), (19), a result by Wasserman, [Wa], and an equivariant partition of unity argument, there is a G-invariant smooth map A defined on a neighborhood of K io such that A coincides with A on a neighborhood of K io , The critical ^-orbits on levels between ct-(ε(δ)/2) and d + (ε(δ)/2) are nondegenerate, Proof. This is of course a replica of the proof of the corresponding properties of α s . Note that by (30) K io has property (19) with respect to the minus-gradient flow associated to A. Equations (34) and (35) follow from the fact that (31) holds, so that there can be only a finite number of critical orbits between levels Z-ε(δ)/2 and 3+ ε(δ)/2. (lϊdi = d i + ι for some i, then there would be infinitely many orbits on level Our aim is to show that there exists a critical point of A in K io on level d { having index 2(fc -1 + /). From this Proposition 2 will follow easily. For this we have to recall some facts from equivariant Morse theory [Bo, Hi], as well as some local results concerning the Poincare polynomial of a nondegenerate orbit. The reader can also use the note by Viterbo [V]. Combining a local version (in K io ) of Lemma 7 with Lemma 20 and using a localization technique in K io similar to the procedure within this chapter (however somewhat simpler) together with the nondegeneracy we obtain

Lemma 21. For d t as in Lemma 20 there exists a critical point u { of A in K io on level ά { such that
* where N t -+G* u { denotes the negative bundle and N ( is the negative bundle with the zero-section deleted.
We need now some information about the Morse index of the u t .

Lemma 22. The Morse index of u t as given in Lemma 21 is
By the nondegeneracy of u t the nullity is exactly one: Proof Denote by N t x the fibre over xsG*u t and consider the trivial vectorbundle The isotropy group G x of x is a Έ b / = ordG x . Let g be a generator for G x . Then gN i>x = N itX and G x acts on the vectorbundle (39) in the obvious way. Of course we take the standard action on S 00 . p commutes with the action and taking quotients we obtain where L 00 is an infinite dimensional lens-space. Clearly we have the commutative diagram (iV ; x£ G )/G = * ζ I I

{G*u l xE G )/G=+L°w
here the horizontal maps are isomorphisms. (So we have a vectorbundle isomorphism.) Now ζ-^L 00 is Q-orientable iff JV ί ->G*w ί is Q-orientable. We start with computing H G (G * w f ). We have (G * u t x E G )/G -L 00 = S^/G;, = S^/Z,.

I. Ekeland and H. Hofer
By a result in [B] we infer So (43) gives we obtain again by a result in [B] x BcVGJ ~ [H(iV i;< x £ G )] Z ' j (S 00 contractible). If all geΈ, induce an orientation preserving (op) map, we have with dimJV,-, x = α, if one is orientation reversing (or), fl z (if or).
So if Z; = G^ acts orientation preserving on N t which is equivalent to N t -*G* M, is orientable, we infer combining (44), (45), (46), If N i -+G*u i is not orientable then a similar argument based on (47) gives Now by assumption HQ (k~ί + i \N i ,N i )φ0. So we must be in case (48) with 2(fc-1 +ί) = a. This proves (37). Equation (38) is clear. •

Proof of Proposition 2. Since
A is arbitrarily C 00 -close to Ά, we find in view of Lemma 22 for ie{k,...,k-\-j} a critical point x t of A on level 3 such that Since δ > 0 is arbitrarily given, (5) follows immediately. Π

The Index Interval
We shall show in this section that σ(S) is a compact interval in (0, + oo) and that the map S-+σ(S) is continuous.  (1)

fc->oo
We have where <ϊ^<i is the closest point of discontinuity for α s on the right of d. Defining d^d similarly we obtain Hence (4) and (5) imply our assertion. •

IILί. Index Sequence and Winding Number
Let SeJf and pick Γe$~(S). Denote by x:R-^S a solution of x = JH\x) with x(0) e Γ, where H = H s . Consequently x(t) e Γ for all t e R. As we have already seen the minimal period T of x satisfies T= F(Γ). We study now the linearisation of x -JH'(x) along x, which is Definition 8. Two times ί t < t 2 are called conjugate along x if the linearised problem (LHS) possesses a solution y: [t u ί 2 ]-^lR 2 " satisfying y(tι) = y(ί 2 )' The multiplicity of ί 2 with respect to t 1 is the number of linearly independent solutions of (LHS) satisfying y(t ί ) = y(t 2 ). If t ί =0, we define m(ί) for ί>0 as follows: [ 0 if ί is not conjugate to 0. multiplicity of ί if ί is conjugate to 0. Now we are in the position to associate to Γ e &~(S) an index sequence as follows

0<s<kV(Γ)
In [E1-E3] the reader will find the basic properties of the index sequence. An alternative but equivalent definition of the index sequence can be given as follows.
where) is a map from the unit circle {zeC| |z| = l} in C into the non-negative integers, which is described in detail in [E2]. Equation (3) implies that zctoo k 2π We call ΐ Γ the mean index of Γ. Now using results in [C-Z 1, C-Z 2] we can relate ΐ Γ to a winding number. In [C-Z 1] Conley and Zehnder introduced an index based on a winding number and related to previous work by Duistermaat [Du] and Cushman-Duistermaat [Cu-Du]. From facts which can be found in [C-Zl, p. 651] and formula (1.17) in [C-Z 2] we have for a constant C > 0 independent of Γ (note that our Δ Γ is \ times Conley-Zehnder's A) Since, as shown in [C-Z 2, p. 652] A Γ (kV(Γ)) = kA Γ (V(Γ)l we infer combining (5) with the following: Proof. Using (5) and (6) we have

k'
Taking the limit gives (7). Π In the following we study in more detail the quantity ΐ Γ to obtain information concerning γ(Γ) and y(Γ)
Proof F x possesses a (d + zf )-dimensional subspace X such that Q 1 \X ^ 0. X admits the Q x -orthogonal decomposition where X λ is spanned by the functions in the kernel of Q { and X 2 is spanned by the eigenfunctions belonging to negative eigenvalues. Let {y u ...,y d \ be a (^-orthogonal basis ioτX ι and {y d+1 ,..., yd + ύ} a (^-orthogonal basis for X 2 . We define YjCF 2 ϊoτj=ί,2,3 by Then the Yj are mutually Q 2 -orthogonal in F 2 and a simple calculation shows Moreover Q 2 () ; ) = 0 implies y £ 1^ if j e Y x 0 7 2 ®7 3 . Since Y γ does not contain an eigenfunction since y is constant on (V(Γ) . Here we perturb R(t 0 ) by changing t 0 to some neighboring t, but the argument is quite similar. By standard perturbation theory, there is a C^-map ί->w(ί), defined on a neighborhood of U, such that w(ί) is the only eigenvalue of R(t) close to e iθo . Since R(t) is symplectic and w(t) is a simple eigenvalue, it cannot leave the unit circle, so w(ί) = e ιθ(t \ We can also choose for each t an eigenvector y(t) in such a way that the map t^y(t) is C 1 . Now write
We take the Hermitian product with Jy(t). The left-hand side vanishes since

ί(y(tl Jy(t))θ(t) = (H"(x(t))y(tl y(ή).
The right-hand side is positive. It is known that the Hermitian form -U does not vanish on the eigenvector y(t\ and by definition its sign defines the Krein-sign of the eigenvalue e is Krein-negative. Hence the result. • Before proceeding, we must make an excusion into index-theory. Take w on the unit circle and t > 0. Consider the Hermitian form {Qy, y) = ί <JM ΛΦdτ + ] <#*"( -Jx(τ))Jy(τ% Jy(τ)>d 0 0 on the complex Hubert space This form is the sum of a positive definite term (for vv+ 1) and a compact term. Hence it has a finite index. We call it j(w, ί). Note that j(w) =j(w, T) in our previous notation. Clearly j(w, t) cannot change without Q degenerating, which happens only if w is an eigenvalue of R(t).
Definition 10. Let w be on the complex unit circle. We call £>0 w-conjugate to 0 along x if w is an eigenvalue of R(t). Note that Definition 8 is concerned with 1-conjugate times ί. Denote by m(w; ί l5 1 2 ) for t x < t 2 the number of s £ (ί l5 1 2 ) which are w-conjugate to 0, each counted with multiplicity. (The multiplicity is of course defined similar to that in Definition 8.) Assume t is not w-conjugate to 0 and w +1, thenj is constant in a neighborhood of (w, ί). If w= 1 and t is not 1-conjugate to 0, we have limj(e ίθ ,ί) = /(U) + » We can split L 2 into LQ©(C 2 ", where L 2 0 is the space of (C 2 "-valued L 2 -functions with mean value zero and (C 2/I denotes the space of constant functions. The restriction of Q to (C 2n has index n and the restriction of Q to L 2 0 has index j(l, ί). If w is close enough to 1 the index of Q will be j(l,ί) + n. Thus we have proved Lemma 29. If n = 2 then if t is not 1-conjugate to zero: \imj(e w ,t)=j(l,t) + n. • 0ΦO Lemma 30. Assume w = l is α double eigenvalue of R{T). Then there are neighborhoods V of 1 and U of T and a continuous map t -» θ(t) from U to IR such that (i) 0(ί) Φ 0 /or ί + Γ <md <? ίβ(ί) eVVteU. (ii) 77zέ? restriction of θ(t) to U\{T} is C 1 . (iii) For ίe 1/ e ιθ(ί) and e~ι θ(ί) are t/ze on/j; eigenvalues of R(t) belonging to V.
Proof T is clearly conjugate to 0 with multiplicity 2 as we have previously seen. Conjugate points are known to be isolated [E2, E3] so that there is a neighborhood U' of T with We consider the equation The left-hand side is a polynomial in w with smooth coefficients in t. For t = T there is a double root w= 1. Choose a disk V around w= 1 containing no other root. Then there exists an open neighborhood UcU'ofT such that whenever t e U and t+T. Eq. (9) has two simple roots in V. Since R(t) is symplectic these roots must either be both real ρ(ί) and ρ(t)' 1 with 0 < ρ(t) ^ 1 (10) or both on the unit circle e m) and e~m ) with 0^θ{t)<π.
The functions ρ(ί) and θ(t) must be C 1 on t/\{Γ}. This leaves us with four possibilities (a) real roots for all t e U. (b) real roots for t < T, complex roots for t > T.
(c) complex roots for t < T, real roots for t > T.
(d) complex roots for all t + T. We may choose U to be an interval containing T. By the preceding lemma θ(t) will have a constant sign on each of the half-intervals Un{t<T} and Un{t> T}.It follows that a complex eigenvalue w = ^ί θ can occur at most once on each side of T. In other words, for each w e V with |w| = 1 and, w φ 1, Eq. (9), now considered as an equation in ί has at most two solutions t ί and t 2 in l/ 5 one with t 1 < T and one with t 2 >T. If there are exactly two we have case (d).
Since neither t γ nor t 2 is 1-conjugate to zero, it follows that there is a neighborhood W of 1 contained in V with j(w,ί 2 )=Xw,ί 1 ) + 2, weW.
So, whenever WEW and wφl, Eq. (9) must have two solutions in (S l5 S 2 ) C U. We are therefore in case (d) and Lemma 30 is proved.
• Still in the case n = 2we have

Lemma 31. Assume ker(#(T) -Id) is two-dimesnίonal. Then
Proof Take t < T in U and consider θ(t) which was defined in Lemma 30. We have θ(t) > 0 and Θ(T) = 0, so e m) is Krein-negative by Lemma 29. Fix θ ί e (0, π) so that for all t e U with ί < T the only eigenvalue of R(ή of the form ^i θ , 0 < θ ^ θ x is 0(ί). Set β 2 W = 2βW and W!=e ίθl and w 2 (ί) = e iβ2(ί) . We have Between w x and w 2 (ί) there is a single Floquet-multiplier e m \ which is Kreinnegative. The change in j(-,t) is then +1, see [El]: Now let t~>T. Since i^(ί) never has eigenvalue w 1? we have
Proof Since j(w) ^ 3 if w close to 1 and the value of j(w) can drop by at most 1 for wφl (since there can be at most one simple multiplier wφl on the upper half circle) we infer y'(w)^2 for w=f =

IV.2. Proof of the Main Theorem
We have already proved Theorem 1 (i). Moreover we know by Lemma 25 that infσ(S) = lim mϊk\ct k \, supσ(S) = lim sup/c|4l (1) Since we have already seen after the statement of Theorem 2 that Theorem 1 (ii) is a corollary of Theorem 2, we have only to prove Theorem 2. Moreover Theorem 2(i) has been already proved in III.l.
Constructing (k t ) inductively assume k t has been constructed such that We shall now construct k ι + ί such that We find fc* > fe z such that * / +1 * Using the monotonicity of (<? k ) we find for αeN, Since there exists α 0 ^ 0 such that 466 I. Ekeland and H. Hofer we see that the balls ( )WJ) o l+l cover σ(S). Hence we find k ι + 1 e {fc*,...,&* + α 0 } with the desired property.
• Proof of Theorem 2 (iii). We have to show that for Se ffl the following inequality holds: X yiry^l Vε>0.