Marco Mazzucchelli

ON THE MULTIPLICITY OF NON-ITERATED PERIODIC BILLIARD TRAJECTORIES

Keywords: Mathematics Subject Classification. 37J45, 55R80. Key words and phrases. Billiards, Morse theory, Iteration of periodic trajectories. 1

We introduce the iteration theory for periodic billiard trajectories in a compact and convex domain of the Euclidean space, and we apply it to establish a multiplicity result for non-iterated trajectories.

Introduction

Billiard dynamics describes the motion of a particle moving without friction in a compact domain of R N +1 , for N ≥ 1, while being subject to a singular potential which is identically zero in the interior of the domain and +∞ on the boundary. This implies that the particle moves on straight lines with constant speed until it reaches the boundary of the domain, where it bounces with specular reflection and with no loss of energy. In this paper, we investigate the multiplicity of certain periodic billiard trajectories in strictly convex domains of R N +1 enclosed by a smooth hypersurface, which is therefore diffeomorphic to the unit sphere S N . Throughout the paper, by strict convexity we will always mean that the second fundamental form of the boundary is everywhere positive definite.

Historically, the first multiplicity result for periodic billiard trajectories was proved by Birkhoff [START_REF] Birkhoff | On the periodic motions of dynamical systems[END_REF] for convex plane (i.e. N = 1) billiards. The result asserts that, for each pair of coprime positive integers n ≥ 2 and r ≤ n/2, there are at least two distinct periodic billiard trajectories with n bounce points and rotation number r. The plane case is special, since the billiard dynamics can be described by means of an area preserving twist-map on the annulus, and nowadays Birkhoff's Theorem can be proved by means of Aubry-Mather theory, see [START_REF]Mather sets for twist maps and geodesics on tori[END_REF] and references therein. For higher dimensional billiards the problem is essentially harder, and estimates for the number of bounce trajectories with prescribed number of bounce points have been proved by Farber and Tabachnikov [START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF][START_REF] Farber | Periodic trajectories in 3-dimensional convex billiards[END_REF], who corrected and extended an earlier proof by Babenko [START_REF] Babenko | Periodic trajectories of three-dimensional Birkhoff billiards[END_REF]. Further extensions, together with more sophisticated multiplicity results, have been proved by Farber [START_REF] Farber | Topology of billiard problems. I, II[END_REF] and by Karasev [START_REF] Karasev | Periodic billiard trajectories in smooth convex bodies[END_REF]. Roughly, these results can be summarized by saying that, for each n odd, the number of periodic billiard trajectories whose number of bounce points divides n grows at least linearly in n.

The arguments in these papers are based on critical point theory. In fact, billiard trajectories can be characterized by a variational principle, which in the periodic case goes as follows. If S is the smooth boundary of our convex billiard table and n ∈ N, the length functional L ×n , defined on the n-fold cross product of S, computes the perimeter of the n-gon inscribed by a given sequence of points q = (q 0 , ..., q n-1 ) ∈ S ×n . This functional is clearly continuous, and actually smooth when restricted to the so-called cyclic configuration space G ×n (S), the space of those q's such that q j = q j+1 for all j ∈ Z n . The critical points q ∈ G ×n (S) of L ×n are precisely the n-periodic sequences of bounce points of billiard trajectories. Notice that the dihedral group D n acts by permutations on the cyclic configuration space G ×n (S), and all the points that belong to a same orbit of its action refer to the same geometric closed curve. Now, the lack of compactness of G ×n (S) does not represent a real obstacle for applying the machinery of critical point theory with the length functional L ×n or, more precisely, with the functional -L ×n (see [START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF]Section 4] or the paragraph after Proposition 2.1). In particular, each D n -equivariant homology or cohomology class of G ×n (S) gives rise to a billiard periodic trajectory (as critical point of L ×n ) and, at least in the non-degenerate case, linearly independent classes produce different critical points. In [START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF], the equivariant cohomology algebra H * Dn (G ×n (S); Z 2 ) has been computed for each n odd. Its rank gives, in the non-degenerate case, a lower bound for the number of periodic billiard trajectories whose number of bounce points divides n. In the degenerate case, by Lusternik-Schnirelmann theory, such a lower bound is given by the cup-length of H * Dn (G ×n (S); Z 2 ) plus one. In this paper, we proceed along a different line: the result we prove concerns the multiplicity of periodic orbits whose number of bounce points lies in the set of powers of a given prime number p. Our proof will use a minimum amount of information on the homology of the configuration space: we will only need that, for each n ∈ N, the homology group H * (G ×n (S); Z 2 ) is nontrivial in degree N -1, where N = dim(S). Via Morse theory, for each n ∈ N, a nonzero homology class in H N -1 (G ×n (S); Z 2 ) generates a periodic billiard trajectory γ n whose number of bounce points divides n. More precisely, if the number of its bounce points is n/m, the generated critical point q n = (q n,0 , ..., q n,n ) ∈ G ×n (S) of L ×n is the sequence of bounce points of the m-fold iteration of γ n . Since we only make use of the homology of G ×n (S) in degree N -1, all the γ n 's (i.e. all the associated critical points of the length functionals L ×n ) have Morse coindex less than or equal to N -1 and are not local maxima for L ×n .

The main issue here is to prove that, varying n, we obtain infinitely many distinct trajectories γ p n . In order to prove this, we develop an iteration theory for billiard periodic trajectories, a discrete version of the one for closed geodesics (see e.g. [START_REF] Bott | On the iteration of closed geodesics and the Sturm intersection theory[END_REF][START_REF]Periodic geodesics on compact Riemannian manifolds[END_REF][START_REF] Bangert | Homology generated by iterated closed geodesics[END_REF]), that may also have independent interest. More specifically, we investigate the behavior of the Morse indices and of the local homology of billiard periodic trajectories under iteration, and we prove a discrete version of Bangert and Klingenberg's homological vanishing under iteration, a tool that can be used in certain situations to assert the existence of infinitely many closed geodesics.

The main result of the paper is the following.

Theorem 1.1. Consider the billiard dynamics inside a strictly convex subset enclosed by a smooth hypersurface of R N +1 , where N ≥ 2. For each prime number p at least one of the following two statements is satisfied:

• there is some n ∈ N such that infinitely many periodic billiard trajectories bounce p n times; • there is a sequence {γ α | α ∈ N} of geometrically distinct periodic billiard trajectories such that: (i) each γ α bounces p nα times for some n α ∈ N, (ii) each γ α has Morse coindex less than or equal to N and it is not a local maximum for the length functional L ×p nα .

We stress that Theorem 1.1 does not follow from the mere knowledge of the cohomology of the cyclic configuration space (and thus, it does not follow from [FT02b, Theorem 1]). In fact, the Poincaré polynomial associated to H * Dn (G ×n (S); Z 2 ) is given by

P N,n (t) = (t (n-1)(N -1) -1)(t N -1)(t N + 1) (t 2(N -1) -1)(t -1) =   (n-3)/2 j=0 t 2(N -1)j   N -1 k=0 t k (t N + 1)
provided N = dim(S) ≥ 3 and n is odd, see

N j=0 dim H j Dn (G ×n (S); Z 2 ) = P N,n (0) + P ′ N,n (0) + 1 2 P ′′ N,n (0) + ... + 1 N ! P (N ) N,n (0)
= N + 1.

The iteration theory developed in the current paper is thus needed to conclude that, by varying n in the set of powers of any given prime number, the n-periodic billiard trajectories that are found by Morse theory are not all iterations of a finite numbers of lower periodic ones.

Beside the considerations on the Morse coindex, by choosing p = 2 in Theorem 1.1 we obtain that every convex billiard table admits infinitely many (geometrically distinct) periodic billiard trajectories whose numbers of bounce points are powers of 2. To the best of the author's knowledge, this assertion does not follows from any known multiplicity result in the literature.

1.1. Organization of the paper. In section 2 we recall the basic definitions concerning billiards and the variational principle for periodic billiard trajectories. In section 3 we introduce the iteration theory for periodic billiard trajectories: in subsection 3.1 we discuss the behavior of the Morse indices under iteration, while in subsection 3.2 and 3.3 we investigate the behavior of the local homology groups of periodic billiard trajectories under iteration. Section 4 is devoted to the proof of Theorem 1.1. 1.2. Acknowledgments. The author wishes to thank Alberto Abbondandolo, Sergei Tabachnikov and the anonymous referee for encouraging and for useful remarks on a preliminary version of the paper. This research has been supported by the Max Planck Institute for Mathematics in the Sciences (Leipzig, Germany) and by the ANR project "KAM faible".

Preliminaries

Throughout this paper, S will be a smooth hypersurface in R N +1 , where N ∈ N = {1, 2, 3, ...}, enclosing a compact and strictly convex domain U S . Hereafter, strict convexity must be intended in the differentiable sense: the second fundamental form of S is everywhere positive definite. In particular, S is diffeomorphic to the unit N -sphere S N ⊂ R N +1 . We are interested in the billiard dynamics in U S . A curve γ inside U S is a billiard trajectory if it is a piecewise straight curve with constant speed | γ|, except at the instants t in which it hits the hypersurface S, where it bounces according to the usual law of reflection: the component of the velocity that is normal to the boundary instantaneously changes sign, whereas the tangential component is preserved. A billiard trajectory γ is periodic with period T > 0 if it is a curve of the form γ : R/T Z → U S .

Billiard periodic orbits are characterized by a well-known variational principle which we are going to recall. For each n ∈ N, let us denote by S ×n the n-fold product S × ... × S. We consider the open subset G ×n (S) ⊂ S ×n given by G ×n (S) = q = (q 0 , ..., q n-1 ) ∈ S ×n | q j = q j+1 ∀j ∈ Z n , which we will refer to as the cyclic configuration space (or simply the configuration space). The dihedral group D n , seen as a group of permutations of Z n , acts on G ×n (S) by σ • (q 0 , ..., q n-1 ) = (q σ(0) , ..., q σ(n-1) ), ∀q ∈ G ×n (S), σ ∈ D n .

For each q ∈ G ×n (S), we denote by γ q the unique curve in U S with prescribed speed (say, parametrized by arc-length) that is piecewise straight and bounces periodically in the points q 0 , ..., q n-1 (see Figure 1). Notice that each point in the D n -orbit of q is associated to the same geometric curve γ q . We denote by L ×n : S ×n → R the length functional defined by L ×n (q) = j∈Zn |q j+1 -q j |, ∀q = (q 0 , ..., q n-1 ) ∈ S ×n , namely, L ×n (q) is the length of the closed curve γ q . This functional is continuous, and it is smooth on the configuration space G ×n (S), whereas is not even differentiable in the complement of G ×n (S) (since norms are not differentiable at the Closed curve γq and corresponding sequence of bounce points q = (q 0 , ..., q 3 ).

origin). A straightforward computation shows that

dL ×n (q) v = j∈Zn qj -qj-1 |qj -qj-1| - qj+1-qj |qj+1-qj | q ′ j , v j , ∀q ∈ G ×n (S), v ∈ T q S ×n .
A point q ∈ G ×n (S) is then a critical point of the length functional L ×n if and only if, for each j ∈ Z n , the associated point q ′ j ∈ R N +1 is orthogonal to the tangent space T qj S. This amounts to requiring that the curve γ q satisfies the reflection law at the bounce points, and therefore q is a critical point of L ×n if and only γ q is a periodic billiard trajectory.

Having this variational principle, one is tempted to study the multiplicity of periodic billiard trajectories with a prescribed number n of bounce points by means of critical point theory, more specifically by means of Morse theory or Lusternik-Schnirelmann theory for the length functional L ×n : G ×n (S) → R. However, one immediately faces the problem of the lack of compactness of the configuration space G ×n (S). A possible solution has been suggested by Farber and Tabachnikov [START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF], who extended earlier works in [START_REF] Croft | On the Steinhaus billiard table problem[END_REF][START_REF] Babenko | Periodic trajectories of three-dimensional Birkhoff billiards[END_REF][START_REF] Kozlov | Billiards, a genetic introduction to the dynamics of systems with impacts[END_REF] for the twodimensional case. The idea is to restrict L ×n to the compact subspace G ×n ǫ (S) ⊂ G ×n (S), where ǫ > 0 and

G ×n ǫ (S) = q ∈ G ×n (S) j∈Zn |q j -q j-1 | ≥ ǫ n , (2.1)
by virtue of the following statement.

Proposition 2.1 (Proposition 4.1 in [START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF]). For each sufficiently small ǫ > 0 the following claims hold:

(i) G ×n ǫ (S) is a smooth manifold with boundary; (ii) the inclusion G ×n ǫ (S) ⊂ G ×n (S) is a homotopy equivalence; (iii) all the critical points of L ×n : G ×n (S) → R are contained in G ×n ǫ (S); (iv) at every point of ∂G ×n ǫ (S), the gradient of L ×n points inward.

This proposition guarantees that, for any ǫ > 0 sufficiently small, the functional L ×n : G ×n ǫ (S) → R satisfies the so-called "general boundary conditions" for Morse theory, see [Cha93, Section 6.1]. Moreover, if we perform Morse theory using the gradient flow of L ×n (instead of its anti-gradient flow, as it would be more common), point (iv) of the proposition implies that the boundary of G ×n ǫ (S) does not enter into play while applying the principles of Morse theory.

The homology and cohomology of cyclic configuration spaces have been studied by many authors, see e.g. [START_REF] Arnold | The cohomology ring of the group of dyed braids[END_REF][START_REF] Cohen | Artin's braid groups, classical homotopy theory, and sundry other curiosities, Braids[END_REF][START_REF]On configuration spaces, their homology, and Lie algebras[END_REF][START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF]. In particular, in [START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF] the cohomology and the D n -equivariant cohomology rings of G ×n (S) with Z 2 coefficients have been completely determined. For our purposes, we only need to recall that the Poincaré polynomial of H * (G ×n (S); Z 2 ), namely the polynomial

B N,n (t) = nN j=0 dim H j (G ×n (S); Z 2 ) t j , is equal to B N,n (t) = (t N + 1)(t (n-1)(N -1) -1) t N -1 -1 = (t N + 1)   n-2 j=0 t (N -1)j   , (2.2) see [FT02b,

Iteration theory for periodic billiard trajectories

3.1. Morse indices of iterated periodic billiard trajectories. For each n, m ∈ N, consider the embedding ψ ×m : S ×n ֒→ S ×nm given by ψ ×m (q) = q ×m := (q, ..., q ×m ), ∀q ∈ S ×n .

This map is clearly smooth (being the restriction of the linear "diagonal" embedding R nN ֒→ R nmN ), and it restricts as a map G ×n (S) ֒→ G ×nm (S) that we will still denote by ψ ×m . This latter map has a clear interpretation in terms of piecewise straight closed curves associated to the points of the configuration spaces: if γ q and γ q ×m are the closed curves associated to q and q ×m respectively, then γ q ×m is the m-fold iterate of γ q . For this reason we call ψ ×m the (m-fold) iteration map.

By the characterization of the critical points of the length functional as bounce points of periodic billiard trajectories, it is clear that q ∈ G ×n (S) is a critical point of L ×n if an only if, for some (and thus for all) m ∈ N, its iteration q ×m is a critical point of L ×nm . We denote by ind(q), coind(q) and nul(q) the Morse index, the Morse coindex and the nullity of the length functional L ×n at q. We recall that these are nonnegative integers defined respectively as the dimension of the negative eigenspace, of the positive eigenspace and of the kernel of the Hessian of L ×n at q. In this section we investigate the properties of the sequences {ind(q ×m ) | m ∈ N}, {coind(q ×m ) | m ∈ N} and {nul(q ×m ) | m ∈ N}. This of course is reminiscent of the iteration theory for the Morse indices of closed geodesics, which has essentially been pioneered by Bott in his celebrated paper [START_REF] Bott | On the iteration of closed geodesics and the Sturm intersection theory[END_REF], and further extended by many authors also to more general Maslov-type indices, see [Lon02, Part IV] and the bibliography therein for a detailed account. The results that we are going to present can be considered as discrete-time versions of those contained in [Bot56, Section 1]. In view of the application to the multiplicity of periodic billiard trajectories in section 4 we draw, as a consequence of this iteration theory, the following iteration inequalities analogous to the one established by Liu and Long in [LL98, LL00] (see also [START_REF]Index theory for symplectic paths with applications[END_REF]page 213]) for a Maslov-type index.

Proposition 3.1 (Iteration inequalities). For each critical point q of L ×n , the following claims hold.

(i) The nullity of the iterations of q is uniformly bounded by 2N , i.e. nul(q ×m ) ≤ 2N.

(ii) There is a nonnegative real number ind(q), the mean Morse index of L ×n at q, defined by

ind(q) = lim m→∞ 1 m ind(q ×m ),
such that the following inequalities are verified:

m ind(q) -2N ≤ ind(q ×m ) ≤ m ind(q) + 2N -nul(q ×m ).
Analogously, there is a nonnegative real number coind(q), the mean Morse coindex of L ×n at q, defined by

coind(q) = lim m→∞ 1 m coind(q ×m ),
such that the following inequalities are verified:

m coind(q) -2N ≤ coind(q ×m ) ≤ m coind(q) + 2N -nul(q ×m ).
The proof of this proposition will be carried out at the end of this subsection. The reader might skip the following paragraphs and go directly to subsection 3.2 on a first reading.

To begin with, let us write down an expression for the Hessian of L ×n at the critical point q. For each v, w ∈ T q S ×n we have

hess L ×n (q)[v, w] = j∈Zn 1 |qj+1-qj | v j+1 -v j , w j+1 -w j - qj+1-qj |qj+1-qj | , v j+1 -v j qj+1-qj |qj+1-qj | , w j+1 -w j .
Let us denote by H = H q the self-adjoint endomorphism of T q S ×n associated to the Hessian of L ×n at q, i.e. hess L ×n (q)[v, w] = Hv, w . If we write Hv = ((Hv) 1 , ..., (Hv) n ), then for each j ∈ Z n we have

(Hv) j = - 1 |qj+1-qj | π j • πj (v j+1 -v j ) + 1 |qj -qj-1| π j • πj-1 (v j -v j-1 ), (3.1)
where π j : R N +1 → T qj S and πj : R N +1 → q j+1 -q j ⊥ are orthogonal projectors. This expression shows that H is a second order difference operator. Now, for each m ∈ N and z ∈ S 1 ⊂ C, we consider the vector space of sequences V m,z given by

V m,z = ν = {ν j | j ∈ Z} | ν j ∈ T qj S ⊗ C, ν j+nm = z ν j ∀j ∈ Z ,
and, for each λ ∈ R, the eigenvalue problem

Hν = λν, ν ∈ V m,z . (3.2)
We denote by ind z,λ (q ×m ) the number of complex-linearly independent solutions of this eigenvalue problem. Notice that, since H is a real operator with real eigenvalues, the sequence ν ∈ V m,z is a solution of Hν = λν if and only of the complex conjugate sequence ν ∈ V m,z is a solution of H ν = λ ν, and therefore ind z,λ (q ×m ) = ind z,λ (q ×m ). Now, let us set ind z (q ×m ) := λ<0 ind z,λ (q ×m ), coind z (q ×m ) := λ>0 ind z,λ (q ×m ), nul z (q ×m ) := ind z,0 (q ×m ) The above sums are finite, for ind z,λ (q ×m ) is different from zero only if λ belongs to the (finite) spectrum of H. These integer indices that we have just defined generalize the Morse index, the Morse coindex and the nullity for, as it readily follows from their definition, we have ind(q ×m ) = ind 1 (q ×m ), coind(q ×m ) = coind 1 (q ×m ), nul(q ×m ) = nul 1 (q ×m ).

The reason for considering the eigenvalue problem for the operator H in the complexified setting is that there is a nice way to compute the index ind z,λ (q ×m ) from the indices ind w,λ (q), for every w ∈ S 1 , of the non-iterated critical point. The recipe is given by the following statement which is analogous to [Bot56, Theorem I]. Its proof is a simple application of the Fourier expansion of "periodic" sequences, and we include it here for the reader's convenience. Proposition 3.2 (Theorem I in [START_REF] Bott | On the iteration of closed geodesics and the Sturm intersection theory[END_REF]). For each z ∈ S 1 , λ ∈ R and m ∈ N, the index ind z,λ (q ×m ) satisfies ind z,λ (q ×m ) =

w∈ m √ z ind w,λ (q).
Proof. The operator H maps each vector space V m,z to itself. In fact, consider the shift operator S : V m,z → V m,z given by

(Sν) j = ν j+n , ∀ν = {ν j | j ∈ Z} ∈ V m,z .
By the definition of H (in particular, notice that the index j in equation (3.1) is defined modulo n) we have that SH = HS. Moreover, since H is a real operator, we have that

zHν = Hzν = HS m ν = S m Hν, ∀ν ∈ V m,z ,
which proves the claim. Now, every ν = {ν j | j ∈ Z} ∈ V m,z admits a unique Fourier expansion

ν = w∈ m √ z ν w
where, for each w ∈ m √ z, the sequence ν w = {ν w,j | j ∈ Z} belongs to the vector space V 1,w . One can explicitly compute ν w as

ν w,j = 1 m m-1 h=0 w 1-h ν j+h , ∀j ∈ Z.
By the first part of the proof we have that Hν ∈ V m,z and Hν w ∈ V 1,w . Hence, the unique Fourier expansion of Hν ∈ V m,z is given by

Hν = w∈ m √ z Hν w .
From this, we conclude that ν satisfies the eigenvalue problem Hν = λν if and only if all the ν w 's satisfy the same eigenvalue problem Hν w = λν w .

This proposition tells us that it is enough to study the indices of q ×m with a fixed m ∈ N, say m = 1. To start with, let us investigate the properties of nul z (q). We call z ∈ S 1 a Poincaré point of q when nul z (q) = 0. Proposition 3.3. There are only finitely many Poincaré points z 1 , ..., z r ∈ S 1 and we have r α=1 nul zα (q) ≤ 2N.

Proof. Let v = {v j } be a sequence such that, for each j ∈ Z, the element v j belongs to the tangent space T qj S. By (3.1), v satisfies Hv = 0 if and only if

1 |qj+1-qj | π j • πj (v j+1 -v j ) = 1 |qj -qj-1| π j • πj-1 (v j -v j-1 ), ∀z ∈ Z, (3.3)
where π j : R N +1 → T qj S and πj : R N +1 → q j+1 -q j ⊥ are orthogonal projectors as above. For our convenience, let us rewrite equation (3.3) as

π j • πj (v j+1 ) = π j • πj (v j ) + |qj+1-qj | |qj -qj-1| π j • πj-1 (v j -v j-1 ). (3.4)
Now, since the vector q j+1 -q j ∈ R N +1 is transverse to T qj+1 S (as well as to T qj S), the composition π j • πj restricts to an isomorphism

T qj+1 S ∼ = -→ T qj S.
This shows that we can rewrite equation (3.4) as

v j+1 = A j v j + B j v j-1 , (3.5)
where A j : T qj S → T qj+1 S and B j : T qj-1 S → T qj+1 S are linear maps. Analogously, we can rewrite equation (3.4) as

v j-1 = C j v j+1 + D j v j , (3.6)
where C j : T qj+1 S → T qj-1 S and D j : T qj S → T qj-1 S are linear maps. Equations (3.5) and (3.6) show that every solution v = {v j } of Hv = 0 is completely determined by two of its subsequent points, say (v 0 , v 1 ), and conversely any choice of (v 0 , v 1 ) uniquely determine a solution v. Moreover, v depends linearly on (v 0 , v 1 ). Let us denote by Φ the linear endomorphism of T q0 S ⊕ T q1 S given by Φ(v 0 , v 1 ) = (v n , v n+1 ), and let us extend it as a complex linear endomorphism of (T q0 S ⊕ T q1 S) ⊗ C. Now, let us take z ∈ S 1 . By its definition, the integer nul z (q) is equal to the complex dimension of the kernel of (Φ -zId). This establishes the proposition.

Remark 3.1. If z 1 , ..., z r are the Poincaré points of q, their m th powers z m 1 , ..., z m r are the Poincaré points of q ×m , for each m ∈ N.

The next statement summarizes the properties of ind z (q) and coind z (q).

Proposition 3.4. The functions z → ind z (q) and z → coind z (q) are locally constant on S 1 \ {z 1 , ..., z r }, where z 1 , ..., z r are the Poincaré points, and lower semicontinuous on S 1 . Moreover, the jump of these functions at any Poincaré point z α is bounded in absolute value by nul zα (q), i.e.

ind zα (q) ≤ lim

z→z ± α ind z (q) ≤ ind zα (q) + nul zα (q), coind zα (q) ≤ lim z→z ± α coind z (q) ≤ coind zα (q) + nul zα (q).
Proof. For each z ∈ S 1 , let us denote by σ z ⊂ R the spectrum of the operator

H : V 1,z → V 1,z
. This spectrum satisfies the following continuity property: for each interval (a, b) ⊂ R ∪ {±∞} such that a and b do not belong to σ z , there is a neighborhood of z in S 1 such that, for each z ′ in this neighborhood, a and b do not belong to σ z ′ and moreover

λ∈(a,b) ind z,λ (q) = λ∈(a,b)
ind z ′ ,λ (q). Now, assume that z ∈ S 1 is not a Poincaré point, namely 0 does not belong to σ z . Then, by the above continuity property, 0 does not belong to σ z ′ for each z ′ in some neighborhood of z, and moreover

ind z (q) = λ<0 ind z,λ (q) = λ<0 ind z ′ ,λ (q) = ind z ′ (q).
Finally, assume that z ∈ S 1 is a Poincaré point, and let us fix a sufficiently small ǫ > 0 so that [-ǫ, ǫ] ∩ σ z = {0}. Then, by the above continuity property, -ǫ and ǫ do not belong to σ z ′ for each z ′ in some neighborhood of z, and we have

ind z ′ (q) = λ<-ǫ ind z ′ ,λ (q) + λ∈(-ǫ,0) ind z ′ ,λ (q) = λ<-ǫ ind z,λ (q) + λ∈(-ǫ,0) ind z ′ ,λ (q) = ind z (q) + λ∈(-ǫ,0) ind z ′ ,λ (q).
This proves that ind z (q) ≤ ind z ′ (q). Moreover ind z ′ (q) = ind z (q) + λ∈(-ǫ,0)

ind z ′ ,λ (q) ≤ ind z (q) + λ∈(-ǫ,ǫ) ind z ′ ,λ (q) = ind z (q) + λ∈(-ǫ,ǫ) ind z,λ (q) = ind z (q) + nul z (q).
The statement regarding coind z (q) is established in the same way.

Proof of Proposition 3.1. Point (i) follows from Propositions 3.2 and 3.3. As for point (ii), let us fix m ∈ N and denote by z 1 , ..., z r ∈ S 1 the Poincaré points of q, so that z m 1 , ..., z m r are the Poincaré points of q ×m . By Proposition 3.4, for each w, z ∈ S 1 we have ind w (q ×m ) + nul w (q ×m ) ≤ ind z (q ×m ) + r α=1 nul z m α (q ×m ) (3.7) By Propositions 3.2 and 3.3 we have

r α=1 nul z m α (q ×m ) = r α=1 nul zα (q) ≤ 2N,
and, together with (3.7), we obtain ind w (q ×m ) + nul w (q ×m ) ≤ ind z (q ×m ) + 2N. (3.8) Now, by Propositions 3.2 and 3.4, we have

ind(q) = lim m→∞ ind(q ×m ) m = lim m→∞ 1 m w m =1 ind w (q) = 1 2π 2π 0 ind e iθ (q) dθ.
Notice that 1 2π 2π 0 ind e iθ (q ×m ) dθ = ind(q ×m ) = m ind(q), and moreover, since nul z (q ×m ) = 0 for every z ∈ S 1 \ {z m 1 , ..., z m r }, we have 1 2π 2π 0 nul e iθ (q ×m ) dθ = 0. Now, by setting z = 1 and integrating w on S 1 in (3.8), we get m ind(q) ≤ ind(q ×m ) + 2N.

Then, by setting w = 1 and integrating z on S 1 in (3.8), we get ind(q ×m ) + nul(q ×m ) ≤ m ind(q) + 2N. This proves point (ii) for the Morse index. The proof for the Morse coindex is analogous.

3.2.

Local homology of iterated periodic billiard trajectories. This section is devoted to prove that, if the Morse indices of an isolated critical point of the length functional are preserved by iteration, then the same is true for its local homology. As for the previous subsection, there is a clear parallel with the theory of closed geodesics: in fact, this result has been established for closed geodesics by Gromoll and Meyer [START_REF]Periodic geodesics on compact Riemannian manifolds[END_REF], and further extended to more general Lagrangian systems by Long [START_REF] Long | Multiple periodic points of the Poincaré map of Lagrangian systems on tori[END_REF], Lu [START_REF] Lu | The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems[END_REF] and the author [START_REF] Mazzucchelli | The Lagrangian Conley conjecture[END_REF]. In the following, we prove the result after recalling the notion of local homology.

From now on, all the homology groups are assumed to have coefficients in Z 2 . For technical reasons (see the discussion after Proposition 2.1) we will work with minus the length functional, that is, with super-levels of the length functional. The local homology of -L ×n at an isolated critical point q is the homology group C (q) defined by C * (q) := H * (G ×n (S) >c ∪ {q}, G ×n (S) >c ), where c = L ×n (q) and G ×n (S) >c = {q ′ ∈ G ×n (S) | L ×n (q ′ ) > c}. Recall that the dihedral group D n acts by coordinates-permutations on the cyclic configuration space G ×n (S) and L ×n is invariant under its action, see Section 2. The local homology of -L ×n at the D n -orbit of q is defined by

C * (D n • {q}) := H * (G ×n (S) >c ∪ D n • {q}, G ×n (S) >c ).
By excision, it is straightforward to verify that the local homology of D n • {q} is the direct sum of the local homology of each element in the D n -orbit of q, and in particular the inclusion induces a homology monomorphism

C * (q) ֒→ C * (D n • {q}).
We also recall that the local homology C k (q), and therefore C k (D n • {q}) as well, is possibly nontrivial only if coind(q) ≤ k ≤ coind(q) + nul(q), and that q is a local maximum of L ×n if and only if

C k (q) = Z 2 if k = 0, 0 if k = 0.
For a proof of these results as well as for more details on the local homology groups, see [START_REF] Gromoll | On differentiable functions with isolated critical points[END_REF] or [START_REF] Chang | Infinite-dimensional Morse theory and multiple solution problems[END_REF]Chapter I]. The reader should keep in mind that the Morse coindex of L ×n is the Morse index of -L ×n and vice versa.

The main result of this subsection is the following.

Proposition 3.5. Let q ∈ G ×n (S) be an isolated critical point of L ×n with critical value L ×n (q) = c, and assume that, for some m ∈ N, we have coind(q) = coind(q ×m ), nul(q) = nul(q ×m ).

Then, the iteration map ψ ×m induces the homology isomorphism

ψ ×m * : C * (q) ∼ = -→ C * (q ×m ).
We will prove this proposition by means of the following abstract principle of Morse theory. Here we only quote the statement for a finite dimensional setting, as needed for our purposes.

Proposition 3.6 (Theorem 4.1 in [START_REF] Mazzucchelli | The Lagrangian Conley conjecture[END_REF]). Let U ⊆ R k be an open neighborhood of the origin, F : U → R a smooth functional having the origin as isolated critical point, and V ⊂ R k a vector subspace. Assume that ∇F (x) ∈ V for each x ∈ U ∩ V, and that the Morse index and the nullity of F at the origin are equal to the Morse index and the nullity of the restricted functional F | U∩V at the origin. Then the inclusion U ∩ V ⊂ U induces an isomorphism between the local homology of F | U∩V at the origin and the local homology of F at the origin.

Proof of Proposition 3.5. All we need to do is to reduce our setting in such a way that we satisfy the assumptions of Proposition 3.6. For each j ∈ Z n let us consider a chart φ j : V j → R N for S, where V j is an open neighborhood of q j . Up to shrink the V j 's, we can assume that V j ∩ V j+1 = ∅. We further define smooth functions

ℓ j : V j × V j+1 → R by ℓ j (x j , x j+1 ) = φ -1 j (x j ) -φ -1 j+1 (x j+1 ) , ∀(x j , x j+1 ) ∈ V j × V j+1 .
Then, the product

V := V 0 × ... × V n-1 is an open set of G ×n (S)
, and the map

φ := (φ 0 , ..., φ n-1 ) : V → R nN
is a chart for G ×n (S) centered at q. In this local coordinates the length functional L ×n can be written as

ℓ ×n (x) := L ×n • φ -1 (x) = j∈Zn ℓ j (x j , x j+1 ), ∀x = (x 0 , ..., x n-1 ) ∈ φ(V ).
A straightforward computation shows that the gradient of ℓ ×n at x, with respect to the flat metric of R nN , is given by g = (g 0 , ..., g n-1 ), where

g j = ∂ 1 ℓ j (x j , x j+1 ) + ∂ 2 ℓ j-1 (x j-1 , x j ), ∀j ∈ Z n . (3.9)
Now, consider the map

φ ×m = (φ, ..., φ ×m ) : V ×m → (R nN ) ×m .
This map is a chart for G ×nm (S) centered at q ×m , and in this local coordinates we denote the length functional L ×nm by ℓ ×nm := L ×nm • (φ ×m ) -1 . Now, if we put on (R nN ) ×m the flat metric rescaled by the factor m -1 , the gradient of ℓ ×nm at y is given by g = (g 0 , ..., gnm-1 ), where

gj = ∂ 1 ℓ j mod n (y j , y j+1 ) + ∂ 2 ℓ j-1 mod n (y j-1 , y j ), ∀j ∈ Z nm . (3.10)
Let us denote by Ψ ×m := φ ×m • ψ ×m • φ -1 the iteration map in local coordinates, which turns out to be the restriction of the (linear) diagonal embedding of R nN into (R nN ) ×m . Notice that ℓ ×nm • Ψ ×m = m ℓ ×n . From this, together with (3.9) and (3.10), we infer

(∇ℓ ×nm ) • Ψ ×m = m ∇ℓ ×n .
Therefore, the claim of the proposition follows by applying Proposition 3.6 with U = φ ×m (V ×m ), F = -m -1 ℓ ×nm and the inclusion U ∩ V ⊂ U given by the iteration map Ψ ×m .

Homological vanishing by iteration.

In this section we show how to recover, in the "discrete" setting of billiards, the homological vanishing under iteration. This is a remarkable phenomenon, discovered by Bangert in [Ban80, Section 3] in the study of closed geodesics and further investigated by several authors in [START_REF] Bangert | Homology generated by iterated closed geodesics[END_REF][START_REF] Long | Multiple periodic points of the Poincaré map of Lagrangian systems on tori[END_REF][START_REF] Lu | The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems[END_REF][START_REF] Mazzucchelli | The Lagrangian Conley conjecture[END_REF].

Let q be a critical point of the length functional L ×n with critical value c = L ×n (q). For each ǫ > 0 sufficiently small, the subset G ×n ǫ (S) ⊂ G ×n (S) defined in (2.1) contains q, and therefore for each m ∈ N the subset G ×nm ǫ (S) ⊂ G ×nm (S) contains q ×m . Now notice that, by excision, the local homology of L ×n at q can be equivalently defined as

C * (q) = H * (G ×n ǫ (S) >c ∪ {q} , G ×n ǫ (S) >c ).
Then, let us fix an arbitrary b < c and consider the iteration map restricted as a map of pairs of the form

ψ ×m : (G ×n ǫ (S) >c ∪ {q} , G ×n ǫ (S) >c ) ֒→ (G ×nm ǫ (S) >mb , G ×nm ǫ (S) >mc ).
In homology, this map induces the homomorphism

ψ ×m * : C * (q) → H * (G ×nm ǫ (S) >mb , G ×nm ǫ (S) >mc ). (3.11)
Proposition 3.7 (Homological vanishing by iteration). Assume that q is not a local maximum for L ×n . Then, for each integer p ≥ 2, there exists m = m(q, p) ∈ N that is a nonnegative power of p such that the homomorphism ψ × m * as in (3.11) is the zero one.

The proof is based on a homotopical result that we are going to discuss here. Consider a point q = (q 0 , ..., q n-1 ) in the space G ×n ǫ (S) and, for each j ∈ Z n , a sufficiently small neighborhood U j ⊂ S of q j , so that U := U 0 × ... × U n-1 is an open neighborhood of q in G ×n ǫ (S). For each c ∈ R, we denote by U >c the set of points q ′ in U such that L ×n (q ′ ) > c. Analogously, for each m ∈ N, we denote by U ×m >c the set of points q ′′ in U ×m such that L ×nm (q ′′ ) > c. Notice that the iteration map ψ ×m send U >c into U ×m >mc . Then, for each j ∈ N, we denote by ∆ j the standard j-simplex in R j . Lemma 3.8 (Homotopical vanishing by iteration). Let b < c such that U >c = ∅, and consider the singular simplex σ : (∆ j , ∂∆ j ) → (U >b , U >c ), i.e. [σ] ∈ π j (U >b , U >c ). Then, there exists m = m(σ) ∈ N and, for each integer m ≥ m, a homotopy

σ ⊠m : [0, 1] × (∆ j , ∂∆ j ) → (U ×m >mb , U ×m >mc ) (3.12) such that (i) σ ⊠m (0, •) = σ ×m := ψ ×m • σ, (ii) σ ⊠m (t, x) = σ ⊠m (0, x) for each x ∈ ∂∆ j , (iii) σ ⊠m (1, ∆ j ) ⊂ U ×m >mc . In particular [σ ×m ] = 0 in π j (U ×m >mb , U ×m >mc ).
Proof. Let us begin by explaining the basic construction that will be employed in the proof. Consider a map γ :

[x 0 , x 1 ] → U , where [x 0 , x 1 ] ⊂ R. For each m ∈ N we define a map γ m : [x 0 , x 1 ] → U ×m s L x0(s, z) x1(s, z) z Figure 2.
in the following way: for each k ∈ {0, ..., m -1} and y ∈ [x 0 , x 1 ], denoting q 0 = γ(x 0 ), q 1 = γ(x 1 ) and x = x 0 + (x 1 -x 0 ) k m + y-x0 m , we set γ m (x) := (q 1 , ..., q 1 ×k , γ(y), q 0 , ..., q 0 ×m-k-1

).

The length of γ m (x) is bounded from below as

L ×nm (γ m (x)) ≥ (k -1)L ×n (q 1 ) + (m -k -2)L ×n (q 0 ) ≥ (m -3) min L ×n (γ(x 0 )), L ×n (γ(x 1 )) . (3.13)
Now, consider the singular simplex σ of the statement. We want to decompose its domain ∆ j as a "continuous" family of paths, and then apply the above construction to each path separately. Let L ⊆ R j be the axis passing through the origin and the barycenter of ∆ j ⊂ R j . For each s ∈ [0, 1] we denote by s∆ j the rescaled j-simplex given by {sz | z ∈ ∆ j }. For each z ∈ s∆ j , we denote by [x 0 (s, z), x 1 (s, z)] ⊂ s∆ j the maximum segment that contains z and is parallel to L. This notation is summarized, for j = 2, in Figure 2.

We define the homotopy σ ⊠m : [0, 1] × ∆ j → U by

σ ⊠m (s, z) =    (σ| [x0(s,z),x1(s,z)] ) m (z) if z ∈ s∆ j , σ ×m (z) if z ∈ s∆ j .
This homotopy clearly satisfies properties (i-ii) in the statement. Then, for each s ∈ [0, 1] and z ∈ s∆ j , by the estimate in (3.13) we have

L ×nm (σ ⊠m (s, z)) ≥ (m -3) min L ×n (σ(x 0 (s, z))), L ×n (σ(x 1 (s, z))) ,
while for each z ∈ ∆ j \ s∆ j we have

L ×nm (σ ⊠m (s, z)) = L ×nm (σ ×m (z)) = m L ×n (σ(z)).
Let us choose δ > 0 sufficiently small so that σ(∆ j ) ⊂ U >b+δ and σ(∂∆ j ) ⊂ U >c+δ . For each (s, z) ∈ [0, 1] × ∆ j we obtain

L ×nm (σ ⊠m (s, z)) ≥ (m -3)(b + δ), L ×nm (σ ⊠m (1, z)) ≥ (m -3)(c + δ).
This implies that, for m ∈ N sufficiently big, the homotopy σ ⊠m has the form (3.12) and satisfies (iii).

Proof of Proposition 3.7. The proof is based on the homotopical vanishing principle of Lemma 3.8, together with the homotopical invariance property of singular homology as stated in [BK83, Lemma 1].

Let us fix the integer p ≥ 2 of the statement and set

K p = {p n | n ∈ N ∪ {0}}.
Since the local homology group C * (q) is finitely generated (see [START_REF] Gromoll | On differentiable functions with isolated critical points[END_REF]Lemma 2]), all we need to prove is that, for each homology class [µ] ∈ C * (q) and for some m = m([µ]) ∈ K p , we have

ψ × m * [µ] = 0 in H * (G ×n m ǫ (S) > mb , G ×n m ǫ (S) > mc
). Let U be the open neighborhood of q introduced in the paragraph preceding Lemma 3.8. By excision, we can assume that the relative cycle µ representing [µ] has support contained in U . We denote by Σ(µ) the collection of singular simplices in µ and all their lower dimensional faces. For each σ : ∆ j → U contained in Σ(µ) we will define m = m(σ) ∈ K p and a homotopy

σ • m : [0, 1] × ∆ j → U × m > mb such that (i) σ • m(0, •) = σ × m, (ii) σ • m(1, ∆ j ) ⊂ U × m > mc , (iii) if σ(∆ j ) ⊂ U >c , then σ • m(s, •) = σ × m for each s ∈ [0, 1], (iv) (σ • F i ) • m = σ • m(•, F i (•))
for each i ∈ {0, ..., j}, where F i : ∆ j-1 → ∆ j is the standard affine map onto the i th face of ∆ j .

For each m ∈ K p greater than m, we define a homotopy σ

•m : [0, 1] × ∆ j → U ×m >mb by σ •m := ψ ×m/ m • σ • m.
This homotopy satisfies the analogous properties (i-iv) in period m. Notice that property (iv) implicitly requires that m(σ • F i ) ≤ m(σ) for each i ∈ {0, ..., j}. Now, take a sufficiently big m ∈ K p so that σ •m is defined for all σ ∈ Σ(µ). Then, by means of the above homotopies, the relative cycle µ ×m = ψ ×m • µ is homologous to a relative cycle whose support is contained in U ×m >mc ⊂ G ×m ǫ (S) >mc . In fact, this latter relative cycle is obtain from µ ×m by homotoping each singular simplex σ ×m ∈ Σ(µ ×m ) to σ •m (1, •) via the homotopy σ •m . This implies that

ψ ×m * [µ] = [µ ×m ] = 0 in H * (G ×nm ǫ (S) >mb , G ×nm ǫ (S) >mc ).
In order to finish the proof, we only need to build the above homotopies satisfying (i-iv). The idea is to apply Lemma 3.8 subsequently to all the faces of the singular simplices in Σ(µ), starting from the 0-dimensional faces and then going up to the higher dimensional ones.

Let us proceed by induction, starting by assuming that µ is a 0-relative cycle. Then, Σ(µ) is simply a finite set of points in U >b . Let w be one of these points. If w ∈ U >c we are already done: we simply set m = m(w) = 1 and w • m(s) = w for each s ∈ [0, 1]. In the other case, w ∈ U >b \ U >c , we take an arbitrary continuous path γ : [0, 1] → U such that γ(0) = w and γ(1) ∈ U >c . Notice that such a path exists, since the critical point q of the statement is not a local minimum and therefore, since L ×n (q) = c, the subset U >c is not empty. Then, consider m = m(γ) ∈ N and the associated homotopy

γ ⊠ m : [0, 1] × [0, 1] → U × m
given by Lemma 3.8, such that γ ⊠ m(s, 0) = w × m, γ ⊠ m(s, 1) = γ(1) × m and γ ⊠ m(1, s) ∈ U × m > mb for each s ∈ [0, 1]. Without loss of generality we can assume that m ∈ K p , and we set

w • m = γ ⊠ m(1, •).
When µ is a j-relative cycle, with j ≥ 1, we can apply the inductive hypothesis: for every nonnegative integer j ′ ≤ j -1 and for each j ′ -singular simplex σ ∈ Σ(µ) we obtain m = m(σ) ∈ K p and a homotopy σ • m satisfying the above properties (i-iv). Now, consider a j-singular simplex σ ∈ Σ(µ). If σ(∆ j ) ⊂ U >c we simply set m = m(σ) := 1 and σ • m(s, •) := σ for each s ∈ [0, 1]. Otherwise, if σ(∆ j ) ⊂ U >c , we denote by m′ = m′ (σ) the maximum of the m(ν)'s for all the proper faces ν of σ. For each m ∈ K p greater than or equal to m′ , every such ν has an associated homotopy ν •m satisfying the above condition (i-iv). For technical reasons, let us assume that ν

•m (s, •) = ν •m ( 1 2 , •) for each s ∈ [ 1 2 , 1].
Patching together the homotopies of the proper faces of σ, we obtain

σ •m : ([0, 1 2 ] × ∂∆ j ) ∪ ({0} × ∆ j ) → U ×m >mb , such that σ •m (0, •) = σ ×m and σ •m (•, F i (•)) = (σ • F i ) •m for each i = 0, ..., j. By retracting [0, 1 2 ] × ∆ j onto ([0, 1 2 ] × ∂∆ j ) ∪ ({0} × ∆ j ) we can extend the homotopy σ •m , obtaining σ •m : [0, 1 2 ] × ∆ j → U ×m >mb . (3.14) Notice that σ • m′ ( 1 2 , •) is a singular simplex of the form σ • m′ ( 1 2 , •) : (∆ j , ∂∆ j ) → (U × m′ > m′ b , U × m′ > m′ c
). Let us briefly denote this singular simplex by σ, and consider m′′ = m(σ) and the homotopy σ⊠ m′′ given by Lemma 3.8, so that

σ⊠ m′′ (0, •) = σ× m′′ = σ • m′′ m′ ( 1 2 , •), σ⊠ m′′ (1, ∆ j ) ⊂ U × m′ m′′
> m′ m′′ c . Then, we set m = m(σ) := m′ m′′ and we extend the homotopy in (3.14

) to [0, 1]×∆ j by σ • m(s, •) := σ⊠ m′′ (2s -1, •), ∀s ∈ [ 1 2 , 1].

Proof of the main result

In this section we carry out the proof of Theorem 1.1, stated in the introduction. Let us adopt the notation of Section 2, so that our billiard table is the strictly convex compact subset U S enclosed by a smooth hypersurface S ⊂ R N +1 , with N ≥ 2. The proof of Theorem 1.1 will be by contradiction: let us fix, once for all, a prime number p ∈ N and let us assume that the following two conditions hold:

(F1) for each n ∈ N, there are only finitely many periodic billiard trajectories bouncing p n times; (F2) there are only finitely many periodic billiard trajectories γ 1 , ..., γ r satisfying properties (i-ii) in the statement.

Let us define K p := {p n | n ∈ N ∪ {0}}. For each α ∈ {1, ..., r}, let us denote by qα = (q α,1 , ..., qα,nα ) ∈ G ×nα (S) the sequence of bounce points of γ α . Let n ∈ K p be the maximum of the n α 's, and let us set q α := q×n/nα α ∈ G ×n (S), ∀α ∈ {1, ..., r}.

Remark 4.1. For each critical point q of the length functional L ×n , the function m → coind(q ×m ), where m ∈ K p , is monotone increasing (although not strictly). In fact, the differential of the iteration map ψ ×m at q maps (injectively) any positive eigenspace of hess L ×n (q) to a positive eigenspace of hess L ×nm (q ×m ). This implies that, by Assumption (F2), for all m ∈ K p the set of critical points of L ×nm with Morse coindex less than or equal to N is given by the D nm -orbits of those q ×m α 's such that coind(q ×m α ) ≤ N . As a first step in our proof, let us establish the following claim. We refer the reader to Section 3.2 for the definition and properties of local homology groups. From now on, all the homology groups are assumed to have coefficients in Z 2 .

Claim 1. There exists q ∈ {q 1 , ..., q r } with coind(q) = 0 and an infinite subset K ′ p ⊂ K p such that, for each m ∈ K ′ p , the local homology group C N -1 (q ×m ) is nontrivial.

Proof. For each m ∈ K p , let us fix a sufficiently small ǫ = ǫ(m) > 0 and consider the space G ×nm ǫ (S) introduced in Section 2. By Proposition 2.1(iii), this space contains all the critical points of L ×nm . Proposition 2.1(ii) implies that the inclusion G ×nm ǫ (S) ֒→ G ×nm (S) induces an isomorphism in homology, and therefore by (2.2) we infer that dim C k (q ′ ), (4.2) where the sum in the right hand side runs over all the critical points q ′ of L ×nm . Notice that only those q ′ such that coind(q ′ ) ≤ k ≤ coind(q ′ ) + nul(q ′ ) (4.3) may possibly give a nonzero contribution. Now, choosing k = N -1, by Remark 4.1 the elements in the D nm -orbit of the q ×m α 's are the only critical points q ′ of L ×nm that may satisfy (4. Now, if all the q α 's had nonzero mean Morse coindex coind(q α ), by the iteration inequality in Proposition 3.1(ii) we would have coind(q ×m α ) > N -1, providing m ∈ K p is big enough. However, this would imply that C N -1 (D nm • {q ×m α }) = 0 for each α ∈ {1, ..., r}, contradicting (4.4). Hence, some of the q α 's, say q 1 , ..., q s where s ≤ r, satisfy coind(q α ) = 0. Up to choosing m big enough the inequality in (4.4) reduces to 0 = s α=1 dim C N -1 (D nm • {q ×m α }).

This implies that at least one q among q 1 , ..., q s satisfies C N -1 (D nm • {q ×m }) = 0, and thus C N -1 (q ×m ) = 0 as well, for infinitely many m ∈ K p . Now, by Proposition 3.1, the indices coind(q ×m ) and nul(q ×m ) are uniformly bounded for all m ∈ K ′ p , and hence we can choose an infinite subset K ′′ p ⊂ K ′ p such that coind(q ×m ) and nul(q ×m ) are constant in m ∈ K ′′ p . Without loss of generality, let us assume that 1 belongs to K ′′ p (equivalently, set m := min K ′′ p and rename q to be q ×m and K ′′ p to be m -1 K ′′ p ). We set c := L ×n (q) and we fix an arbitrary real number b < c such that none of the q α 's has critical value in the open interval (b, c), i.e. L ×n (q α ) ∈ (b, c) for all α ∈ {1, ..., r}. Now, for each m ∈ K ′′ p , let us choose ǫ = ǫ(m) > 0 small enough so that G ×n ǫ (S) and G ×nm ǫ (S) satisfy the assertions (i-iv) of Proposition 2.1. Then, let us consider the iteration map restricted as a map of pairs of the form

  Figure1. Closed curve γq and corresponding sequence of bounce points q = (q 0 , ..., q 3 ).

  Theorem 4 and Remarks 3.2 and 3.3].

H

  1(i,iv), the functional -L ×nm : G ×nm ǫ (S) → R satisfies the "general boundary conditions" for Morse theory (see [Cha93, Section 6.1]), and we have the Morse inequality dim H k (G ×nm ǫ (S)) ≤ q ′

  3). Hence, by (4.1) and by the Morse inequality (4.2) we infer 0 = r α=1 dim C N -1 (D nm • {q ×m α }). (4.4)

ψ

  ×m : (G ×n ǫ (S) >c ∪ {q} , G ×n ǫ (S) >c ) ֒→ (G ×nm ǫ (S) >mb , G ×nm ǫ (S) >mc ). (4.5) Claim 2. For each m ∈ K ′′ p , the iteration map in (4.5) is injective in degree (N -1) homology, i.e. ψ ×m * : C N -1 (q) ֒→ H N -1 (G ×nm ǫ (S) >mb , G ×nm ǫ (S) >mc ), ∀m ∈ K ′′ p Proof. Let us consider an arbitrary m ∈ K ′′ p . By Assumption (F1), we can choose b ′ ∈ (mb, mc) sufficiently close to mc such that the only critical value of L ×nm in the interval (b ′ , mc] is mc. By Morse theory (see e.g. [Cha93, Theorem 4.2]), the inclusion (G ×nm ǫ (S) >c ∪ {q ×m }, G ×nm ǫ (S) >c ) ֒→ (G ×nm ǫ (S) >b ′ , G ×nm ǫ (S) >mc ) is injective in homology, namely it induces the monomorphism C * (q ×m ) ֒→ H * (G ×nm ǫ (S) >b ′ , G ×nm ǫ (S) >mc ). (4.6) Now, let us examine the long exact sequence of the triple(G ×nm ǫ (S) >mb , G ×nm ǫ (S) >b ′ , G ×nm ǫ (S) >mc ),which we can write as the following exact triangle:H * (G ×nm ǫ (S) >b ′ , G ×nm ǫ (S) >mc ) T H * (G ×nm ǫ (S) >mb , G ×nm ǫ (S) >b ′ ) >mb , G ×nm ǫ (S) >mc ) o oHere, ∂ * is the boundary homomorphism which lower the grade * by one, and the other homomorphisms are simply induced by inclusions. Now, by our choice of b and b ′ , none of the q ×m α 's has critical value inside the interval (mb, b ′ ] and, by assumption (F2), all the critical values of L ×nm inside (mb, b ′ ] correspond to critical points

In[START_REF]Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards[END_REF] Theorem 7] the formula of the Poincaré polynomial contains a typo.

that either are local maxima or have Morse coindex greater than N . This, together with the Morse inequalities, implies that the group H j (G ×nm ǫ (S) >mb , G ×nm ǫ (S) >b ′ ) is trivial in degree j = N -1 and j = N . Hence, the above exact triangle implies that the diagonal homomorphism is an isomorphism in degree N -1, i.e. the inclusion induces the homology isomorphism

This isomorphism and the monomorphism in (4.6) fit in the following commutative diagram, where all the homomorphisms are induced by inclusions:

This forces the inclusion-induced homomorphism ι to be injective, i.e.

Now, by Proposition 3.5, the iteration map induces the homology isomorphism

By composing the monomorphism ι with the isomorphism ψ ×m * we obtain the claim.

For each m ∈ K ′′ p , the critical point q ×m of L ×nm is not a local maximum. In fact, C N -1 (q ×m ) is nontrivial whereas the local homology groups of a local maximum are nontrivial only in degree zero (we recall that N ≥ 2). Therefore, by the homological vanishing principle in Proposition 3.7, the homomorphism

is zero provided m ∈ K ′′ p is sufficiently big. This contradicts Claim 2, and concludes the proof of Theorem 1.1.


Marco Mazzucchelli

ON THE MULTIPLICITY OF ISOMETRY-INVARIANT GEODESICS ON PRODUCT MANIFOLDS

Keywords: Mathematics Subject Classification. 58E10, 53C22. Key words and phrases. isometry-invariant geodesics, closed geodesics, Morse theory

We prove that on any closed Riemannian manifold (M 1 × M 2 , g), with rank H 1 (M 1 ) = 0 and dim(M 2 ) ≥ 2, every isometry homotopic to the identity admits infinitely many isometry-invariant geodesics.

Introduction

On a closed Riemannian manifold (M, g) equipped with an isometry I, a natural problem consists in searching for 1-dimensional submanifolds M that are complete geodesics invariant by I. More precisely, is an I-invariant geodesic when it can be parametrized with a geodesic γ : R → of constant positive speed and there exists τ > 0 such that I(γ(t)) = γ(t + τ ) for all t ∈ R. A special instance of this definition is when I is the identity, in which case invariant geodesics are precisely closed geodesics. The study of isometry-invariant geodesics was initiated by Grove [START_REF] Grove | Condition (C) for the energy integral on certain path spaces and applications to the theory of geodesics[END_REF][START_REF]Isometry-invariant geodesics[END_REF], who established several existence and multiplicity results. Further investigations are due to, among others, Grove and Tanaka [START_REF] Grove | On the number of invariant closed geodesics[END_REF][START_REF]On the number of invariant closed geodesics[END_REF], Tanaka [START_REF] Tanaka | On the existence of infinitely many isometry-invariant geodesics[END_REF], Hingston [START_REF] Hingston | Isometry-invariant geodesics on spheres[END_REF] and Rademacher [START_REF] Rademacher | Metrics with only finitely many isometry invariant geodesics[END_REF].

While any closed Riemannian manifold possesses closed geodesics, it is not always the case that it possesses isometry-invariant geodesics. The easiest example is probably the flat torus T 2 = [0, 1] 2 /{0, 1} 2 , on which the rotation I(x, y) = (1-y, x) is an isometry without invariant geodesics. Moreover, even if we require an isometry to be isotopic to the identity (e.g. the time-1 map of a Killing vector field), it may still have only finitely many invariant geodesics. For instance, on the standard Riemannian sphere S 2 , any rotation of an angle θ around the axis joining north and south poles has only the equator as invariant geodesic unless θ is a multiple of π. This is in contrast with the case of closed geodesics: celebrated results due to Bangert [START_REF]On the existence of closed geodesics on two-spheres[END_REF], Franks [START_REF] Franks | Geodesics on S 2 and periodic points of annulus homeomorphisms[END_REF], and Hingston [START_REF]On the growth of the number of closed geodesics on the two-sphere[END_REF] imply that any Riemannian S 2 has infinitely many closed geodesics. It is a long standing conjecture in Riemannian geometry that any closed Riemannian manifold (of dimension at least 2) has infinitely many closed geodesics. In [START_REF] Tanaka | On the existence of infinitely many isometry-invariant geodesics[END_REF], Tanaka extended a celebrated result by Gromoll and Meyer [START_REF] Gromoll | Periodic geodesics on compact Riemannian manifolds[END_REF] to the setting of isometry-invariant geodesics, asserting that any isometry I of a closed simply connected Riemannian manifold (M, g) possesses infinitely many closed geodesics provided the homology of a suitable space of I-invariant curves is sufficiently rich. By a result of Vigué-Poirrier and Sullivan [START_REF] Vigué-Poirrier | The homology theory of the closed geodesic problem[END_REF], this latter assumption is verified if I is homotopic to the identity and the cohomology of M is not a truncated polynomial ring in one variable.

In this paper we shall prove the following result that complements the one of Tanaka, and extends another important closed geodesics result due to Bangert and Klingenberg [BK83, Corollary 3].

Theorem 1.1. Let (M, g) be a Riemannian manifold such that M is homeomorphic to a product M 1 × M 2 of closed manifolds with rank H 1 (M 1 ) = 0 and dim(M 2 ) ≥ 2. Then every isometry I of (M, g) that is homotopic to the identity admits infinitely many I-invariant geodesics.

The starting point in the study of this kind of results is the crucial observation due to Grove [START_REF]Isometry-invariant geodesics[END_REF]Theorem 2.4] that any isometry with only finitely many invariant geodesics does not have non-closed ones. In view of this fact, the main issue in the proof of multiplicity results consists in identifying several iterations of a same closed isometry-invariant geodesic detected as distinct critical points of an energy function. The classical tools for dealing with this problem, the iteration theory for Morse indices [START_REF] Bott | On the iteration of closed geodesics and the Sturm intersection theory[END_REF][START_REF]Index theory for symplectic paths with applications[END_REF] and local homology groups [START_REF] Gromoll | Periodic geodesics on compact Riemannian manifolds[END_REF] of closed geodesics, have been ingeniously extended to the setting of isometry-invariant geodesics by Grove and Tanaka [START_REF] Grove | On the number of invariant closed geodesics[END_REF][START_REF]On the number of invariant closed geodesics[END_REF][START_REF] Tanaka | On the existence of infinitely many isometry-invariant geodesics[END_REF]. In order to prove our main theorem, we will combine this machinery together with an extension to the isometry-invariant setting of a homological technique of Bangert and Klingenberg [START_REF] Bangert | Homology generated by iterated closed geodesics[END_REF].

Further generalizations of closed geodesics results may be possible. Specifically, remarkable results due to Bangert and Hingston [START_REF] Bangert | Closed geodesics on manifolds with infinite Abelian fundamental group[END_REF] show that every closed Riemannian manifold with infinite abelian fundamental group must have infinitely many closed geodesics. To the best of the author's knowledge, a generalization of such result to the isometry-invariant case has not been investigated yet. Indeed, even when the fundamental group has rank larger than 1, it is not clear how to conclude that there are infinitely many geodesics invariant by a general isometry homotopic to the identity. We plan to study this problem further in the future.

1.1. Organization of the paper. In Section 2 we review the variational principle, Morse indices and local homology of isometry-invariant geodesics. In Section 3, after recalling Grove and Tanaka's results on the local homology of iterated geodesics, we prove an isometry-invariant version of the homological vanishing result of Bangert and Klingenberg, and we derive an application to the multiplicity of isometryinvariant geodesics. Finally, in Section 4 we prove Theorem 1.1. 1.2. Acknowledgments. I wish to thank the anonymous referee for his or her careful reading of the manuscript, and for providing useful comments.

Preliminaries

2.1. The variational setting. Throughout this paper, (M, g) will be a closed Riemannian manifold equipped with an isometry I. Isometry-invariant geodesics can be detected by the following well-known variational principle. Consider the path space Λ(M ;

I) of all W 1,2 loc curves ζ : R → M such that I(ζ(t)) = ζ(t + 1) for all t ∈ R. We recall that a curve ζ has W 1,2
loc -regularity when it is absolutely continuous, weakly differentiable, and the function t → g( ζ(t), ζ(t)) is locally integrable. The space Λ(M ; I) is a Hilbert manifold, and the tangent space T ζ Λ(M ; I) is given by all the W 1,2 loc vector fields X along ζ such that I * (X(t)) = X(t + 1) for all t ∈ R. We can equip Λ(M ; I) with a complete Riemannian metric G given by

G(X, Y ) = 1 0 g(X(t), Y (t)) + g( Ẋ(t), Ẏ (t)) dt, ∀X, Y ∈ T ζ Λ(M ; I),
where the dots in this expression denote the covariant derivative along ζ. The energy function E : Λ(M ; I) → R is given by

E(ζ) = 1 0 g( ζ(t), ζ(t)) dt.
This function is C ∞ , and satisfies the Palais-Smale condition: any sequence

{ζ n } ⊂ Λ(M ; I) such that E(ζ n ) is uniformly bounded and G(∇E(ζ n ), ∇E(ζ n ))
→ 0 admits a subsequence converging toward a critical point of E. The critical points of E are precisely the g-geodesics γ : R → M parametrized with constant speed g( γ, γ) and such that I(γ(t)) = γ(t + 1). We refer to reader to Grove [START_REF] Grove | Condition (C) for the energy integral on certain path spaces and applications to the theory of geodesics[END_REF] for more details about the variational principle associated to isometry-invariant geodesics. Strictly speaking, any fixed point of the isometry I would be a stationary Iinvariant geodesic. However, in this paper, we will only consider geodesics with positive energy. If γ is a critical point of E, for each t 0 ∈ R the translated curve t → γ(t+t 0 ) is also a critical point of E corresponding to the same geometric curve. If E(γ) > 0, the family orb(γ) of translated curves associated to γ forms a critical orbit of E. Notice that orb(γ) ∼ = S 1 if γ is a periodic curve, otherwise orb(γ) ∼ = R.

2.2. Indices of the critical orbits of the energy. Counting the critical orbits of E with positive critical value by means of topological methods does not provide a count for the number of I-invariant geodesics: a closed I-invariant geodesic ∼ = S 1 gives rise to infinitely many critical orbits of E. Indeed, let γ : R → be a parametrization of with constant speed, basic period p ≥ 1, and such that I(γ(0)) = γ(1). For every real number k = 0 we define γ k : R → M to be the curve γ k (t) = γ(kt). All the parametrized curves γ mp+1 , where m ∈ N, belong to Λ(M ; I) and are critical points of E corresponding to the same oriented geodesic : the curve γ mp+1 | [0,1] joins γ(0) with γ(1) after winding around for m times. An essential observation due to Grove [Gro74, Theorem 2.4] claims that if I has a non-closed invariant geodesic, then it admits uncountably many invariant geodesics. This implies that, in order to study the multiplicity of I-invariant geodesics, we can assume that all of these geodesics are closed. A way to identify critical orbits of E corresponding to the same geometric curve is to look at their Morse index, a strategy that was first introduced by Bott [Bot56] in the study of closed geodesics. If γ is a critical point of E, its Morse index ind(γ) and nullity nul(γ) are defined respectively as the dimension of the negative eigenspace and the dimension reduced by 1 of the kernel of the Hessian of E at γ. All the curves in the critical orbit of γ share the same index and nullity. In [START_REF]On the number of invariant closed geodesics[END_REF][START_REF] Tanaka | On the existence of infinitely many isometry-invariant geodesics[END_REF], Grove and Tanaka developed an analogue of Bott's theory in the much harder setting of isometry invariant geodesics. For our applications, we will need the following statement.

Proposition 2.1 (Grove-Tanaka [START_REF]On the number of invariant closed geodesics[END_REF][START_REF] Tanaka | On the existence of infinitely many isometry-invariant geodesics[END_REF]). Let γ be a critical point of E that is a periodic curve of basic period p > 0. Then m -1 ind(γ mp+1 ) converges to a nonnegative number ind(γ) as m → ∞. Moreover, if ind(γ) = 0 then ind(γ mp+1 ) = 0 for all m ∈ N.

Another important index of homological nature is the local homology of E at orb(γ), defined as the relative homology group

C * (E, orb(γ)) := H * ({E < c} ∪ orb(γ), {E < c}),
where c = E(γ), and H * denotes the singular homology functor with rational coefficients. The interplay between the local homology and the Morse indices of a critical orbit can be summarized by saying that the graded group C * (E, orb(γ)) is always trivial in degree less than ind(γ) or greater than ind(γ)+nul(γ)+1. Local homology groups are the "building blocks" for the homology of the path space Λ(M ; I). More precisely, for all b > c = E(γ) such that the interval (c, b) does not contain critical values of E, the inclusion induces an injective homomorphism C * (E, orb(γ)) → H * ({E < b}, {E < c}).

Here, we also allow b to be equal to +∞, in which case {E < b} = Λ(M ; I). For a general interval [a, b] ⊂ (0, +∞] and any homological degree d, we have the Morse inequality

rank H d ({E < b}, {E < a}) ≤ orb(γ) rank C d (E, orb(γ)),
where the sum on the right-hand side runs over all the critical orbits orb(γ) such that a ≤ E(γ) < b.

Bangert-Klingenberg Lemmas for isometry-invariant geodesics

In their seminal paper [START_REF] Bangert | Homology generated by iterated closed geodesics[END_REF], Bangert and Klingenberg showed that any sufficiently iterated closed geodesic with average Morse index 0, and that is not a global minimum of the energy in his free homotopy class, cannot arise as a minimax point generated by a (relative) homology class of the free loop space. The important consequence of this result is that, whenever there is a closed geodesic with these properties that is homologically visible, the Riemannian manifold must contain infinitely many closed geodesics. The proof of this result is based on a homotopic technique introduced earlier by Bangert [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF], and further employed in different settings by Bangert-Hingston [START_REF] Bangert | Closed geodesics on manifolds with infinite Abelian fundamental group[END_REF], Hingston [START_REF]On the growth of the number of closed geodesics on the two-sphere[END_REF], Bangert [START_REF]On the existence of closed geodesics on two-spheres[END_REF], Long [START_REF] Long | Multiple periodic points of the Poincaré map of Lagrangian systems on tori[END_REF], Lu [START_REF] Lu | The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems[END_REF], and the author [START_REF]The Lagrangian Conley conjecture[END_REF][START_REF]On the multiplicity of non-iterated periodic billiard trajectories[END_REF]. In this section we apply results due to Grove and Tanaka [START_REF] Grove | On the number of invariant closed geodesics[END_REF][START_REF]On the number of invariant closed geodesics[END_REF][START_REF] Tanaka | On the existence of infinitely many isometry-invariant geodesics[END_REF] in order to establish the analogue of Bangert and Klingenberg's result in the context of closed isometry-invariant geodesics.

3.1. Local homology of iterated orbits. In the setting of Section 2, let γ be a critical point of E that is a periodic curve of basic period p ≥ 1 and average index ind(γ) = 0. Since in this paper we are looking for infinitely many I-invariant geodesics, we can assume that each orb(γ mp+1 ), where m ∈ N, is isolated in the set of critical points of E. In [GT78, Tan82], Grove and Tanaka showed that, up to isomorphism, there are only finitely many different groups in the family {C * (E, γ pm+1 ) | m ∈ N}, a statement established earlier by Gromoll and Meyer [START_REF] Gromoll | Periodic geodesics on compact Riemannian manifolds[END_REF] in case I = id. For later purposes, we need to rephrase their results, and we refer to the reader to their papers for a detailed proof.

For syntactic convenience, for every τ > 0 let us consider the Hilbert manifold

Λ τ (M ; I) = γ ∈ W 1,2 loc (R; M ) I(γ(t)) = γ(t + τ ) ∀t ∈ R ,
and the energy function

E τ : Λ τ (M ; I) → R defined by E τ (ζ) = 1 τ τ 0 g( ζ(t), ζ(t)) dt.
There is an obvious diffeomorphism of Hilbert manifolds Ψ τ : Λ τ (M ;

I) → Λ(M ; I) given by Ψ τ (ζ) = ζ τ , where ζ τ (t) = ζ(τ t). Moreover E • Ψ τ = τ 2 E τ . With this notation, for each m ∈ N, the curve γ is the critical point of E mp+1 corresponding to the critical point γ mp+1 of E. If c = E mp+1 (γ), we denote the local homology of E mp+1 at orb(γ) by C * (E mp+1 , orb(γ)) := H * ({E mp+1 < c} ∪ orb(γ), {E mp+1 < c}).
The diffeomorphism Ψ mp+1 induces a homology isomorphism of this group with the local homology C * (E, orb(γ mp+1 )).

We first consider the case in which the period p of γ is irrational, treated in [Tan82, Section 3]. For all µ ∈ N, we define the Hilbert manifold

Λ m,µ := Λ mp+1 (M ; I) ∩ Λ µp (M ; id) = ζ ∈ W 1,2 loc (R; M ) I(ζ(t -mp -1)) = ζ(t) = ζ(t + µp) ∀t ∈ R . It readily follows from the definition that Λ m,µ = Λ n,µ if m ≡ n mod µ. Moreover, for all ζ ∈ Λ m,µ ∩ C ∞ (R; M ), the function t → g( ζ(t), ζ(t)
) is constant, being both µp-periodic and (mp + 1)-periodic with (µp) -1 (mp + 1) irrational. Since smooth curves are dense in Λ m,µ , if we denote by 1 ι m,µ : Λ m,µ → Λ mp+1 (M ; I) the inclusion of the corresponding spaces, we have that E mp+1 • ι m,µ = E µp | Λ m,µ . According to the following lemma, which is a variation of [Tan82, Lemma 3.2], some of the maps ι m,µ induce an isomorphism of the corresponding local homology groups of orb(γ).

Lemma 3.1. There exists a bounded function µ : N → N such that, for all but finitely many m ∈ N, the inclusion ι m,µ(m) induces a homology isomorphism

ι m,µ(m) * : C * (E µ(m)p | Λ m,µ(m) , orb(γ)) ∼ = -→ C * (E mp+1 , orb(γ)).
Proof. Throughout this proof, let us adopt the extensive notation ind(F, x) and nul(F, x) to denote Morse index and nullity of a function F at a critical point x.

We recall that our curve γ is supposed to have average Morse index 0. By Proposition 2.1, ind(E mp+1 , γ) = 0 for all m ∈ N. Since ind

(E µp | Λ m,µ , γ) ≤ ind(E mp+1 , γ), we infer ind(E µp | Λ m,µ , γ) = ind(E mp+1 , γ) = 0, ∀µ, m ∈ N.
In the proof of [Tan82, Lemma 3.2], Tanaka showed that there exists a bounded function µ : N → N such that, for all m ∈ N large enough, the null spaces of the Hessians of E µ(m)p | Λ m,µ and E mp+1 at γ are the same. In particular, there exists

m 0 ∈ N such that nul(E µ(m)p | Λ m,µ , γ) = nul(E mp+1 , γ), ∀m ≥ m 0 .
Now, let us equip the Hilbert manifold Λ mp+1 (M ; I) with the Riemannian metric

G mp+1 (X, Y ) = mp+1 0 g(X(t), Y (t)) + g( Ẋ(t), Ẏ (t)) dt, (3.1) ∀X, Y ∈ T ζ Λ mp+1 (M ; I). (3.2)
We denote by ∇E mp+1 the gradient of the energy E mp+1 with respect to this Riemannian metric. A standard computation shows that, if ζ ∈ Λ mp+1 (M ; I) is periodic with period q, then ∇E mp+1 (ζ) is a q-periodic vector field along ζ. In particular ∇E mp+1 (ζ) belongs to the tangent space T ζ Λ m,µ for all ζ ∈ Λ m,µ . Summing up, for all m ≥ m 0 , the submanifold Λ m,µ(m) of Λ mp+1 (M ; I) is invariant under the gradient flow of Λ mp+1 (M ; I), and the Morse index and nullity of E mp+1 at γ do not change when we restrict the function to the submanifold Λ m,µ(m) . By a standard argument in Morse theory (see e.g. [Maz11a, Theorem 5.1.1]), our statement follows.

Let us now consider the case in which the period p is rational, for which the reference is [GT78, Sections 2-3]. Let a and b be relatively prime positive integers such that p = a/b. Notice that I a (γ(t)) = γ(t) for all t ∈ R, namely the geometric curve γ(R) is contained in fix(I a ). We recall that the fixed points set of an isometry is a collection of closed totally-geodesic submanifolds of (M, g). For a fixed value of m, consider three positive integers τ , α and θ with the following properties:

• τ -1 (mp + 1) is a positive integer,

• a divides α, in particular I α (γ(t)) = γ(t) for all t ∈ R, • I θ (γ(t)) = γ(t + τ ) for all t ∈ R,
• τ -1 θ(mp + 1) ≡ 1 mod α. These properties readily imply that γ ∈ Λ τ (fix(I α ); I θ ) and there is an inclusion j m,τ,α,θ : Λ τ (fix(I α );

I θ ) → Λ mp+1 (M ; I) that is a smooth embedding of Hilbert manifolds. Moreover E mp+1 • j m,τ,α,θ = E τ | Λ τ (fix(I α );I θ ) .
The following Lemma is a variation of [GT78, Lemma 2.9] together with [GT78, Proposition 3.6].

Lemma 3.2. There exist bounded functions τ : N → N, α : N → N and θ : N → N such that, for all m ∈ N, the inclusion j m,τ (m),α(m),θ(m) induces a homology isomorphism

j m,τ (m),α(m),θ(m) * : C * (E τ (m) | Λ τ (m) (fix(I α(m) );I θ(m) ) , orb(γ)) ∼ = -→ C * (E mp+1 , orb(γ)).
Proof. We proceed as in the proof of Lemma 3.1. For any given m ∈ N, if τ , α and θ are integers as above, our assumption on the average Morse index of γ implies

that ind(E τ | Λ τ (fix(I α );I θ ) , γ) = ind(E mp+1 , γ) = 0.
By [GT78, Lemma 2.9], there exist bounded functions τ : N → N, α : N → N and θ : N → N such that, for all m ∈ N, we have

nul(E τ (m) | Λ τ (m) (fix(I α(m) );I θ(m) ) , γ) = nul(E mp+1 , γ) = 0, ∀m ∈ N.
Let G mp+1 be the standard Riemannian metric on Λ mp+1 (M ; I), already introduced in (3.1), and ∇E mp+1 the gradient of the energy E mp+1 with respect to this Riemannian metric. Since fix(I α(m) ) is a collection of totally-geodesic submanifolds of M , the proof of [GT78, Proposition 3.5] shows that ∇E mp+1 (ζ) is tangent to Λ mp+1 (fix(I α(m) ); I) for all ζ ∈ Λ mp+1 (fix(I α(m) ); I). Moreover, a standard computation shows that

∇E mp+1 (ζ) is tangent to Λ τ (m) (M ; I θ(m) ) for all ζ ∈ Λ τ (m) (M ; I θ(m) ) ∩ Λ mp+1 (M ; I). Since Λ τ (m) (fix(I α(m) ); I θ(m) ) = Λ mp+1 (fix(I α(m) ); I) ∩ Λ τ (m) (M ; I θ(m) ),
we conclude that Λ τ (m) (fix(I α(m) ); I θ(m) ) is a submanifold of Λ mp+1 (M ; I) that is invariant by the gradient flow of E mp+1 , and the Morse index and nullity of γ do not change when we restrict E mp+1 to this submanifold. As in the proof of Lemma 3.1, a standard argument in Morse theory implies our statement.

3.2. Bangert's construction. In the proof of the main results of this section (Lemmas 3.3-3.4 and Proposition 3.5) we will need a homotopy constructed by Bangert in [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF], that we shall now review with our notation.

Γ m ( m-1+s m )(t) = Γ(1)( 2 1+s t), ∀t ∈ [0, 1+s 2 p], Γ m ( m-1+s m )(t) = Γ(1)(t + 1-s 2 p) ∀t ∈ [ 1+s 2 p, (m + s-3 2 )p], Γ m ( m-1+s m )(t) = Γ(2s + 4m -5 -4 p t)((m -1)p), ∀t ∈ [(m + s-3 2 )p, (m + s-5 4 )p], ( ) Γ m ( m-1+s m )(t) = Γ(s)(t + 1-s 4 p), ∀t ∈ [(m + s-5 4 )p, (m + s-1 4 )p], Γ m ( m-1+s m )(t) = Γ( 4 p t + 1 -4m)(mp), ∀t ∈ [(m + s-1 4 )p, mp], ( )
see Figure 1(d). If the original path Γ is only continuous, the curves s → Γ(s)(t) are continuous as well, but not W 1,2 loc . However, we can still define a continuous path Γ m : [a, b] → Λ mp (M ; id) in a similar way. All we have to do is modify the equations marked with ( ) in the construction. Each of those equations is of the form Γ m (s

)(t) = Γ((-1) i 4 p t + s )(t ), ∀t ∈ [t 0 , t 1 ], (3.4)
for suitable values i ∈ {1, 2}, s , s ∈ [0, 1], t ∈ [0, mp], and an interval [t 0 , t 1 ] ⊂ [0, mp]. Let δ = δ(Γ) > 0 be such that, for all r ∈ R and r 0 , r 1 ∈ [0, 1] with |r 0 -r 1 | ≤ 4 p δ, the distance between the points Γ(r 0 )(r) and Γ(r 1 )(r) is less than the injectivity radius of (M, g). We set λ : [t 0 , t 1 ] → M to be the piecewise smooth curve such that

• t → g( λ(t), λ(t)) is constant, • for each non-negative integer k such that (k + 1)δ < |t 1 -t 0 |, the curve λ| [t0+kδ,t0+ (k+1)
δ] is the (unique) length-minimizing geodesic joining the points Γ((-1) i 4 p (t 0 + kδ) + s )(t ) and Γ((-1) i 4 p (t 0 + (k + 1)δ) + s )(t ). • if k is the maximal non-negative integer such that kδ < |t 1 -t 0 |, the curve λ| [t0+kδ,t1] is the (unique) length-minimizing geodesic joining the points Γ((-1) i 4 p (t 0 + kδ) + s )(t ) and Γ((-1) i 4 p t 1 + s )(t ). In the definition of Γ m , we replace equation (3.4) with

Γ m (s )(t) = λ(t), ∀t ∈ [t 0 , t 1 ].
In this way, Γ → Γ m defines a continuous map of the form

C 0 ([a, b]; Λ p (M ; id)) → C 0 ([a, b]; Λ mp (M ; id)).
Notice that, if we see the path Γ as a path in Λ mp (M ; id) via the inclusion Λ p (M ; id) → Λ mp (M ; id), then Γ is homotopic to Γ m with fixed endpoints. A possible homotopy h Γ : [0, 1] × [a, b] → Λ mp (M ; id) can be defined by setting

h Γ (r, s) =    Γ(s) if s ∈ [r, 1], (Γ| [0,r] ) m (s) if s ∈ [0, r].
(3.5)

The main property of this construction is that, by taking m large, the energy of each mp-periodic curve Γ m (s) can be made almost as small as the energy of Γ(0) and Γ(1). More precisely where C Γ > 0 is a constant depending continuously on Γ, but not on m.

E mp (Γ m (s)) = 1 mp mp 0 g( d dt Γ m (s)(t), d dt Γ m (s)(t)) dt ≤ 1 mp (m -2) max s ∈{0,1} p 0 g( d dt Γ(s )(t), d dt Γ(s )(t)) dt + CΓ < max {E p (Γ(0)), E p (Γ(1))} + CΓ mp , (3.6) s Γ(0) Γ(s) Γ(1) Γ(s)(0) Γ(s)(p) Γ(s)(mp) Γ(0)(p) Γ(0)(mp) Γ(1)(0) Γ(1)(kp) Γ(1)(mp) Γ(0)(mp) (a) Γ(1)(p) Γ(s)((k + 1)p) Γ(s)(kp) Γ(0)((k + 1)p) Γ(1)(0) Γ(1)((m -1)p) Γ(1)(mp) Γ(1)(p) Γ(s)(mp) Γ(s)((m -1)p) (b) (c) (d)
3.3. Bangert-Klingenberg lemmas. Let K ⊂ N be an infinite subset, p, p 0 ∈ N, and Ω a Hilbert manifold contained in

Λ p0 (M ; id) ∩ m∈K Λ mp+1 (M ; I)
as a Hilbert submanifold of each space involved in the intersection. Assume also that the energy functions E p0 and E mp+1 , for all m ∈ K, coincide on Ω, i.e.

E p0 (ζ) = 1 p 0 p0 0 g( ζ(t), ζ(t)) dt = 1 mp + 1 mp+1 0 g( ζ(t), ζ(t)) dt = E mp+1 (ζ), ∀ζ ∈ Ω, m ∈ K.
Examples of such Ω's are the manifolds Λ m0,µ0 and Λ τ0 (fix(I α0 ); I θ0 ) for suitable value of the integer parameters and for suitable K, see Section 3.1.

The next statements in this section were originally established by Bangert and Klingenberg [START_REF] Bangert | Homology generated by iterated closed geodesics[END_REF] in the special case I = id. Lemma 3.3. Consider c > 0, and let Ω be the union of the connected components of Ω having non-empty intersection with the sub-level {E p0 | Ω < c}. For each m ∈ K, consider the homomorphism

ι m * : H * (Ω , {E p0 | Ω < c}) → H * (Λ mp+1 (M ; I), {E mp+1 < c}) induced by the inclusion. Then, for each h ∈ H * (Ω , {E p0 | Ω < c}) we have ι m * (h) = 0 provided m is large enough.
Proof. The statement is straightforward in homological degree zero, since the group H 0 (Ω , {E p0 | Ω < c}) is trivial. Now, consider a non-zero h ∈ H d (Ω , {E p0 | Ω < c}) for some d ≥ 1 (if it exists). Let µ be a relative cycle representing h. We denote by Σ j (µ) the set of singular simplexes that are j-dimensional faces of the simplexes in the chain µ.

By modifying µ within the same homology class h if necessary, we can assume that all the 0-simplexes in Σ 0 (µ) are already contained in the sublevel {E p0 | Ω < c}. We set m 0 := 1 and, for all κ ∈ Σ 0 (µ), we define the maps h κ , h κ : [0, 1] × ∆ 0 → Ω as h κ (t, z) = h κ (t, z) := κ(z). Here, ∆ 0 is the 0-dimensional standard simplex, i.e. a point.

For each degree j ∈ {1, ..., d} we will find m j ∈ N and, for each σ ∈ Σ j (µ), two homotopies

h σ : [0, 1] × ∆ j → Λ mj p0 (M ; id) (3.7) h σ : [0, 1] × ∆ j → Ω (3.8)
with the following properties.

(i) m j-1 divides m j , (ii) h σ (0, •) = h σ (0, •) = σ, (iii) h σ (s, z)(0) = h σ (s, z)(0) for all (s, z) ∈ [0, 1] × ∆ j , (iv) E mj p0 (h σ (1, z)) < c for all z ∈ ∆ j , (v) h σ (s, f k (z)) = h σ•f k (s, z) and h σ (s, f k (z)) = h σ•f k (s, z
) for all k ∈ {0, ..., j}, where f k : ∆ j-1 → ∆ j is the standard affine map onto the k-th face of ∆ j .

Let us assume that we have such a family of homotopies. Let m ∈ K be a (large) integer that we will fix later, and let m be the maximal multiple of m d such that m p 0 is less than of equal to m p + 1. For each j ≥ 1 and singular simplex σ ∈ Σ j (µ) we define the homotopy

h σ : [0, 1] × ∆ j → Λ m p+1 (M ; I) by h σ (s, z)(t) =    h σ (s, z)(t) t ∈ [0, m p 0 ], h σ (s, z)(t) t ∈ [m p 0 , m p + 1].
(3.9)

Notice that h σ (s, z) is a well-defined curve in Λ m p+1 (M ; I). Indeed

h σ (s, z)(m p 0 ) = h σ (s, z)(0) = h σ (s, z)(0) = h σ (s, z)(m p 0 ),
I(h σ (s, z)(0)) = I(h σ (s, z)(0)) = h σ (s, z)(m p + 1),
which proves that h σ (s, z) is an I-invariant curve with time-shift m p + 1. The energy of h σ (1, z) can be estimated as follows:

E m p+1 (h σ (1, z)) = 1 m p + 1 m p0 0 g( d dt h σ (1, z)(t), d dt h σ (1, z)(t)) dt + m p+1 m p0 g( d dt h σ (1, z)(t), d dt h σ (1, z)(t)) dt ≤ 1 m p + 1 m p 0 E m p0 (h σ (1, z)) + m d p 0 E m d p0 (h σ (1, z)) = 1 m p + 1 m p 0 E mj p0 (h σ (1, z)) ≤c-σ +m d p 0 E m d p0 (h σ (1, z)) ≤cσ ≤ c -σ + m d p 0 c σ m p + 1 ,
where σ > 0 is a quantity given by condition (iv), and

c σ := max z∈∆ j E m d p0 (h σ (1, z))
. Therefore, if we choose m ∈ K large enough, for each j ≥ 1 and singular simplex σ ∈ Σ j (µ), we have

E m p+1 (h σ (1, z)) < c.
By the homotopy invariance property for representatives of relative homology classes (see [BK83, Lemma 1] or [Maz11a, page 146]), the existence of the family of homotopies defined in (3.9) implies that µ, seen as a relative chain in the pair (Λ m p+1 (M ; I), {E m p+1 < c}), is homologous to a relative chain contained in the sub level {E m p+1 < c}. In particular [µ] = 0 in H d (Λ m p+1 (M ; I), {E m p+1 < c}).

In order to complete the proof we only have to find suitable integers m j and construct the homotopies (3.7) and (3.8) satisfying properties (i-v). We do this inductively on the homological degree of the involved singular simplexes, starting in degree 1 and going up with the dimension.

Let σ ∈ Σ 1 (µ). Notice that ∆ 1 = [0, 1] and Ω ⊂ Λ p0 (M ; id). Thus, let us consider σ as a continuous path of the form σ : [0, 1] → Λ p0 (M ; id). By our assumptions on the elements of Σ 0 (µ), we have E p0 (σ(z)) < c for z = 0 and z = 1. Let m be a positive integer that we will fix later, and consider the continuous path

σ m : [0, 1] → Λ mp0 (M ; id)
obtained by applying Bangert's construction of Section 3.2 to σ. We define the map h σ : [0, 1] × ∆ 1 → Λ mp0 (M ; id) to be a homotopy as in equation (3.5), i.e.

h σ (s, z) =    (σ| [0,s] ) m (z) if z ∈ [0, s], σ(z) if z ∈ [s, 1].
In particular h σ (0,

•) = σ, h σ (1, •) = σ m , and h σ (s, z) = σ(z) for all s ∈ [0, 1] and z ∈ ∂∆ 1 = {0, 1}. We define h σ : [0, 1] × ∆ 1 → Ω to be the unique map such that h σ (s, z)| [0,p0] = h σ (s, z)| [0,p0] , ∀s ∈ [0, 1], z ∈ ∆ 1 .
By an estimate as in (3.6), if we fix m ∈ N large enough, for all σ ∈ Σ 1 (µ) and z ∈ ∆ 1 we obtain E mp0 (σ m (z)) < c. Thus, we set m 1 := m. Notice that h σ , h σ and m 1 satisfy assumptions (i-v) listed above when j = 1. Now, let us proceed iteratively with the construction: assuming we are done up to degree j -1, we show how to make the next step for j-simplexes in Σ j (µ). We consider m ∈ N that is a (large) multiple of m j-1 , and we will fix it later. Up to a minor modification in the previous steps, we can assume that, for each i < j, κ ∈ Σ i (µ) and s ∈ [ 1 2 , 1], we have h κ (s,

•) = h κ (1, •) and h κ (s, •) = h κ (1, •). Let σ ∈ Σ j (µ)
. We begin by putting together the homotopies of the faces of σ, in such a way that we obtain a continuous map

h ∂σ : [0, 1] × ∂∆ j → Λ mp0 (M ; id).
Notice that E mp0 (h ∂σ (s, z)) < c for all s ∈ [0, 1] and z ∈ ∂∆ j . Moreover h ∂σ (s,

•) = h ∂σ (1, •) for all s ∈ [ 1 2 , 1]. Now consider a retraction r : [0, 1 2 ] × ∆ j → ([0, 1 2 ] × ∂∆ j ) ∪ ({0} × ∆ j ). We define h σ : [0, 1 2 ] × ∆ j → Λ mj-1p0 (M ; id) ⊂ Λ mp0 (M ; id) by h σ := h ∂σ • r. Set σ := h σ ( 1 2 , •
), and see it as a map of the form σ : ∆ j → Λ mj-1p0 (M ; id).

As in [Lon00, page 461], let L ⊆ R j be the 1-dimensional vector subspace generated by the vector pointing to the barycenter of standard j-simplex ∆ j ⊂ R j . For each s ∈ [0, 1] we denote by s∆ j the rescaled j-simplex given by {sz | z ∈ ∆ j }. For each z ∈ s∆ j , we define [a(s, z), b(s, z)] to be the maximum segment inside s∆ j that contains z and is parallel to L. Notice that a and b are continuous functions on their domains. We define the other piece of homotopy

h σ : [ 1 2 , 1] × ∆ j → Λ mp0 (M ; id) by h σ (s, z) =    (σ| [a(2s-1,z),b(2s-1,z)] ) m (z) if z ∈ (2s -1)∆ j , σ(z) if z ∈ (2s -1)∆ j .
Basically here we are piecing together Bangert's homotopies (described in Section 3.2) of each path σ|

[a(1,z),b(1,z)] : [a(1, z), b(1, z)] → Λ mj-1p0 (M ; id). We define h σ : [0, 1] × ∆ 1 → Ω to be the unique map such that h σ (s, z)| [0,p0] = h σ (s, z)| [0,p0] , ∀s ∈ [0, 1], z ∈ ∆ j .
If we fix m ∈ N to be a sufficiently large multiple of m j-1 , an estimate as in (3.6) implies that E mp0 (h σ (1, z)) < c for all σ ∈ Σ j (µ) and z ∈ ∆ j . We set m j := m. As for the case in degree 1, we have that h σ , h σ and m j satisfy assumptions (i-v) listed above.

From Lemma 3.3 we can infer an analogous statement concerning the local homology of periodic I-invariant geodesics.

Lemma 3.4. Let γ be a critical point of E that is periodic of basic period p ≥ 1. Assume that there exist a degree d ≥ 2 and an infinite set K ⊂ N such that, for all m ∈ K, the local homology C d (E, orb(γ mp+1 )) is non-trivial. Then, there are arbitrarily large m ∈ K such that the homomorphisms

j m * : C d (E, orb(γ mp+1 )) → H d (Λ(M ; I), {E < E(γ mp+1
)}) induced by the inclusion are not injective.

Proof. Since the local homology C d (E, orb(γ mp+1 )) is non-trivial, we infer that ind(γ mp+1 ) ≤ d for all m ∈ K, and therefore ind(γ) = 0. Thus we can apply the results of Section 3.1. Let us employ the (equivalent) variational setting introduced there. Our assumption on the local homology can be rephrased by saying that C d (E mp+1 , orb(γ)) is non-trivial for all m ∈ K. Let us consider separately the cases in which the period p of γ is irrational or rational.

If p is irrational, let us consider the bounded function µ : N → N of Lemma 3.1. By the pigeonhole principle we can find m 0 , µ 0 ∈ N and an infinite subset K ⊂ K such that m ≡ m 0 mod µ 0 and µ(m) = µ 0 for all m ∈ K . As we remarked right before Lemma 3.1, we have that Λ m,µ0 = Λ m0,µ0 ⊂ Λ mp+1 (M ; I), and E µ0p = E mp+1 on this space. Lemma 3.1 implies that the inclusion induces an isomorphism

C * (E µ0p , orb(γ)) ∼ = -→ C * (E mp+1 , orb(γ)), ∀m ∈ K .
We set p 0 := µ 0 p, and recall that every ζ ∈ Λ m0,µ0 is a p 0 -periodic curve. We also set Ω := Λ m0,µ0 . In the other case, when p is rational, we apply the pigeonhole principle to the bounded functions τ, α, θ : N → N of Lemma 3.2. We find τ 0 , α 0 , θ 0 ∈ N and an infinite subset K ⊂ K such that τ (m) = τ 0 , α(m) = α 0 and θ(m) = θ 0 for all m ∈ K . We recall that, for all m ∈ K , we have that Λ τ0 (fix(I α0 ); I θ0 ) ⊂ Λ mp+1 (M ; I), and the functions E τ0 and E mp+1 coincide on Λ τ0 (fix(I α0 ); I θ0 ). Lemma 3.2 implies that the inclusion induces an isomorphism

C * (E τ0 | Λ τ 0 (fix(I α 0 );I θ 0 ) , orb(γ)) ∼ = -→ C * (E mp+1 , orb(γ)), ∀m ∈ K .
We set p 0 := τ 0 α 0 . As before, every ζ ∈ Λ τ0 (fix(I α0 ); I θ0 ) is a p 0 -periodic curve, and we set Ω := Λ τ0 (fix(I α0 ); I θ0 ).

In either case (p rational or irrational), Ω satisfies the assumptions required in Lemma 3.3. Let Ω be the union of the connected components of Ω that intersect the sub-level {E p0 < c}, where c = E p0 (γ). Notice that the orbit of γ is not a local minimum of the energy E p0 | Ω , otherwise we would have

C d (E p0 | Ω , orb(γ)) = H d (orb(γ)) H d (S 1 ) = 0, whereas C d (E p0 | Ω , orb(γ)
) is non-trivial by our assumptions. Therefore orb(γ) is contained in Ω , and we have the following commutative diagram where all the homomorphisms are induced by inclusions.

C d (E p0 | Ω , orb(γ)) ∼ = / / C d (E mp+1 , orb(γ)) jm * H d (Ω , {E p0 | Ω < c}) ι m * / / H d (Λ mp+1 (M ; I), {E mp+1 < c})
Our statement follows from Lemma 3.3.

After these preliminaries, we can now state and prove the main result of this section.

Proposition 3.5. Let γ be a critical point of E that is periodic of basic period p ≥ 1. If there exists a degree d ≥ 2 such that, for infinitely many m ∈ N, the local homology C d (E, orb(γ mp+1 )) is non-trivial, then the Riemannian manifold (M, g) contains infinitely many I-invariant geodesics.

Proof. We prove the proposition by contradiction, assuming that there are only finitely many I-invariant geodesics 1 , ..., r . By [Gro74, Theorem 2.4], all these geodesics must be closed. Let γ i : R → M be a parametrization of i with constant speed, period p i ≥ 1, and such that I(γ i (t)) = γ i (t + 1) for all t ∈ R. Thus the critical orbits of the energy E corresponding to i are all the orb(γ mpi+1 i ), for m ∈ N. We set

D i = d ∈ N C d (E, orb(γ mpi+1 i )) = 0 for infinitely many m ∈ N , ∀i = 1, ..., r.
Up to renaming the I-invariant geodesics, we can assume that D i = ∅ for i = 1, ..., s, whereas D i = ∅ for all i > s. Notice that, by the assumptions of the theorem, s ≥ 1. We also set

d i = max D i , ∀i = 1, ..., s.
Notice that d i is finite. Indeed, the fact that the local homology C * (E, orb(γ mpi+1 i ) is non-trivial in a fixed degree for arbitrarily large m implies that ind(γ i ) = 0. Therefore, by Proposition 2.1, ind(γ mpi+1 i ) = 0 for all m ∈ N, and we conclude that

d i ≤ nul(γ mpi+1 i ) + 1 ≤ 2 dim(M ).
Let i be such that d i = max{d 1 , ..., d s }. In particular d i ≥ d ≥ 2, where d is the integer in the statement of the proposition. We reset γ := γ i , p := p i and d := d i . Notice that this new γ still satisfies the assumptions of the proposition (with respect to the new p and d). Fix m ∈ N sufficiently large so that, for all j > s, we have the following:

• if ind(γ j ) = 0 then C * (E, orb(γ mpj +1 j )) is trivial for all integers m ≥ m, • if ind(γ j ) > 0 then ind(γ mpj +1 j ) > d + 2 for all integers m ≥ m.
We set e := max E(γ mpj +1 j

) s < j ≤ r .

For all j > s, the inequality E(γ mpj +1 j

) > e implies m > m. Choose another large enough m ≥ m such that E(γ mp+1 ) > e for all integers m ≥ m . Now, fix an integer m ≥ m , and set c = c(m) := E(γ mp+1 ). Take = (m) > 0 small enough so that the interval (c, c + ] does not contain critical points of E. In particular, the inclusion induces an injective homomorphism

C * (E, orb(γ mp+1 )) → H * ({E < c + }, {E < c}). ( 3
{E < c} ⊂ {E < c} ∪ orb(γ mp+1 ) ⊂ {E < c + } we infer that H d+1 ({E < c + }, {E < c} ∪ orb(γ mp+1
)) is trivial. This, together with (3.12) and the long exact sequence of the triple

{E < c} ∪ orb(γ mp+1 ) ⊂ {E < c + } ⊂ Λ(M ; I),
implies that H d+1 (Λ(M ; I), {E < c} ∪ orb(γ mp+1 )) is trivial. Finally this, together with the long exact sequence of the triple

{E < c} ⊂ {E < c} ∪ orb(γ mp+1 ) ⊂ Λ(M ; I),
implies that the inclusion induces an injective homomorphism

C d (E, orb(γ mp+1 )) → H d (Λ(M ; I), {E < E(γ mp+1 )}).
Since this is true for any integer m ≥ m , it contradicts the assertion of Lemma 3.4.

Proof of Theorem 1.1

The proof of Theorem 1.1 goes along the following lines. By applying Morse theory to the energy function E, we find infinitely many critical points that correspond to either infinitely many (geometrically distinct) I-invariant geodesics, or to a single I-invariant geodesic satisfying the assumptions of Proposition 3.5. However, this latter proposition implies that there are infinitely many I-invariant geodesics. We go over this argument in the following.

Since rank H 1 (M 1 ) = 0, we can find a smooth 1-cycle s such that s m = 0 in H 1 (M 1 ), ∀m ∈ N. (4.1)

Let σ : R → M 1 be a 1-periodic smooth curve such that σ| [0,1] is a parametrization of s. For m ∈ N, we define a smooth map Σ m : M 2 → Λ(M ; id) by Σ m (q)(t) = (σ m (t), q). We denote by C m the connected component of the free loop space Λ(M ; id) containing Σ m (M 2 ). By (4.1) we have that

C m ∩ C n = ∅ if m = n.
Consider a smooth homotopy I t : M → M , where t ∈ [0, 1], such that I 0 = id and I 1 = I. This homotopy induces a continuous map ι : Λ(M ; id) → Λ(M ; I) in the following way. For all ζ ∈ Λ(M ; id) and t ∈ [0, 1] we set We can assume that M 2 is an orientable manifold. If this is not true, we proceed as follows. We replace M 2 by its orientable 2-fold covering M 2 . Our manifold M admits a 2-fold covering M that is homeomorphic to the 2-fold covering M 1 × M 2 of M 1 × M 2 . It suffices to lift the Riemannian metric g and the isometry I to M , and carry over the proof of Theorem 1.1 for ( M , g). Indeed, infinitely many I-invariant geodesics on M project down to infinitely many I-invariant geodesics on M .

By our orientability assumption on M 2 , we have that H d (M 2 ) is non-trivial for d = dim(M 2 ). Thus H d (D m ) is non-trivial as well. Since we are looking for infinitely many I-invariant geodesics, we can assume that all the critical orbits of E are isolated (otherwise we are already done). By the Morse inequalities, there exists a critical point γ m contained in the connected component D m and such that the local homology C d (E, orb(γ m )) is non-trivial.

Summing up, we have found infinitely many critical orbits orb(γ m ), where m ∈ N, whose local homology is non-trivial in the fixed degree d = dim(M 2 ) ≥ 2. If these critical orbits correspond to only finitely many (geometrically distinct) Iinvariant geodesics, then there must be a critical point γ of E which is periodic with period p ≥ 1 and such that, for infinitely many m ∈ N, the critical orbit orb(γ m ) is the critical orbit orb(γ µp+1 ) for some µ = µ(m). In particular γ satisfies the assumptions of Proposition 3.5, and therefore (M, g) must have infinitely many I-invariant geodesics.

  Fix a period p ∈ N, and consider a smooth path Γ : [a, b] → Λ p (M ; id), i.e. each curve t → Γ(s)(t) is p-periodic. For each m ∈ N, we define an associated continuous path Γ m : [a, b] → Λ mp (M ; id) (3.3) in the following way. For the sake of simplicity, let us assume that [a, b] = [0, 1]. For each s ∈ [0, 1], we set see Figure 1(c). Finally, for each s ∈ [0, 1], we set

Figure 1 .

 1 Figure 1. (a) Path Γ : [0, 1] → Λ p (M ; id); the shaded square contains the portion of each curve t → Γ(s)(t) for t ∈ [0, mp]. (b-d) Curve t → Γ m (s )(t) for t ∈ [0, mp] and suitable values of s ; the little horizontal arrows show the direction in which the curve is pulled as s grows.

  and thus h σ (s, z)| [0,m p+1] is a continuous curve obtained by joining the W 1,2 curves h σ (s, z)| [0,m p0] and h σ (s, z)| [m p0,m p+1] . Moreover, since h σ (s, z) ∈ Ω, we have I(h σ (s, z)(0)) = h σ (s, z)(m p + 1), and therefore

  if t ∈ [0, 1/2], I 2t-1 (ζ(0)) if t ∈ [1/2, 1],and we extend ι(ζ) to the whole real line in such a way thatI(ι(ζ)(t)) = ι(ζ)(t + 1), ∀t ∈ R.It is easy to see that ι is a homotopy equivalence (see [Gro73, Lemma 3.6]). If we denote by D m the connected component of Λ(M ; I) containing ι(C m ), we have that D m ∩ D n = ∅. Let ev : D m → M 2 be the evaluation map given by ev(ζ) = ζ 2 (0) for all ζ ∈ D m , where ζ 2 denotes the M 2 -factor of the curve ζ : R → M 1 ×M 2 . Notice that ev•ι•Σ m is the identity on M 2 . Therefore ev is a left inverse for ι • Σ m , and this latter map induces an injective homomorphism (ι • Σ m ) * : H * (M 2 ) → H * (D m ).

  .10) Moreover, since E does not have any critical orbit with critical value larger than e and non-zero local homology in degree d + 1, the Morse inequalities imply

	H d+1 {E < c + }, {E < c} = 0,	(3.11)
	H d+1 Λ(M ; I), {E < c + } = 0.	(3.12)
	By (3.10), (3.11), and the long exact sequence of the triple

In some sense, the map ι m,µ , as well as the map j µ,τ,α,θ introduced below before Lemma 3.2, plays the same role as the iteration map in the theory of closed geodesics.

Γ m ( s m )(t) = Γ(s)( 2t 2-s ), ∀t ∈ [0, (1 -s 2 )p], Γ m ( s m )(t) = Γ(4 -s -4 p t)(p), ∀t ∈ [(1 -s 2 )p, (1 -s 4 )p], ( ) Γ m ( s m )(t) = Γ(0)(t + s 4 p), ∀t ∈ [(1 -s 4 )p, (m -s 4 )p], Γ m ( s m )(t) = Γ( 4 p t + s -4m)(mp), ∀t ∈ [(m -s 4 )p, mp], ( ) see Figure 1(b). For each s ∈ [0, 1] and k ∈ {1, ..., m -2}, we set Γ m ( k+s m )(t) = Γ(1)(2t), ∀t ∈ [0, p 2 ], Γ m ( k+s m )(t) = Γ(1)(t -p 2 ) ∀t ∈ [ p 2 , (k -1 2 )p], Γ m ( k+s m )(t) = Γ(4k -1 -4 p t)(kp), ∀t ∈ [(k -1 2 )p, (k -1+s 4 )p], ( ) Γ m ( k+s m )(t) = Γ(s)(t -1+s 4 p), ∀t ∈ [(k -1+s 4 )p, (k + 3-s 4 )p], Γ m ( k+s m )(t) = Γ(4k + 3 -4 p t)((k + 1)p), ∀t ∈ [(k + 3-s 4 )p, (k + 3 4 )p], ( ) Γ m ( k+s m )(t) = Γ(0)(t + 1 4 p), ∀t ∈ [(k +34 )p, (m -1 4 )p], Γ m ( k+s m )(t) = Γ( 4 p t -4m + 1)(mp), ∀t ∈ [(m -1 4 )p, mp], ( )