Leonardo Macarini
email: leonardo@impa.br

Marco Mazzucchelli
email: marco.mazzucchelli@ens-lyon.fr

ISOMETRY-INVARIANT GEODESICS AND THE FUNDAMENTAL GROUP, II

Keywords: Mathematics Subject Classification. 58E10, 53C22 Isometry-invariant geodesics, closed geodesics, Morse theory

We show that on a closed Riemannian manifold with fundamental group isomorphic to Z, other than the circle, every isometry that is homotopic to the identity possesses infinitely many invariant geodesics. This completes a recent result in [Maz15] of the second author.

Introduction

In this paper, we complete the study began in [START_REF]Isometry-invariant geodesics and the fundamental group[END_REF] of isometry-invariant geodesics on closed Riemannian manifolds with infinite abelian fundamental group. Isometry-invariant geodesics play the role of closed geodesics in a Riemannian setting with symmetry. Given an isometry I of a closed connected Riemannian manifold (M, g), a geodesic γ : R M is called I-invariant if I(γ(t)) = γ(t + τ ) for some positive τ > 0 and for all t ∈ R. Intuitively, these curves should be the closed geodesics of the possibly singular quotient M/I.

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] in the 1970s. The problem admits a variational description, which generalizes the one of closed geodesics: isometry-invariant geodesics are the critical points of an energy function defined on a space of invariant paths. If the considered isometry is homotopic to the identity, this space of invariant paths is homotopy equivalent to the free loop space. This may induce someone to naively conjecture that all multiplicity results for closed geodesics remain true for isometry-invariant geodesics, provided the isometry is homotopic to the identity. A quite sophisticated argument 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]The isometry-invariant geodesics problem: Closed and open, Geometry and Topology[END_REF] shows that this is the case for Gromoll and Meyer's theorem: every closed Riemannian manifold with non-monogenic rational cohomology admits infinitely many isometry-invariant geodesics. This result is proved by cleverly exploiting the richness of the homology of the free loop space. However, there are multiplicity results, such as the existence of infinitely many closed geodesics on Riemannian 2-spheres [START_REF]On the existence of closed geodesics on two-spheres[END_REF][START_REF] Franks | Geodesics on S 2 and periodic points of annulus homeomorphisms[END_REF][START_REF] Hingston | On the growth of the number of closed geodesics on the two-sphere[END_REF], whose proofs need arguments that go beyond the abundance of the homology of the free loop space. These results may fail for isometry-invariant geodesics: for instance, a non-trivial rotation on a round 2-sphere has only one invariant geodesic.

A famous theorem of Bangert and Hingston implies that closed Riemannian manifolds with infinite abelian fundamental group always possess infinitely many closed geodesics. As in the case of the 2-sphere, the proof of this result combines general minimax techniques from Morse theory with the investigation of homotopy groups of the free loop space and more ad-hoc arguments. In a previous paper of the second author [START_REF]Isometry-invariant geodesics and the fundamental group[END_REF], Bangert and Hingston's theorem was extended to the isometry-invariant setting under the extra assumption that the infinite abelian fundamental group was not cyclic. The main result of this paper completes the extension of the result.

Theorem 1.1. Let (M, g) be a closed connected Riemannian manifold different from the circle and with fundamental group isomorphic to Z. Every isometry of (M, g) that is homotopic to the identity possesses infinitely many invariant geodesics.

The proof of this result is not a mere generalization of Bangert and Hingston's one, but requires crucial new ingredients. Actually, Theorem 1.1 will be a corollary of a more general result, Theorem 5.1. This latter statement asserts the existence of infinitely many isometry-invariant geodesics, provided the space of invariant curves has infinitely many connected components with enough non-trivial homotopy in a fixed positive degree. The proof will be based on the technical Lemma 4.1 asserting that, for any positive degree d, a sufficiently iterated periodic invariant geodesic is not a d-dimensional mountain pass. We believe that this lemma has independent interest, and for instance a version of it for the Lagrangian free-period action functional (see [START_REF] Abbondandolo | Lectures on the free period Lagrangian action functional[END_REF]) might find application to the multiplicity problem for periodic orbits of Tonelli Lagrangian systems with prescribed energy.

1.1. Organization of the paper. In Section 2, we recall the variational setting of the energy function for isometry-invariant geodesics, and we quote the results of Grove and Tanaka that describe the Morse theoretic properties of iterated periodic isometry-invariant geodesics. In Section 3 we introduce a finite dimensional reduction of the space of invariant curves by means of Morse's broken geodesics approximations, and we focus on the properties of this reduction that will be needed later on. Section 4, which is the core of the paper, is devoted to the proof of the above mentioned main technical lemma. Finally, in Section 5, we prove the main result of the paper, Theorem 5.1, and carry over the proof of Theorem 1.1 as a corollary of it. 1.2. Acknowledgements. This project began during a research stay of the authors at IMPA (Rio de Janeiro, Brazil). The authors thank the Brazilian-French Network in Mathematics, as well as Henrique Bursztyn, for providing financial support, and IMPA for its stimulating working environment. The second author also acknowledges support by the ANR projects WKBHJ (ANR-12-BS01-0020) and COSPIN (ANR-13-JS01-0008-01).

Preliminaries

2.1. The space of isometry-invariant curves. Let I be an isometry of a closed connected Riemannian manifold (M, g). We consider the following space of curves on which I acts as a translation of time τ > 0

Λ τ (M ; I) := ζ ∈ W 1,2 loc (R; M ) I(ζ(t)) = ζ(t + τ ) ∀t ∈ R . We denote by E τ : Λ τ (M ; I) → R the energy function E τ (ζ) = 1 τ τ 0 g ζ(t) ( ζ(t), ζ(t)) dt,
whose critical points are precisely those smooth geodesics of (M, g) that are contained in Λ τ (M ; I), that is, the I-invariant geodesics of (M, g) with a suitable parametrization. It is well known that, with the usual W 1,2 Riemannian metric on path spaces, Λ τ (M ; I) is a complete Hilbert-Riemannian manifold, and E τ satisfies the Palais-Smale condition. We refer the reader to [START_REF] Grove | Condition (C) for the energy integral on certain path spaces and applications to the theory of geodesics[END_REF] for the background on this functional setting. Since any reparametrization with constant speed of a geodesic is still a geodesic, the particular value τ is not conceptually relevant here. We will often set τ = 1 and, in such case, omit τ from the notation.

The real line R acts on Λ(M ; I) by translation, i.e. t • ζ = ζ(t + •) for all t ∈ R and ζ ∈ Λ(M ; I). The energy function E is invariant by this action, and therefore its critical points come in orbits. An open I-invariant geodesic corresponds to a unique critical orbit of E, whereas a p-periodic I-invariant geodesic γ with positive energy corresponds to a countable sequence of embedded critical circles {orb(γ mp+1 ) | m ∈ N}, where γ mp+1 (t) = γ((mp + 1)t). Notice that there may also be uninteresting critical points of E: the fixed points of I, which come in totally geodesic submanifolds of (M, g). As critical points of E, these stationary curves have zero critical value and, by compactness, are contained in finitely many connected components of Λ(M ; I). In this paper, as usual, we will only be interested in I-invariant geodesics with positive energy, and we will consider two critical orbits orb(γ 1 ) and orb(γ 2 ) of the energy E as distinct I-invariant geodesics if and only if γ 1 (R) = γ 2 (R), that is, if and only if they define different immersed submanifolds of (M, g).

2.2.

Periodic isometry-invariant geodesics. By a classical theorem of Grove [Gro74, Thm. 2.4], the closure of the critical orbit of an open I-invariant geodesic contains uncountably many other critical orbits of E. Therefore, for the study of multiplicity results, we will always assume that all I-invariant geodesics are periodic curves.

Consider the sequence of critical circles {orb(γ mp+1 ) | m ∈ N} of the energy function E associated to an I-invariant geodesic of minimal period p > 0. Notice that p ≥ 1. Indeed, if p ∈ (0, 1), there exists τ ∈ (0, p] such that I(γ(t)) = γ(t+τ ) for all t ∈ R, but this is not possible since the critical circle of the curve γ τ (t) := γ(τ t) does not belong to {orb(γ mp+1 ) | m ∈ N}. In the following, we wish to summarize the Morse-theoretic properties of the sequence {orb(γ mp+1 ) | m ∈ N}. We denote by ind(E, γ) and nul(E, γ) the Morse index and the nullity of E at γ. The following lemma was proved by Grove and Tanaka [GT78, Lemma 2.8] under the assumption that the isometry I has finite order, and extended to the general case by Tanaka [Tan82, Lemma 1.8].

Lemma 2.1 (Grove-Tanaka). Either ind(E, γ mp+1 ) = 0 for all non-negative integers m, or ind(E, γ mp+1 ) → ∞ as m → ∞.

Notice that, from the point of view of Morse theory, the critical point γ mp+1 of E is equivalent to the critical point γ of E mp+1 . Indeed, the diffeomorphism

ψ mp+1 : Λ mp+1 (M ; I) ∼ = -→ Λ(M ; I), ψ mp+1 (ζ) = ζ mp+1 , satisfies E • ψ mp+1 = (mp + 1) 2 E mp+1 , and in particular ind(E, γ mp+1 ) = ind(E mp+1 , γ).
The following lemma follows from the arguments in [GT78, Sections 2-3] and [Tan82, Sections 2-3]. We provide its proof for the reader's convenience.

Lemma 2.2 (Grove-Tanaka

). There exist finitely many Hilbert manifolds Ω 1 , ..., Ω n , positive real numbers q 1 , ..., q n that are multiples of p, real numbers q 1 , ..., q n satisfying q i ∈ [0, q i ) for all i, and a partition M 1 ∪ ... ∪ M n of the set N \ {1, 2, ..., n 0 }, for some n 0 ∈ N, such that for all m ∈ M i the following properties hold: (i) Ω i is a complete Hilbert submanifold of both Λ qi (M ; id) and Λ mp+1 (M ; I), and is invariant by the gradient flow of

E mp+1 ; (ii) E qi | Ωi = E mp+1 | Ωi ; (iii) nul(E mp+1 , γ) = nul(E mp+1 | Ωi , γ); (iv) mp + 1 ≡ q i mod q i .
Remark 2.1. Lemma 2.2 was stated in the previous paper of the second author [Maz15, Lemma 2.3] with two unfortunate typos: the numbers q 1 , ..., q n were said to be integers instead of real numbers, and the manifolds Ω i were said to be contained in Λ qi (M ; I) instead of Λ qi (M ; id). Both typos were rather obvious from the context, since the period p is not necessarily rational, and the lemma was employed correctly in the paper.

Remark 2.2. If the isometry I is the identity, that is, if we are in the closed geodesics setting, Lemma 2.2 reduces to a classical result due to Gromoll and Meyer [START_REF] Gromoll | Periodic geodesics on compact Riemannian manifolds[END_REF]: in this case we have p = 1, each q i is a positive integer dividing m -1 for all the numbers m ∈ M i , the numbers q i are all equal to zero, and the manifolds Ω i are simply the free loop spaces Λ qi (M ; id).

Proof of Lemma 2.2. We will treat the cases in which the period p is rational and irrational separately. We begin with the rational case: p = a/b for some relatively prime integers a, b ∈ N. We recall that the fixed point set of an isometry of a closed Riemannian manifold is a disjoint union of finitely many closed totally geodesic submanifolds, see [START_REF] Kobayashi | Transformation groups in differential geometry[END_REF]page 59]. Notice that, for all multiples s of the numerator a and for all m ∈ N, the I-invariant geodesic γ belongs to the Hilbert submanifold Λ mp+1 (fix(I s ); I) ⊂ Λ mp+1 (M ; I). By [Tan82, Lemma 2.1], there is such s ∈ N and some n 0 ∈ N such that nul(E mp+1 , γ) = nul(E mp+1 | Λ mp+1 (fix(I s );I) , γ), ∀m ∈ N with m ≥ n 0 .

Let ∇E mp+1 denote the gradient of the energy function E mp+1 : Λ mp+1 (M ; I) → R with respect to its usual Riemannian metric. The argument in the proof of [GT78, Proposition 3.5] implies that the gradient ∇E mp+1 (ζ) is tangent to the submanifold Λ mp+1 (fix(I s ); I) for all ζ ∈ Λ mp+1 (fix(I s ); I). The isometry I has order s on fix(I s ). Therefore, from now on, we can work inside the manifold fix(I s ) and apply the arguments in [GT78, Section 2], which are valid for isometries of finite order. For any fixed m ∈ N, consider a positive rational number τ , and two positive integers q, r ∈ N such that

• (mp + 1)/τ ∈ N, • q is a multiple of s, • I r (γ(t)) = γ(t + τ ),
• (mp + 1)r/τ ≡ 1 mod q. These conditions imply γ ∈ Λ τ (fix(I q ); I r ) ⊂ Λ mp+1 (M ; I) ∩ Λ q (M ; id),

E mp+1 | Λ τ (fix(I q );I r ) = E q | Λ τ (fix(I q );I r ) = E τ | Λ τ (fix(I q );I r ) .
By [GT78, Lemma 2.9], there exist positive rational numbers τ 1 , ..., τ n , q 1 , ..., q n , positive integers r 1 , ..., r n , and a partition M 1 ∪ ... ∪ M n of the set of positive integers N such that, for all i = 1, ..., n and m ∈ M i , the four points above are verified by τ := τ i , q := q i , and r := r i , and we have nul(E mp+1 | Λ mp+1 (fix(I s );I) , γ) = nul(E mp+1 | Λ τ (fix(I q );I r ) , γ), ∀m ∈ M i .

Another application of the argument in the proof of [GT78, Proposition 3.5] implies that the gradient ∇E mp+1 (ζ) is tangent to the submanifold Λ τ (fix(I q ); I r ) for all ζ ∈ Λ τ (fix(I q ); I r ). If we set Ω i := Λ τi (fix(I qi ); I ri ), the arguments given so far prove points (i), (ii), and (iii) of the lemma. Notice that, for each i = 1, ..., n, there are finitely many positive rational numbers q 1 , ..., q u ∈ [0, q i ) such that, for each m ∈ M i , mp + 1 is congruent to an element of {q 1 , ..., q u } modulo q i . Therefore, point (iv) follows by taking a suitable subpartition of M 1 ∪ ... ∪ M n . Assume now that the period p is irrational. We follow the arguments in [Tan82, Section 3]. For all m ∈ N and for all multiples q of the period p, the intersection Λ m,q := Λ mp+1 (M ; I) ∩ Λ q (M ; id) is a totally geodesic submanifold of Λ mp+1 (M ; I), and a standard computation shows that the gradient ∇E mp+1 (ζ) is tangent to Λ m,q for all ζ ∈ Λ m,q . Fix one such ζ. For each > 0 there exists δ ∈ [0, ] such that

δ 0 g ζ(t) ( ζ(t), ζ(t)) dt < .
Since p is irrational, the quotient (mp + 1)/q is irrational as well. Therefore, there exist a, b ∈ N such that 0 ≤ a(mp + 1) -bq ≤ δ. We infer that

a(mp + 1)E mp+1 (ζ) -bqE q (ζ) ∈ [0, ],
and therefore

E mp+1 (ζ) ≤ a(mp + 1) bq E mp+1 (ζ) ≤ E q (ζ) + bq , E q (ζ) ≤ a(mp + 1) bq E mp+1 (ζ) ≤ 1 + bq E mp+1 (ζ).
Since > 0 can be taken arbitrarily small, we have proved that

E mp+1 | Λ m,q = E q | Λ m,q
By [Tan82, Lemma 3.2], we infer that there exist positive real numbers q 1 , ..., q n that are multiples of p, and a partition

M 1 ∪ ... ∪ M n of the set N \ {1, 2, ..., n 0 }, for some n 0 ∈ N, such that nul(E mp+1 , γ) = nul(E mp+1 | Λ m,q i , γ), ∀m ∈ M i .
Notice that Λ m,qi = Λ m ,qi if m ≡ n mod q i /p. Therefore, up to replacing the partition M 1 ∪...∪M n with a subpartition, we can assume that Ω i := Λ m,qi = Λ m ,qi for all i = 1, ..., n and m, m ∈ M i . This proves points (i), (ii), and (iii) of the lemma in the case where p is irrational. As before, point (iv) follows if we take a further suitable subpartition of M 1 ∪ ... ∪ M n .

Let us equip Ω i with the complete Riemannian metric pulled back from the one of the space Λ mp+1 (M ; I) via the inclusion, for an arbitrary m ∈ M i . The restricted energy function E qi | Ωi satisfies the Palais-Smale condition. Indeed, E qi | Ωi = E mp+1 | Ωi and the gradient ∇E mp+1 is tangent to Ω i ; this implies that all Palais-Smale sequences for E qi | Ωi are Palais-Smale sequences for E mp+1 as well, and therefore are compact.

For a given m ∈ M i , let h ∈ N be such that mp + 1 = hq i + q i . Points (i) and (iv) of Lemma 2.2 imply that, for all ζ ∈ Ω i , we have that

I(ζ(t)) = ζ(t + mp + 1) = ζ(t + q i ), ∀t ∈ R, and
mp+1 0 g ζ(t) ( ζ(t), ζ(t)) dt = h qi 0 g ζ(t) ( ζ(t), ζ(t)) dt + q i 0 g ζ(t) ( ζ(t), ζ(t)) dt.
If q i > 0, this can be rephrased by saying that Ω i is contained in Λ q i (M ; I) and

(mp + 1)E mp+1 | Ωi = hq i E qi | Ωi + q i E q i | Ωi ,
which, together with Lemma 2.2(ii), implies that

E mp+1 | Ωi = E qi | Ωi = E q i | Ωi , ∀m ∈ M i .

Broken geodesics

3.1. A space of broken geodesics. In Section 4, we will make use of the well-known property that closed sublevel sets of the energy function are homotopy equivalent to a compact manifold with boundary. This compact manifold is a subset of Morse's space of broken geodesics, which we now recall. We refer the reader to [START_REF] Mazzucchelli | Critical point theory for Lagrangian systems[END_REF]chap. 4] and to the references therein for more background details.

Consider a vector τ = (τ 0 , τ 1 , ..., τ k ) ∈ R k+1 such that 0 = τ 0 < τ 1 < ... < τ k . We denote by Υ ⊂ C([0, τ k ]; M ) the subspace of all continuous curves ζ : [0, τ k ] → M such that ζ(0) = ζ(τ k ) and, for all i = 0, ..., k -1, ζ| [τi,τi+1
] is a geodesic of length less than the injectivity radius of (M, g).

The space Υ is diffeomorphic to an open neighborhood of the diagonal in the k-fold product M × ... × M via the map ζ → (ζ(τ 1 ), ζ(τ 2 ), ..., ζ(τ k )). As such, it inherits the Riemannian metric g ⊕ ... ⊕ g. A curve s → ζ s is a geodesic of (Υ , g ⊕ ... ⊕ g) if and only if, for all i = 0, ..., k -1, the curve s → ζ s (τ i ) is a geodesic of (M, g).
For some k ∈ {0, ..., k -1}, we introduce the subspace Υ ⊂ Υ of all ζ's such that

I(ζ(0)) = ζ(τ k ). It is easy to see that Υ is an embedded smooth submanifold of Υ . Indeed, if k = 0, we have that Υ = F -1 (graph(I)), where F : Υ → M × M is the smooth submersion given by F (ζ) = (ζ(0), ζ(τ k )). If k = 0, we have that Υ = G -1 (fix(I))
, where G : Υ → M is the smooth submersion given by G(ζ) = ζ(0), and fix(I) if the fixed point set of I, which is a disjoint union of finitely many closed totally geodesic submanifolds of M , see [Kob95, page 59]. In both cases, Υ itself is a possibly disconnected, complete, and totally geodesic submanifold of Υ . We denote by dist M : M × M → [0, ∞) the Riemannian distance on M , and we define the function dist

Υ : Υ × Υ → by dist Υ (ζ 0 , ζ 1 ) := max i=0,...,k-1 dist M (ζ 0 (τ i ), ζ 1 (τ i )).
This function is a distance on the connected components of Υ equivalent to the one induced by the Riemannian metric g ⊕ ... ⊕ g.

Two energy functions.

On Υ we will need to consider two energy functions

F τ k : Υ → [0, ∞) and F τ k : Υ → [0, ∞).
The first one is defined by

F τ k (ζ) := 1 τ k τ k 0 g ζ(t) ( ζ(t), ζ(t)) dt = 1 τ k k-1 i=0 dist M (ζ(τ i ), ζ(τ i+1 )) 2 τ i+1 -τ i . If k = 0, that is, if τ k > 0,
the second one is defined analogously by

F τ k (ζ) := 1 τ k τ k 0 g ζ(t) ( ζ(t), ζ(t)) dt = 1 τ k k -1 i=0 dist M (ζ(τ i ), ζ(τ i+1 )) 2 τ i+1 -τ i .
If τ k = 0, this second energy function will not be relevant, and we simply set it to be F τ k ≡ 0. Let us fix, once for all, an energy bound b > 0. Notice that

dist M (ζ(τ i ), ζ(τ i+1 )) 2 ≤ (τ i+i -τ i )τ k F τ k (ζ).
Therefore, if we choose τ such that

max{τ i+1 -τ i | i = 0, ..., k -1} < injrad(M, g) 2 /(9τ k b), (3.1) we have dist M (ζ(τ i ), ζ(τ i+1 )) < injrad(M, g)/3, ∀ζ ∈ {F τ k ≤ b}, i = 0, ..., k -1.
This implies that the sublevel set {F τ k ≤ b} is compact. Actually, all sublevel sets {F τ k ≤ b } are compact provided b < 9b. On the other hand, the sublevel set {F τ k ≤ b} is never compact, because the energy function F τ k is independent of ζ(τ i ) for all i > k . Nevertheless, with the above choice of τ , the intersection

{F τ k ≤ b} ∩ {F τ k ≤ b} is compact.
3.3. Polydiscs of broken geodesics. We fix a radius R ∈ (0, injrad(M, g)/3) small enough so that the Riemannian balls of M of radius less than or equal to R are geodesically convex: any pair of points in one such ball are joined by a unique shortest geodesic that is entirely contained in the ball (see e.g. [dC92, page 76]).

We will be dealing with polydisc neighborhoods of elements γ

∈ {F τ k ≤ b} of the form W (γ, r) := ζ ∈ Υ dist M (ζ(τ i ), γ(τ i )) < r ∀i = 1, ..., k ,
where r ∈ (0, R]. This polydisc is simply the ball with radius r centered at γ for the distance dist Υ . Since the sublevel set {F τ k ≤ b} is compact, the restriction of the function F τ k to it is uniformly continuous. Hence, we can require the upper bound R for the radii to be small enough (depending on b) so that the closure of each of the above polydiscs is compact in Υ.

A useful property of W (γ, r) is that every pair of points in it can be joined by a unique shortest geodesic of (Υ, g ⊕ ... ⊕ g). More specifically, for each x, y ∈ M with dist M (x, y) < injrad(M, g), we denote by

γ x,y : [0, 1] → M (3.2)
the unique shortest geodesic of (M, g) such that γ(0) = x and γ(1) = y. It is well known that γ x,y depends smoothly on (x, y), see e. Indeed, by the geodesic convexity of Riemannian balls of radius r, for all i ∈ Z k and s

∈ [0, 1] we have dist M (ζ s (τ i ), γ(τ i )) < r. Moreover dist M (ζ s (τ i ), ζ s (τ i+1 )) < dist M (ζ s (τ i ), γ(τ i )) + dist M (γ(τ i ), γ(τ i+1 )) + dist M (γ(τ i+1 ), ζ s (τ i+1 )) < r + injrad(M, g)/3 + r < injrad(M, g).
Therefore, the whole curve s → ζ s is well defined and lies inside W (γ, r). In other words, W (γ, r) is geodesically convex.

3.4. Deformation to the space of broken geodesics. Now, let γ ∈ Λ(M ; I) be an Iinvariant geodesic that is periodic with minimal period p ≥ 1. Consider the objects given by Grove and Tanaka's Lemma 2.2: a period q := q i that is a multiple of p, a set of integers M = M i ⊂ N, a non-negative number q = q i ∈ [0, q) such that mp + 1 ≡ q mod q, and a Hilbert manifold Ω = Ω i ⊂ Λ q (M ; id) ∩ Λ mp+1 (M ; I). We will choose our energy bound b of the previous section larger than E(γ), and the vector τ that enters the definition of Υ such that τ k = q and τ k = q.

For all m ∈ M, we will see the space Υ as a submanifold of the Hilbert manifold Λ mp+1 (M ; I) via the embedding

ω m : Υ → Λ mp+1 (M ; I)
uniquely defined by

ω m (ζ)(hq + t) = ζ(t), ∀h ∈ N, t ∈ [0, q] with hq + t ∈ [0, mp + 1].
The map ω m will play the role of the classical m-fold iteration map from the theory of closed geodesics. The various energy functions are related by

E mp+1 • ω m (ζ) = 1 mp+1 mp+1 q qF q (ζ) + q F q (ζ) = F q (ζ) + q mp+1 F q (ζ) -F q (ζ) .
(3.3) Now, we extend the time-intervals given by the vector τ in a suitable periodic fashion. We define the bi-infinite sequence {ν j | j ∈ Z} as follows. First, we set

ν hk+i := hq + τ i , ∀h ∈ N, i ∈ {0, ..., k} with hq + τ i ∈ [0, mp + 1].
This defines ν j for all j ∈ {0, ..., j }, where j = mp+1 q k + k . Notice that ν j = mp + 1. We complete the definition by setting

ν hj +j = h(mp + 1) + ν j , ∀h ∈ Z, j ∈ {0, ..., j -1}.
The sequence {ν j | j ∈ Z} is precisely the one such that, for all ζ ∈ Υ and i ∈ Z, the restriction

ω m (ζ)| [νi,νi+1] is a geodesic of length less than injrad(M, g). If ζ ∈ Ω satisfies E mp+1 (ζ) = E q (ζ) ≤ b, then for all time values t 0 , t 1 ∈ R such that 0 ≤ t 1 -t 0 < injrad(M, g) 2 /(9qb), we have dist M (ζ(t 1 ), ζ(t 0 )) ≤ t1 t0 g ζ(t) ( ζ(t), ζ(t)) dt ≤ (t 1 -t 0 )qb < injrad(M, g)/3.
This, together with (3.1), allows us to define a continuous homotopy The image of the time-1 map r 1 lies inside ω m (Υ), i.e.

r s : {E mp+1 ≤ b} ∩ Ω → {E mp+1 ≤ b}, s ∈ [0, 1], (3.4
r 1 ({E mp+1 ≤ b} ∩ Ω) ⊂ ω m (Υ). (3.6)
We recall that E mp+1 | Ω = E q | Ω , and if q > 0 we further have

E mp+1 | Ω = E q | Ω .
This, together with (3.5), implies that

max{F q • r 1 (ζ), F q • r 1 (ζ)} ≤ E mp+1 (ζ), ∀ζ ∈ Ω. (3.7)
4. The main lemma 4.1. Statement and outline of the proof. We say that an I-invariant geodesic γ ∈ crit(E) with minimal period p ≥ 1 is isolated when orb(γ mp+1 ) is isolated in crit(E) for all m ∈ M. The following lemma will be the crucial ingredient for our main Theorem 5.1.

Lemma 4.1. Let γ ∈ crit(E) be an isolated I-invariant geodesic with minimal period p ≥ 1, and fix a degree d ≥ 1. For all m ∈ N large enough and for all sufficiently small neighborhoods V of orb(γ mp+1 ) the following holds: given any compact domain K ⊂ R d and any continuous map u :

K → {E < E(γ mp+1 )} ∪ V such that u(∂K) ∩ V = ∅, there exists a homotopy u s : K → Λ(M ; I) such that u 0 = u, u 1 (K) ⊂ {E < E(γ mp+1 )}, and
u s ≡ u outside u -1 (V ) for all s ∈ [0, 1].
The special case in which the compact domain K is 1-dimensional was established in [Maz15, Lemma 2.5], and was inspired by an analogous result due to Bangert [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF] (see also [AMMP14, Theorem 2.6] for a similar result in the context of the Lagrangian free-period action functional). When K has dimension at least 2, the situation becomes much more complicated, and we now give an outline of the argument leading to the proof.

There are two cases to consider. The first, easy one, is when the mean index of the geodesic γ is positive, that is to say, the Morse index ind(E mp+1 , γ) tends to infinity as m → ∞. In particular, if m is large, this index will be larger than the dimension of K, and the assertion of the lemma follows from standard arguments of non-linear analysis. This case will be treated in Lemma 4.5.

The second, difficult, case, is when the Morse index ind(E mp+1 , γ) is zero for all m ∈ N. Here, we need to employ Grove and Tanaka's Lemma 2.2, which gives us finitely many Hilbert manifolds Ω 1 , ..., Ω n such that, for all m ∈ N, the space of I-invariant curves Λ mp+1 (M ; I) contains one of the Ω i 's, and the Morse theory of the function E mp+1 : Λ mp+1 (M ; I) → R on a neighborhood V of the critical circle orb(γ) is completely described by its restriction to the submanifold Ω i . The existence of a given map u as in the lemma implies that γ is not a global minimum of E mp+1 in its connected component of Ω i . By the broken geodesics approximation of Section 3.4, any sublevel set of the energy on the space Ω i can be deformed to a finite dimensional submanifold Υ of broken I-invariant geodesics. We introduce a sufficiently fine triangulation Σ of the connected component of Υ containing the original I-invariant geodesic γ. By a well-known technique due to Bangert and Klingenberg, for m large enough, every simplex σ 0 ∈ Σ can be deformed with a homotopy σ s inside the space Λ mp+1 (M ; I) such that the simplex σ 1 is contained in the sublevel set {E mp+1 < E(γ)}, and all faces of σ 0 that were already contained in this sublevel set are not moved by the homotopy. Now, we take m large enough so that these homotopies are defined for all the simplexes in Σ. A map u as in the lemma can be equivalently seen of the form u : K → {E mp+1 < E(γ)} ∪ V , where V ⊂ Λ mp+1 (M ; I) is an open neighborhood of γ. We take a fine triangulation Σ of the domain K. We show that it is possible to deform u inside a subset of u -1 (V ) such that every simplex σ ∈ Σ therein is mapped by u to one of the simplexes of Σ, while the remaining simplexes of Σ are mapped inside the sublevel set {E mp+1 < E(γ)} as they were originally. Finally, Bangert and Klingenberg's homotopies allow to deform u| u -1 (V ) to a map that takes values inside {E mp+1 < E(γ)}. The details of this argument will be carried over in the next four subsections, and will culminate with the proof of Lemma 4.3, which is Lemma 4.1 in the case where γ has zero mean index.

4.2. Small neighborhoods of the critical circle. Throughout Section 4, we will work with an isolated I-invariant geodesic γ ∈ Λ(M ; I) with minimal period p ≥ 1 and zero mean index, meaning that ind(E mp+1 , γ) = 0 for all m ∈ N. The case of positive mean index will be treated in Section 4.6. We set

c := E mp+1 (γ) = E(γ) = 1 0 g γ(t) ( γ(t), γ(t)) dt,
and we stress that this action is independent of the order of iteration m ∈ N. Let Ω = Ω i be one of the Hilbert manifolds given by Grove and Tanaka's Lemma 2.2, q := q i the basic period given there, and M = M i the subset of integers m such that claims (i-iv) of the lemma hold. We equip Ω with the complete Riemannian metric pulled back from the space Λ mp+1 (M ; I) via the inclusion, for an arbitrary m ∈ M. We denote by φ t the anti-gradient flow of the energy E q | Ω , which is precisely the restriction to Ω of the anti-gradient flow of E mp+1 . Lemma 4.2. For every sufficiently small open neighborhood U ⊂ Ω of orb(γ) the following hold.

(i) For all > 0 there exists τ > 0 such that φ

t (U ∩ {E q | Ω ≤ c -}) ∩ U = ∅ for all t ≥ τ . (ii) There exist > 0 and a smaller open neighborhood U of orb(γ) whose closure is contained in U such that φ t (U ) \ U ⊂ {E q | Ω < c -} for all t ≥ 0.
Proof. We denote by U (ρ) ⊂ Ω the open tubular neighborhood of orb(γ) of radius ρ.

The energy E q | Ω satisfies the Palais-Smale condition. Therefore, for all 0 < ρ 1 < ρ 2 such that the closure of U (ρ 2 ) does not contain critical circles of E q | Ω other than orb(γ), we have

µ(ρ 1 , ρ 2 ) := inf ∇E q | Ω (ζ) ζ ∈ U (ρ 2 ) \ U (ρ 1 ) > 0.
If ζ ∈ U (ρ 1 ) and t > 0 are such that φ t (ζ) ∈ U (ρ 2 ), the curve φ [0,t] (ζ) must cross the shell U (ρ 2 ) \ U (ρ 1 ) of Riemannian width ρ 2 -ρ 1 . Therefore, if we denote by t ∈ (0, t) the supremum of the times s such that φ s (ζ) ∈ U (ρ 1 ), we have

E q (ζ) -E q (φ t (ζ)) ≥ E q (φ t (ζ)) -E q (φ t (ζ)) = t t ∇E q | Ω (φ s (ζ)) • d ds φ s (ζ) ds ≥ (ρ 2 -ρ 1 ) µ(ρ 1 , ρ 2 ). (4.1)
We fix once for all two such values 0 < ρ 1 < ρ 2 , and we require ρ 2 to be small enough such that U (ρ 2 ) does not intersect the sublevel set {E q | Ω < c-}, for some > 0. The open neighborhood U of the lemma must be small enough so that it is contained in U (ρ 1 ) and disjoint from the sublevel set

{E q | Ω ≤ c-(ρ 2 -ρ 1 ) µ(ρ 1 , ρ 2 )}. Given > 0, we set ν := inf ∇E q | Ω (ζ) ζ ∈ U (ρ 2 ) ∩ {E q | Ω ≤ c -} > 0, τ := /ν 2 .
Let us prove point (i) of the lemma: we claim that, for every ζ ∈ U ∩{E q | Ω ≤ c-} and t ≥ τ , we have that φ t (ζ) ∈ U . Indeed, the curve φ [0,t] (ζ) cannot be entirely contained in U (ρ 2 ), since otherwise

E q (φ t (ζ)) = E q (ζ) - t 0 ∇E q | Ω (φ s (ζ)) 2 ds < c -τ ν 2 = c -.
Hence, by the energy estimate in (4.1), we have

E q (φ t (ζ)) ≤ E q (ζ) -(ρ 2 -ρ 1 ) µ(ρ 1 , ρ 2 ) < c -(ρ 2 -ρ 1 ) µ(ρ 1 , ρ 2 ),
which proves our claim. Now, we choose a radius r 2 > 0 small enough so that U (r 2 ) ⊂ U , and we set := µ(r 2 /2, r 2 )r 2 /4. We also choose a radius r 1 ∈ (0, r 2 /2) small enough so that

U := U (r 1 ) ⊂ {E q | Ω < c + }.
Let ζ ∈ U be a point such that φ t (ζ) ∈ U for some t > 0. In particular, E q (ζ) < c+ and the estimate (4.1) implies

E q (ζ) -E q (φ t (ζ)) ≥ µ(r 2 /2, r 2 ) r 2 /2 = 2 .
This proves point (ii).

4.3. Reference simplexes. In this section, we will also need to consider the real number q = q i ∈ [0, q) given by Lemma 2.2. We fix, once for all, an energy bound b > c = E q (γ) and a vector τ = (τ 0 , ..., τ k ) ∈ R k+1 such that 0 = τ 0 < τ 1 < ... < τ k = q, τ k = q for some k ∈ {0, ..., k -1}, and the inequality (3.1) is satisfied. With the notation of Section 3, we consider the space of broken geodesics Υ associated to τ , and the energy functions F q and F q . We will also consider the polydiscs defined in Section 3.3, whose radii are smaller than or equal to R.

We fix an arbitrarily small > 0. Since the sublevel set {F q ≤ b} is compact, the functions F q and F q are uniformly equicontinuous on this set. Namely, there exists δ ∈ (0, R] such that max

F q (ζ 0 ) -F q (ζ 1 ), F q (ζ 0 ) -F q (ζ 1 ) < , ∀ζ 0 , ζ 1 ∈ {F q ≤ b} with dist Υ (ζ 0 , ζ 1 ) < δ. (4.2)
We denote by Υ the union of those connected components of Υ intersecting the sublevel set {F q < c}. We can find a finite subset Σ 0 = {σ 1 , ..., σ h } ⊂ {F q ≤ b}∩Υ that is δ/5-dense in the sublevel set, that is,

{F q ≤ b} ∩ Υ ⊂ σ∈Σ0 W (σ, δ/5). (4.3)
Notice that, since δ < R, the union of the polydiscs W (σ, δ/5) is relatively compact in Υ. For each pair of (not necessarily distinct) σ i , σ j ∈ Σ 0 such that dist Υ (σ i , σ j ) < δ, we denote by σ ij : [0, 1] → Υ the 1-simplex given by the minimal geodesic of Υ joining σ i and σ j . We denote by Σ 1 the finite collection of all these 1-simplexes. We now inductively define the finite collections Σ d for increasing values of d, starting for d = 2. Given (not necessarily pairwise distinct) σ i0 , ..., σ i d ∈ Σ 0 such that dist Υ (σ ij , σ i l ) < δ for all j, l = 0, ..., d, we define σ i0...i d : ∆ d → Υ to be the d-simplex whose j-th face is the (d -1)-simplex

σ i0... ij ...i d ∈ Σ d-1 ,
and whose interior is defined as follows: we see the standard d-simplex Notice that, by the geodesic convexity of polydiscs of radius less than or equal to R, every d-simplex σ i0...i d ∈ Σ d is entirely contained in the polydisc of radius δ centered at any of its 0-faces. 4.4. Bangert's homotopies. For all m ∈ M, the space of broken geodesics Υ can be seen as a submanifold of Λ mp+1 (M ; I) via the embedding ω m : Υ → Λ mp+1 (M ; I) defined in Section 3.4. In this section, we will further need to assume that our Iinvariant geodesic γ is not a global minimum of the energy E q | Ω inside its connected component of Ω. Since the homotopy r s of equation (3.4) decreases the energy and, in particular, its time-1 map satisfies (3.7), the element r 1 (γ) is not a global minimum of F q , nor of F q if q > 0, in its connected component of Υ. By modifying an argument due to Bangert and Klingenberg [BK83, Thm. 2] we will show that, if m is large, the images of the simplexes constructed in the previous section under the map ω m can be pushed, in a suitable way, below the critical level c.

∆ d = x ∈ [0, 1] d i x i ≤
The main ingredient for such statement is Bangert's well known technique of "pulling one loop at the time". Given any continuous map θ 0 : ∆ j → Λ q (M ; id) ⊂ Λ mq (M ; id), we can deform it with a homotopy θ s : ∆ j → Λ mq (M ; id), s ∈ [0, 1], such that

• θ s | ∂∆ j = θ 0 | ∂∆ j ,
• θ s (x)(0) = θ 0 (y s (x))(0) for a suitable continuous homotopy y s : ∆ j → ∆ j such that y 0 = id,

• E mq • θ 1 ≤ max{E q (θ 0 (x)) | x ∈ ∂∆ j } + const/m
, where const ≥ 0 is a quantity depending only on θ 0 (in particular, independent of m).

The proof of this fact is outlined with our notation and for j = 1 in [Maz14, Sect. 3.2], see in particular Figure 1 therein. The case j > 1 is not harder: it suffices to see the j-simplex as a smooth family of 1-simplexes, and apply to each of them the construction. In the following, we will refer to homotopies of this kind as to Bangert's homotopies.

Lemma 4.3. For every sufficiently large integer m ∈ M we can associate to each j-simplex σ ∈ Σ 0 ∪ ... ∪ Σ d a homotopy

h σ : [0, 1] × ∆ j → Λ mp+1 (M ; I) (4.4)
with the following properties:

(i) h σ (0, •) = ω m • σ, (ii) if σ is contained in {F q < c}, then h σ (s, •) = ω m • σ for all s ∈ [0, 1], (iii) h σ (s, F l (•)) = h σ•F l (s,
•) for all l = 0, ..., j, where F l : ∆ j-1 → ∂∆ j is the affine map onto the l-th face of ∆ j ,

(iv) h σ (1, •) is mapped in {E mp+1 < c}.
Proof. The proof is essentially the same as the one of [Maz14, Lemma 3.3]. Since there are some technical differences in the setting and in the statement, we provide the complete argument for the reader's convenience. We define the embedding

ψ : Υ → Λ q (M ; id)
in the obvious way: for all ζ ∈ Υ, the image ψ(ζ) is the unique q-periodic curve whose restriction to the interval [0, q] is precisely equal to ζ. Notice that, by the definition of Υ, we have that

ψ(ζ)(mp + 1) = ψ(ζ)(q ) = I(ψ(ζ)(0)),
and indeed ψ(ζ)| [0,mp+1] = ω m (ζ)| [0,mp+1] .
Let us construct the homotopies of the lemma, beginning with the 0-simplexes σ ∈ Σ 0 . Since F q (r 1 (γ)) = E q (γ) = c and r 1 (γ) is not a global minimum of F q in its connected component of Υ, there exists a continuous path s → σ s ∈ Υ such that σ 0 = σ and F q (σ 1 ) < c. If F q (σ) < c and F q (σ) < c we choose this path to be the stationary one σ s ≡ σ. We define a map

h σ : [0, 1] × ∆ 0 = [0, 1] → Λ q (M ; id) by h σ (s) = ψ(σ (d+1)s ), if s ∈ [0, 1/(d + 1)], ψ(σ 1 ) if s ∈ [1/(d + 1), 1].
Notice that E q (h σ (s)) = F q (σ 1 ) < c, ∀s ∈ 1 d+1 , 1 . We also set m 0 := 1.

We proceed to construct homotopies for the higher dimensional simplexes iteratively in the degree j, from j = 1 upward. We consider a large enough integer m j that is divisible by m j-1 , so that Λ mj-1q (M ; id) ⊂ Λ mj q (M ; id). The precise value of m j will be fixed in a moment. Consider σ ∈ Σ j . We construct the maps

h σ : [0, 1] × ∆ j → Λ mj q (M ; id) y σ : [0, 1] × ∆ j → ∆ j
as follows. We set h σ (0, •) := ψ • σ and y σ (0, •) := id. For all l = 0, ..., j and

s ∈ [0, 1], we set h σ (s, F l (•)) := h σ•F l (s, •) and y σ (s, F l (•)) := y σ•F l (s, •)
, where the maps h σ•F l and y σ•F l were defined in the previous step of the iterative procedure. Up to now, we have defined the maps h σ and y σ on ({0} × ∆ j ) ∪ ([0, 1] × ∂∆ j ). The previous step of this iterative procedure was carried out in such a way that

h σ (s, •)| ∂∆ j = h σ ( j d+1 , •)| ∂∆ j , ∀s ∈ j d+1 , 1 , and max E mj-1q (h σ ( j d+1 , x)) | x ∈ ∂∆ j < c. (4.5)
We choose a retraction

π j : [0, j] × ∆ j → ({0} × ∆ j ) ∪ ([0, j] × ∂∆ j ),
and we set

h σ (s, x) := h σ (s, π j (x)), y σ (s, x) := y σ (s, π j (x)), ∀[0, j] × ∆ j .
Finally, we define the homotopies s → h σ (s, •) and s → y σ (s, •), for s ∈ j d+1 , j+1 d+1 , by means of a Bangert homotopy associated to the j-simplex

h σ ( j d+1 , •) : ∆ j → Λ mj-1q ,
and we extend them constantly for s ∈ j+1 d+1 , 1 . Summing up, we have constructed the maps h σ and y σ such that

(i') h σ (0, •) = ψ • σ, (ii') h σ (s, F l (•)) = h σ•F l (s, •) for all l = 0, ..., j, (iii') h σ (s, •) = h σ ( j+1 d+1 , •) for all s ∈ j+1 d+1 , 1 , (iv') E mj q (h σ ( j+1 d+1 , •)) ≤ max{E mj-1q (h σ ( j d+1 , x)) | x ∈ ∂∆ j } + const mj /mj-1
, where const ≥ 0 is a quantity depending only on σ,

(v') h σ (s, x)(0) = σ(y σ (s, x))(0).
We choose m j large enough so that, by (iii'), (iv'), and (4.5), we have

E mj q (h σ (s, •)) < c, ∀s ∈ j+1 d+1 , 1 . (4.6)
If σ is mapped inside the sublevel set {F q < c}, then we define the above maps simply by h σ (s, x) := σ(x) and y σ (s, x) := x, and notice that conditions (i'-v') are still verified. Now, consider an integer m ∈ M, that we will require to be large enough in a moment. We set m := mp+1 m d q and q := mp + 1 -m m d q ∈ [0, m d q), so that

mp + 1 = m m d q + q .
Notice that, by Lemma 2.2(iv), q ≡ q mod q.

For all σ ∈ Σ 0 ∪ ... ∪ Σ d , we define the homotopy h σ of the lemma by

h σ (s, x)(t) := h σ (s, x)(t), if t ∈ [0, m m d q] , ψ • σ(y σ (s, x))(t -m m d q) if t ∈ [m m d q, mp + 1] .
Notice that

h σ (s, x)(mp + 1) = ψ • σ(y σ (s, x))(q ) = σ(y σ (s, x))(q ) = I(σ(y σ (s, x))(0)) = I(h σ (s, x)(0)) = I(h σ (s, x)(0)).
Therefore, h σ is a well defined map of the form (4.4). Properties (i-iii) readily follow from its construction. As for property (iv), we have

E mp+1 (h σ (1, x)) = 1 mp + 1 m m d qE m d q (h σ (1, x))
+ q /q qF q (σ(y σ (1, x)))

+ q F q (σ(y σ (1, x))) = E m d q (h σ (1, x)) + c mp + 1
,

where c := q /q qF q (σ(y σ (1, x))) + q F q (σ(y σ (1, x))) -q E m d q (h σ (1, x))
< m d q max{F q • σ} + q max{F q • σ}.

Notice that the quantity c is uniformly bounded independently of m ∈ M. Since

E m d q (h σ (s, x)) < c, if m ∈ M is large enough we have E mp+1 (h σ (1, x)) < c
, which proves property (iv).

4.5. The case of zero mean index. For every integer m ∈ M, Lemma 2.2(iii) implies that nul(E q | Ω , γ) = nul(E mp+1 , γ), and, by our assumption, ind(E q | Ω , γ) = ind(E mp+1 , γ) = 0. Let U ⊂ Ω be a small enough open neighborhood of orb(γ) such that Lemma 4.2 applies to it, and, for some h > 0 and for all ζ ∈ U and vector fields ξ ∈ T ζ Λ mp+1 (M ; I) orthogonal to Ω, we have

HessE mp+1 (ζ)[ξ, ξ] ≥ h ξ 2 ζ . (4.7)
Here, the norm on the right hand side is the one induced by the Riemannian metric on the Hilbert manifold Λ mp+1 (M ; I). Lemma 4.2 gives us a quantity > 0 and a smaller open set U of orb(γ) whose closure is contained in U satisfying the properties stated there. Within the proof of Lemma 4.4, we will also need to fix δ > 0 small enough so that (4.2) holds, and we will consider the δ/5-dense reference simplexes built in Section 4.3. Let U m ⊂ Λ mp+1 (M ; I) be a small tubular neighborhood of U , which is diffeomorphic to an open neighborhood of the zero-section in the normal bundle of U . Consider the associated radial deformation retraction

f s : U m → U m , (4.8)
which satisfies f 0 = id and f 1 (U m ) = U . Since, by Lemma 2.2(i), the gradient of ∇E mp+1 is tangent to Ω along Ω, the restriction of the energy E mp+1 to a fiber of f 1 has a critical point at the intersection of the fiber with Ω. By the convexity (4.7), this critical point is a strict minimum. In particular, if the tubular neighborhood U m is sufficiently small, the deformation f s does not increase the energy, i.e. d ds E mp+1 • f s ≤ 0.

Lemma 4.4. Let γ ∈ crit(E) be an isolated I-invariant geodesic that is periodic with minimal period p ≥ 1, having energy c := E(γ), and satisfying ind(E mp+1 , γ) = 0 for all m ∈ N. Fix a degree d ≥ 1. For all m ∈ N large enough and for all sufficiently small neighborhoods V ⊂ Λ mp+1 (M ; I) of orb(γ) the following holds: given any compact domain K ⊂ R d and any continuous map u : K → {E mp+1 < c} ∪ V such that u(∂K) ∩ V = ∅, there exists a homotopy u s : K → Λ mp+1 (M ; I) such that u 0 = u, u 1 (K) ⊂ {E mp+1 < c}, and u s ≡ u outside u -1 (V ) for all s ∈ [0, 1].

Proof. Consider one of the infinite subsets M of the partition M 1 ∪ ... ∪ M n given by Lemma 2.2. We prove the lemma for any m ∈ M large enough so that the homotopies of Lemma 4.3 exist. The neighborhood V ⊂ Λ mp+1 (M ; I) of the critical circle orb(γ) must be small enough so that it is contained into the neighborhood U m constructed above. Let V V V be slightly smaller neighborhoods of orb(γ) in Λ mp+1 (M ; I) such that u(K) \ V ⊂ {E mp+1 < c}. Here, the symbol means that the closure of the former set in contained in the latter set. Fix > 0 small enough so that

u(K) \ V ⊂ {E mp+1 ≤ c -}.
(4.9)

We set A := u -1 (V ), A := u -1 (V ) and A := u -1 (V ). Since u(∂K) ∩ V = ∅, we have that A A A. Moreover, (4.9) can be rewritten as u(K \ A ) ⊂ {E mp+1 ≤ c -}. We introduce two bump functions: a smooth function χ : K → [0, 1] that is identically equal to 1 on A and has support inside A, and a smooth function χ : K → [0, 1] that is identically equal to 1 on A and has support inside A .

Consider the deformation retraction f s of equation (4.8). We replace our u of the statement with x → f χ (x) • u(x), so that u(A ) ⊂ U ⊂ Ω. We denote by φ s the anti-gradient flow of E mp+1 , and we recall that, by Lemma 2.2(i), Ω is invariant by this flow. By (4.9) we have that

u(∂A ) ⊂ {E mp+1 | Ω < c -}.
By Lemma 4.2, there exists τ > 0 large enough so that φ τ (u(∂A )) ∩ U = ∅ and

φ τ (u(A )) \ U ⊂ {E mp+1 < c -}.
We replace u with x → φ τ χ (x) • u(x).

Summing up, we have deformed u by means of a homotopy supported inside A to a new map, which we still denote by u, satisfying

u(K \ A ) ⊂ {E mp+1 < c}, u(A ) ⊂ Ω, u(∂A ) ∩ U = ∅, u(A ) \ U ⊂ {E mp+1 < c -}.
(4.10)

Now, we make a partition A = C ∪ D in two regions: the open shell C := A \ u -1 (U ), and the compact subset D := A \ C. Consider the homotopy r s of equation (3.4). We further replace u with x → r χ (x) • u(x), so that, by (3.6), we have that u(A ) ⊂ ω m (Υ). Since ω m is an embedding, there exists a unique continuous map v : A → Υ such that

u| A = ω m • v.
Equations (4.10) and (3.7) imply v(C) ⊂ {F q < c -} ∩ {F q < c -}.

(4.11)

Now, consider the simplexes in Σ 1 , ..., Σ d constructed in Section 4.3. We introduce a sufficiently fine triangulation of a compact neighborhood Σ ⊂ A of D, so that the image of every simplex under v is contained in a polydisc of Υ of diameter δ/5 (see Section 3.3). Up to further reducing the size of all simplexes by applying finitely many barycentric subdivisions to Σ , we can assume that the subset Σ given by all the simplexes intersecting D is contained in the interior of Σ . We denote by Σ j , for j = 0, ..., d, the collection of j-dimensional faces of the simplexes in Σ . We choose a map ι 0 : Σ 0 → Σ 0 such that dist(v(σ), ι 0 (σ)) < δ/5 for all σ ∈ Σ 0 . The existence of such a map is guaranteed by equation (4.3). We define maps ι j : Σ j → Σ j in a compatible way: given a j-simplex σ ∈ Σ j whose ordered set of 0-faces is σ 0 , ..., σ j ∈ Σ 0 , its image ι j (σ) is the (unique) simplex in Σ j whose ordered set of 0-faces is ι 0 (σ 0 ), ..., ι 0 (σ j ) ∈ Σ 0 . Notice that such a ι j (σ) exists, since

dist Υ (ι 0 (σ h ), ι 0 (σ l )) ≤ dist Υ (ι 0 (σ h ), v(σ h )) + dist Υ (v(σ h ), v(σ l )) + dist Υ (v(σ l ), ι 0 (σ l )) < 1 5 δ + 1 5 δ + 1 5 δ = 3 5 δ.
In particular, the 0-faces ι 0 (σ 1 ), ..., ι 0 (σ j ) are contained in W (ι 0 (σ 0 ), 3 5 δ). Since this polydisc is geodesically convex, the whole simplex ι j (σ) is contained in W (ι 0 (σ 0 ), 3 5 δ). The C 0 -distance between the simplexes v • σ and ι j (σ) can be estimated as

dist Υ (v(σ(x)), ι j (σ)(x)) ≤ dist Υ (v(σ(x)), v(σ 0 )) + dist Υ (v(σ 0 ), ι 0 (σ 0 )) + dist Υ (ι 0 (σ 0 ), ι j (σ)(x)) < 1 5 δ + 1 5 δ + 3 5 δ = δ.
(4.12)

We define a homotopy h s : Σ → Υ simplex by simplex as follows. Consider any d-simplex σ : ∆ d → Σ belonging to Σ d , and any point x ∈ ∆ d . We set s → h s (σ(x)) to be the unique minimal geodesic of Υ joining h 0 (σ(x)) = v(σ(x)) and h 1 (σ(x)) = ι d (σ)(x). Let χ : A → [0, 1] be a smooth bump function that is identically 1 on Σ and is supported inside Σ . We define a homotopy v

s : A → Υ by v s (x) := h sχ(x) (x), so that v 0 = v, v s | Σ = h s | Σ ,
and v s is identically equal to v outside Σ . Notice that, by the very construction of it, the homotopy v s still satisfies the same estimate as in (4.12), i.e.

dist Υ (v s (σ(x)), ι j (σ)(x)) < δ. This, together with (4.11) and the uniform equicontinuity (4.2), implies that

v s (A \ Σ ) ⊂ v s (C) ⊂ {F q < c} ∩ {F q < c}.
(4.13)

By equation (3.3), we have

ω m • v s (C) ⊂ {E mp+1 < c}.
Now, we apply Lemma 4.3: for every σ ∈ Σ d we obtain a Bangert homotopy h v1(σ) with the properties (i-iv) given there. Notice that, by property (iii), these homotopies coincide on common faces of simplexes in Σ . Moreover, by (4.13) and property (ii), for every σ ∈ Σ d \ Σ d the homotopy h v1(σ) is identically equal to ω m • v 1 • σ. By patching together the homotopies h v1(σ) , we can build a homotopy

w s : A → Λ mp+1 (M ; I), s ∈ [0, 1], so that w 0 = ω m • v 1 , w s is identically equal to ω m • v 1 outside Σ , and
w 1 (A ) ⊂ {E mp+1 < c}.
Finally, we construct the homotopy u s of the lemma as

u s (x) :=      u(x), if x ∈ K \ A , ω m • v 2s (x), if x ∈ A , s ∈ 0, 1 2 , w 2s-1 (x) if x ∈ A , s ∈ 1 2 , 1 . 4.6.
The case of positive mean index. Let now γ be an I-invariant geodesic exactly as in the previous sections, except that it has positive mean index, meaning that ind(E, γ mp+1 ) is not zero for some m ∈ N. Grove and Tanaka's Lemma 2.1 implies that such index tends to infinity as m → ∞. In this case, the assertion of Lemma 4.1 becomes a consequence of a general result from non-linear analysis (see e.g. [Cha93, Thm. 2.1 on page 92] for a similar statement in the isolated critical point case). We provide the proof in our setting of isometry-invariant geodesics, but the reader can easily extract an abstract statement that works with smooth functions having isolated critical circles. Lemma 4.5. Let V ⊂ Λ(M ; I) be a sufficiently small neighborhood of an isolated critical circle orb(γ) ⊂ crit(E). Consider a degree d such that 1 ≤ d < ind(E, γ), a compact domain K ⊂ R d , and a continuous map u : K → {E < E(γ)} ∪ V such that u(∂K) ∩ V = ∅. There exists a homotopy u s : K → Λ(M ; I) such that u 0 = u, u 1 (K) ⊂ {E < E(γ)}, and u s ≡ u outside u -1 (V ) for all s ∈ [0, 1].

Proof. The normal bundle of the critical circle orb(γ) ⊂ Λ(M ; I) splits as the Whitney sum

N (orb(γ)) = N 0 ⊕ N + ⊕ N -,
where the fibers of N 0 are the intersection of null-space of the Hessian of E with the normal bundle N (orb(γ)), while the fibers of N + and N -are the negative and positive eigenspaces respectively of the Hessian of E. Let

π : N (orb(γ)) → orb(γ) S 1
be the projection of the normal bundle onto the base. Let V 0 R ⊂ N 0 , V + R ⊂ N + , and V - R ⊂ N -be the open neighborhoods of the 0-section of radius R > 0. We choose R sufficiently small so that, by means of the exponential map of Λ(M ; I), we can identify

V 0 R ⊕ V + R ⊕ V - R
with an open neighborhood of the critical circle orb(γ), and moreover there exists δ > 0 such that

∂ 2 x -x -E(x 0 , x + , x -)[v, v] < -δ 2 v 2 π(x) , ∀x = (x 0 , x + , x -) ∈ V 0 R ⊕ V + R ⊕ V - R , v ∈ π| -1 N -(π(x)). (4.14)
Here, the norm on the bundle N -is the one associated with the Riemannian metric of Λ(M ; I). By the implicit function theorem, there exists r ∈ (0, R/2) and a bundle map φ :

V 0 r ⊕ V + r → V - R/2
such that, for each (x 0 , x + , x -) ∈ V 0 r ⊕ V + r ⊕ V - R/2 , we have ∂ x -E(x 0 , x + , x -) = 0 if and only if x -= φ(x 0 , x + ). We use φ to define the diffeomorphism onto its image

Φ : V 0 r ⊕ V + r ⊕ V - R/2 → V 0 r ⊕ V + r ⊕ V - R
given by Φ(x 0 , x + , φ(x 0 , x + )+x -). From now on, we will replace the energy function E by the composition E • Φ, so that ∂ x -E(x 0 , x + , x -) = 0 if and only if x -= 0. By equation (4.14) we have

E(x 0 , x + , x -) ≤ E(x 0 , x + , 0) -δ x -2 π(x) , ∀x = (x 0 , x + , x -) ∈ V 0 r ⊕ V + r ⊕ V - R/2 . (4.15) Let η s : V - R/2 \ {0-section} → V - R/2 \ {0-section}, for s ∈ [0, 1], be the radial defor- mation η s (x -) = (1 -s) + s R 2 x -π(0,0,x -)
x -, whose time-1 map η 1 is the radial retraction onto ∂V - R/2 . Since x -→ E(x 0 , x + , x -) is a fiberwise strictly concave function, it decreases along the deformation η s , i.e. d ds E(x 0 , x + , η s (x -)) < 0. Now, let ρ ∈ (0, r/2) be so small that sup

V 0 ρ ⊕V - ρ ⊕V + ρ E < E(γ) + δR/2. (4.16)
We require the neighborhood V of the statement to be small enough so that

V ⊂ V 0 ρ/2 ⊕ V - ρ/2 ⊕ V + ρ/2
. Let u : K → {E < E(γ)} ∪ V be a continuous map such that u(∂K) ∩ V = ∅.

We consider the open set A := u -1 (V ). Notice that there exists a compact subset K ⊂ A and a quantity > 0 such that u(K \ K ) ⊂ {E < E(γ) -}. Therefore, for our purpose, we only need to focus on the restriction u| A = (u 0 , u + , u -) : A → V.

By our assumption, the degree d = dim(K) is smaller than the Morse index ind(E, γ), which is the rank of the bundle N -. Therefore, by the transversality theorem, we can perturb the map u -inside a compact subset of A in order to obtain that u -(K ) does not intersect the 0-section of N -. The perturbation can be chosen arbitrarily small in the C 0 -topology, and in particular so small that the perturbed u still maps the complement of K to the sublevel set {E < E(γ)}, and A to the set V 0 ρ ⊕ V + ρ ⊕ V - ρ . Let χ : K → [0, 1] be a smooth map that is identically 1 on K and has support inside A. The desired homotopy u s : K → Λ(M ; I) will be given by u s (z) := (u 0 (z), u + (z), η χ(z)s • u -(z)), ∀z ∈ A.

Indeed, u s is stationary on the complement of the compact subset supp(χ) ⊂ A, it does not increase the energy, i.e.

d ds E(u s (z)) ≤ 0, and, by equations (4.15) and (4.16), its time-1 map u 1 satisfies E(u 1 (z)) ≤ E(u(z)) -δR/2 < E(γ), ∀z ∈ K .

The homotopic multiplicity result

Let (M, g) be a non-simply connected closed Riemannian manifold. Each connected component of its free loop space Λ(M ; id) is not simply connected. Indeed, the fundamental group of M injects into the fundamental group of the connected components of Λ(M ; id) of the contractible loops; moreover, given any element γ ∈ Λ(M ; id), its orbit under the circle action t • γ = γ(t + •) is a contractible loop in Λ(M ; id) if and only if γ itself is a contractible loop in M .

Let I be an isometry of (M, g). If I is homotopic to the identity (not necessarily through isometries), the space of I-invariant curves Λ(M ; I) is homotopy equivalent to the free loop space Λ(M ; id), see [START_REF] Grove | Condition (C) for the energy integral on certain path spaces and applications to the theory of geodesics[END_REF]Lemma 3.6]. Therefore, all the connected components of Λ(M ; I) are not simply connected. If M were the circle, all connected components of Λ(M ; I) would be homotopy equivalent to the circle. Obviously, in this case, there is only one I-invariant geodesic: the circle M itself. The next theorem, which is the main result of this paper, asserts that there are infinitely many isometry-invariant geodesics whenever the homotopy groups of Λ(M ; I) are richer than in the example of the circle.

Theorem 5.1. Let (M, g) be a connected closed Riemannian manifold equipped with an isometry I. Assume that there exists a degree d ≥ 1 and an infinite sequence of pairwise distinct connected components C n ⊂ Λ(M ; I), for n ∈ N, with nontrivial homotopy group π d (C n ). If d = 1, assume further that these fundamental groups are not cyclic. Then, there exist infinitely many I-invariant geodesics in C := ∪ n∈N C n .

Proof. Since we are looking for infinitely many I-invariant geodesics in C, we can assume that the critical circles of E| C are isolated. In particular, all I-invariant geodesics contained in C must be periodic curves. We choose as base-point of C n an I-invariant geodesic α n whose critical circle is a global minimum of E in the connected component. In general we cannot expect the family of the α n 's to give infinitely many I-invariant geodesics: for instance, in the worst case scenario, α 1 may be an I-invariant geodesic that is periodic with minimal period q ≥ 1, and each α n may be of the form α nq+1 1 . We will prove the theorem by contradiction: we assume that C contains only the I-invariant geodesics γ 1 , ..., γ r (together with infinitely many of their "iterates"). For each of these γ i 's, we denote by p i ≥ 1 its minimal period, and by m i the integer such that for all m ≥ m i the assertion of Lemma 4.1 holds. If n is large enough, say n ≥ n, an iterated I-invariant geodesic γ mpi+1 i belongs to C n only if m ≥ m i . We fix n ≥ n and a non-trivial homotopy class h n ∈ π d (C n , α n ). If d = 1, we assumed that π d (C n , α n ) is not cyclic, and therefore we can choose h n that is not represented by a multiple of the critical circle orb(α m ). The representatives of h n are maps of the form u : (B d , ∂B d ) → (C n , α n ). Notice that every such map cannot have image entirely contained in a small tubular neighborhood N ⊂ C n of orb(α n ). Otherwise, the map u could be deformed inside orb(α n ), which would force d = 1 and h n to be represented by a multiple of the critical circle orb(α n ). Since orb(α n )

  g. [Maz12, Thm. 4.1.2]. For each pair of elements ζ 0 , ζ 1 ∈ W (γ, r), the unique shortest geodesic s → ζ s joining them is given by ζ s (τ i ) = γ ζ0(τi),ζ1(τi) (s), ∀i = 0, ..., k -1.

  1 as a union of affine curves of slope (1, ..., 1) ∈ R d , so that each x ∈ ∆ d belongs to the curve joining the points α(x), ω(x) ∈ ∂∆ d , and the maps α : ∆ d → ∂∆ d and ω : ∆ d → ∂∆ d are continuous; we define the restriction of σ i0...i d to the curve passing through x ∈ ∆ d as the geodesic of Υ joining σ i0...i d (α(x)) and σ i0...i d (ω(x)).

is a strict local minimum of the energy E, we have

(5.1) By Morse theory, c is a critical value of E| Cn . The assumption made in the previous paragraph implies that every critical circle in C n at level c is of the form γ mipi+1 i , where γ i ∈ {γ 1 , ..., γ r } and m i ≥ m i . We denote by V i ⊂ C n a sufficiently small neighborhood of the critical circle orb(γ mipi+1 i ) that does not contain other critical circles of E, and by V the (disjoint) union of the these neighborhoods for all the critical circles in C n at level c. By Morse theory, we can find a representative u :

This contradicts the definition of c in (5.1).

Proof of Theorem 1.1. Let ev : Λ(M ; id) → M be the evaluation map ev(γ) = γ(0). We fix a generator θ of the fundamental group π 1 (M )

Z, and we denote by D n ⊂ Λ(M ; id) the connected component of the free loop space containing the n-fold iteration of the curve θ. Since the closed manifold M is not a circle, it is not homotopy equivalent to a circle. In particular, by Whitehead Theorem, M must have a non-trivial homotopy group in some degree d ≥ 2. We fix d to be a minimal such degree. Bangert and Hingston proved in [START_REF] Bangert | Closed geodesics on manifolds with infinite Abelian fundamental group[END_REF] that there exists k ∈ N and, for all n ∈ N, a non-trivial homotopy class h n ∈ π d-1 (D kn ) such that ev * (h n ) = 0. As we have already remarked at the beginning of this section, all connected components D n are not simply connected, and the evaluation map induces a surjective homomorphism ev * : π 1 (D n ) → π 1 (M ) for all n > 0. Therefore, if the above degree d is equal to 2, Bangert and Hingston's result implies that, for all n > 0, the fundamental group π d-1 (D kn ) is not cyclic. Now, let I be an isometry that is homotopic to the identity. We already recalled that there exists a homotopy equivalence ι : Λ(M ; id) → Λ(M ; I). We denote by C n ⊂ Λ(M ; I) the connected component containing ι(D n ). Notice that these connected components are pairwise distinct, and they have the same homotopy groups as the corresponding D n 's. Therefore, the sequence C kn , for n ∈ N, satisfies the assumptions of Theorem 5.1, which implies that the union C := ∪ n∈N C kn contains infinitely many I-invariant geodesics.


Randal Douc

François Roueff

Philippe Soulier
email: philippe.soulier@u-paris10.fr

On the existence of some ARCH(∞) processes

Keywords: ARCH processes, Fractionaly integrated processes, Long memory MSC: 60G10, 62M10

A new sufficient condition for the existence of a stationary causal solution of an ARCH(∞) equation is provided. This condition allows to consider coefficients with power-law decay, so that it can be applied to the so-called FIGARCH processes, whose existence is thus proved.

Introduction

It can arguably be said that autoregressive conditionnally heteroskedastic (ARCH) and long memory processes are two success stories of the nineties, so that they were bound to meet. Their tentative offspring was the FIGARCH process, introduced by [START_REF] Baillie | Fractionally integrated generalized autoregressive conditional heteroskedasticity[END_REF] without proving its existence, which has remained controversial up to now. More precisely, the FIGARCH(p, d, q) process is the solution of the equations

X n = σ n z n , (1)
σ 2 n = a 0 + I -(I -L) d θ(L) φ(L) X 2 n , (2)
where {z n } is an i.i.d. sequence with zero mean and unit variance, a 0 > 0, d ∈ (0, 1), L is the backshift operator and (I -L) d is the fractional differencing operator:

(I -L) d = I + ∞ j=1 (-d)(1 -d) • • • (j -1 -d) j! L j ,
and θ and φ are polynomials such that θ(0) = φ(0) = 1, φ(z) = 0 for all complex number z in the closed unit disk and the coefficients of the series expansion of 1 -(1 -z) d θ(z)/φ(z) are nonnegative. Then the coefficients {a j } j≥1 defined by ∞ j=1 a j L j = I -(I -L) d θ(L)/φ(L) satisfy a j ∼ cj -d-1 for some constant c > 0 and ∞ j=1 a j = 1.

These processes are subcases of what can be called IARCH(∞), defined as solutions of the equations (1) and

σ 2 n = a 0 + ∞ j=1 a j X 2 n-j , (3)
for some sequence {a j } such that a 0 > 0 and ∞ j=1 a j = 1. The letter I stands for integrated, by analogy to ARIMA processes. An important property of such processes is that a stationary solution necessarily has infinite variance.

Indeed, if σ 2 = E[σ 2 n ] < ∞, then E[X 2 n ] = σ 2 and (3) implies σ 2 = a 0 + σ 2 ,
wich is impossible. If the condition ∞ j=1 a j = 1 is not imposed, a solution to equations ( 1) and ( 3) is simply called an ARCH(∞) process.

A solution of an ARCH(∞) equation is said to be causal with respect to the i.i.d. sequence {z n } if for all n, σ n is F z n-1 measurable, where F z n is the sigmafield generated by {z n , z n-1 , . . .}. Note that to avoid trivialities, here and in the following, σ n is the positive square root of σ 2 n . There exists an important literature on ARCH(∞), IARCH(∞) and FIGARCH processes. For a recent review, see for instance [START_REF] Giraitis | ARCH(∞) models and long memory properties[END_REF]. The known conditions for the existence of stationary causal conditions to ARCH equations are always a compromise between conditions on the distribution of the innovation sequence {z n } and summability conditions on the coefficients {a j , j ≥ 1}. [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] provides a necessary and sufficient condition for the solution to have finite fourth moment. The only rigorous result in the IARCH(∞) case was obtained by [START_REF] Kazakevičius | A new theorem on the existence of invariant distributions with applications to ARCH processes[END_REF]. They prove the existence of a causal stationary solution under the condition that the coefficients a j decay geometrically fast, which rules out FIGARCH processes, and on a mild condition on the distribution of z 0 .

The purpose of this paper is to provide a new sufficient condition for the existence of a stationary solution to an ARCH(∞) equation, which allows power-law decay of the coefficients a j s, even in the IARCH(∞) case. This condition is stated in Section 2. It is applied to the IARCH(∞) case in Section 3 and the existence of a stationary solution to the FIGARCH equation is proved. Further research directions are given in Section 4. In particular, the memory properties of FIGARCH processes are still to be investigated. This is an important issue, since the original motivation of these processes was the modelling of long memory in volatility.

Theorem 1 Let {a j } j≥0 be a sequence of nonnegative real numbers and {z k } k∈Z a sequence of i.i.d. random variables. For p > 0, define

A p = ∞ j=1 a p j and µ p = E[z 2p 0 ] .
If there exists p ∈ (0, 1] such that

A p µ p < 1, (4)
then there exists a strictly stationary solution of the ARCH(∞) equation:

X n = σ n z n , (5)
σ 2 n = a 0 + ∞ j=1 a j X 2 n-j , (6)
given by ( 5) and

σ 2 n = a 0 + a 0 ∞ k=1 j 1 ,...,j k ≥1 a j 1 . . . a j k z 2 n-j 1 . . . z 2 n-j 1 -•••-j k . (7)
The process {X n } so defined is the unique causal stationary solution to equations ( 5) and ( 6)

such that E[|X n | 2p ] < ∞. Proof. Denote ξ k = z 2 k , so that E[ξ p k ] = µ p , and define the [0, ∞]-valued r.v. S 0 = a 0 + a 0 ∞ k=1 j 1 ,...,j k ≥1 a j 1 . . . a j k ξ -j 1 . . . ξ -j 1 -•••-j k (8)
Since p ∈ (0, 1], we apply the inequality (a + b) p ≤ a p + b p valid for all a, b ≥ 0 to S p 0 :

S p 0 ≤ a p 0 + a p 0 ∞ k=1 j 1 ,...,j k ≥1 a p j 1 . . . a p j k ξ p -j 1 . . . ξ p -j 1 -•••-j k .
Then, by independence of the ξ j 's, we obtain

E[S p 0 ] ≤ a p 0 + a p 0 ∞ k=1 j 1 ,...,j k ≥1 a p j 1 . . . a p j k E[ξ p -j 1 . . . ξ p -j 1 -•••-j k ] = a p 0 1 + ∞ k=1 (µ p A p ) k = a p 0 1 -A p µ p , (9)
where we used (4). This bound shows that S 0 < ∞ a.s. and the sequence

S n = a 0 + a 0 ∞ k=1 j 1 ,...,j k ≥1 a j 1 . . . a j k ξ n-j 1 . . . ξ n-j 1 -•••-j k , n ∈ Z ,
is a sequence of a.s. finite r.v.'s. Since only nonnegative numbers are involved in the summation, we may write

∞ j=1 a j S n-j ξ n-j = a 0 ∞ j 0 =1 a j 0 ξ n-j 0 + a 0 ∞ j 0 =1 a j 0 ξ n-j 0 ∞ k=1 j 1 ,...,j k ≥1 a j 1 . . . a j k ξ n-j 0 -j 1 . . . ξ n-j 0 -j 1 -•••-j k = a 0 ∞ k=0 j 0 ,j 1 ,...,j k ≥1 a j 0 . . . a j k ξ n-j 0 . . . ξ n-j 0 -j 1 -•••-j k .
Hence {S n , n ∈ Z} satisfies the recurrence equation

S n = a 0 + ∞ j=1 a j S n-j ξ n-j .
The technique of infinite chaotic expansions used here is standard; it was already used in the proof of (Kokoszka and Leipus, 2000, Theorem 2.1). This proves the existence of a strictly stationary solution for ( 5) and ( 6) by setting

σ 2 n = S n and X n = σ n z n . Using (9), we moreover have E[|X n | 2p ] ≤ µ p a p 0 /(1 - A p µ p ).
Suppose now that {X n } is a strictly stationary causal solutions of the ARCH(∞) equations ( 5) and ( 6). Then, for any q ≥ 1, the following expansion holds:

σ 2 n = a 0 + a 0 q k=0 j 1 ,...,j k ≥1 a j 1 . . . a j k ξ n-j 1 . . . ξ n-j 1 -•••-j k (10) + j 1 ,...,j q+1 ≥1 a j 1 . . . a j q+1 ξ n-j 1 . . . ξ n-j 1 -•••-jq X 2 n-j 1 -•••-j q+1 . (11)
The last display implies that the series on the right-hand side of (10) converges to S n as q → ∞. Denote by R n,q the remainder term in (11

). Since {X n } is a causal solution, X n-j 1 -•••-j q+1 is independent of ξ n-j 1 . . . ξ n-j 1 -•••-jq for all j 1 , . . . , j q+1 ≥ 1. Hence, for any p ≤ 1, E[R p n,q ] ≤ (A p µ p ) q E[X 2p 0 ] . If Assumption (4) holds and E[X 2p 0 ] < ∞, then E[ q≥1 R p n,q ] < ∞ so that, as q → ∞, R n,q → 0 a.s., implying σ 2 n = S n a.s. 2 3 IARCH(∞) processes
IARCH (Integrated ARCH) processes are particular ARCH(∞) processes for which A 1 µ 1 = 1, or, equivalently up to a scale factor,

A 1 = 1 and µ 1 = 1 (12)
To the best of our knowledge, the only rigorous general result on IARCH(∞)

processes was obtained by [START_REF] Kazakevičius | A new theorem on the existence of invariant distributions with applications to ARCH processes[END_REF]. See [START_REF] Giraitis | ARCH(∞) models and long memory properties[END_REF] for a recent review. In Theorem 2.1 of [START_REF] Kazakevičius | A new theorem on the existence of invariant distributions with applications to ARCH processes[END_REF],

it is proved that if

E[| log(z 0 )| 2 ] < ∞ , ( 13
) i a i q i < ∞ for some q > 1 , (14)
equations ( 5)-( 6). Condition (13) on the distribution of z 0 is mild, but the condition ( 14) rules out power-law decay of the coefficients {a j }.

Theorem 1 yields the following sufficient condition for the existence of a IARCH(∞) process.

Corollary 2 If A 1 = 1 and µ 1 = 1, (4) holds for some p ∈ (0, 1] if and only if there exists p * < 1 such that A p * < ∞ and

∞ j=1 a i log(a i ) + E[z 2 0 log(z 2 0 )] ∈ (0, ∞] . (15)
Then, the process defined by ( 5) and ( 7) is a solution of the ARCH(∞) equa-

tion and E[|X n | q ] < ∞ for all q ∈ [0, 2) and E[X 2 n ] = ∞.
Proof. Since a i ≤ 1 for all i ≥ 1, it holds that ∞ j=1 a i log(a i ) ≤ 0 and the convexity of the function x → x log(x) implies E[z 2 0 log(z 2 0 )] ≥ 0.

First assume that there exists p ∈ (0, 1] such that (4) holds. Since A 1 = µ 1 = 1, then necessarily, p < 1 and for all q ∈ [p, 1], A q < ∞. Thus we can define the

function φ : [p, 1] → R by φ(q) = log(A q µ q ) = log ∞ j=1 a q j + log E[z 2q 0 ] .
Hölder inequality implies that the functions q → log ∞ j=1 a q j and q → log E[z 2q 0 ] are both convex on [p, 1]. Thus φ is also convex on [p, 1] and, since φ(p) < 0 and φ(1) = 0, the left derivative of φ at 1, which is given by the left-hand side of (15), is positive (possibly infinite).

Conversely suppose that there exists p * < 1 such that A p * < ∞ and that (15) holds. Then φ is a convex function on [p * , 1] and (15) implies that φ(q) < 0 for q < 1 sufficiently close to 1.

By convexity of φ and since φ(1) = 0, we also get that A p µ p < 1 implies A q µ q < 1 for all q ∈ [p, 1). Then, by Theorem 1, the process {X n , n ∈ Z} defined by ( 7) and ( 6) is a solution to the ARCH(∞) equation and satisfies

E[|X 0 | q ] < ∞ for all positive q < 2. 2.
Comments on Corollary 2.

(i) Condition ( 15) is not easily comparable to conditions ( 13) and ( 14) of [START_REF] Kazakevičius | A new theorem on the existence of invariant distributions with applications to ARCH processes[END_REF]. Condition ( 15) is not necessary to prove the existence of a causal stationary solution if the coefficients a j decay geometrically fast (in particular if there are only finitely many nonvanishing coefficients), as a consequence of [START_REF] Kazakevičius | A new theorem on the existence of invariant distributions with applications to ARCH processes[END_REF] Theorem 2.1); however, this result does not prove that any moments of X n are finite, contrary to Corollary 2.

(ii) It might also be of interest to note that the Lyapounov exponent of the FIGARCH process as defined in [START_REF] Kazakevičius | A new theorem on the existence of invariant distributions with applications to ARCH processes[END_REF] is zero.

So our result proves that such a feature is not in contradiction with strict stationarity.

(iii) In the specific case of IGARCH processes, which are particular parametric subclasses of IARCH(∞) processes, [START_REF] Bougerol | Stationarity of GARCH processes and of some nonnegative time series[END_REF] have a different set of assumptions on the distribution of z 0 : they assume that P(z 2 0 = 0) = 0 and that the support of the distribution of z 2 0 is unbounded.

(iv) The moment E[z 2 0 log(z 2 0 )] can be arbitrarily large (possibly infinite) if the distribution of z 2 0 has a sufficiently heavy tail. It is infinite for instance if the distribution of z 2 0 is absolutely continuous with a density bounded from below by 1/(x 2 log 2 (x)) for x large enough. In that case, condition (15) holds for any sequence {a j } such that A p * < ∞ for some p * < 1. This conditions allows for a power-law decay of the coeffficients a j , for instance a j ∼ cj -δ , for some δ > 1.

Corollary 2 can be used to prove the existence of a causal strictly stationary solution to some FIGARCH(p, d, q) equations. Let us illustrate this in the case of the FIGARCH(0, d, 0) equation, that is ( 5) and ( 6) with d ∈ (0, 1), a 0 > 0 and a j = π j (d) for all j ≥ 1, where Proof. For d ∈ (0, 1] and p ∈ (1/(d + 1), 1], denote

π 1 (d) = d , π j (d) = d(1 -d) • • • (j -1 -d) j! , j ≥ 2 .
H(p, d) = log ∞ j=1 π p j (d) , L(d) = ∞ j=1 π j (d) log(π j (d)) .
For d ∈ (0, 1), π j (d) ∼ cj -d-1 , so that H(p, d) is defined on (1/(d+1 (ii) Tail behaviour of the marginal distribution of GARCH processes have been investigated by [START_REF] Basrak | Regular variation of GARCH processes[END_REF], following [START_REF] Nelson | Stationarity and persistence in the GARCH(1, 1) model[END_REF], but there are no such results in the ARCH(∞) case. Under suitable conditions, we have shown that the squares of the FIGARCH process X 2 n have finite moments of all order p < 1, but necessarily, E[X 2 n ] = ∞. Thus, it is natural to conjecture that perhaps under additional conditions on the distribution of z 0 , the function x → P(X ) is a parametric model, so the issue of estimation of its parameter is naturally raised. Also, if d is linked to some memory property of the process, semi-parametric estimation of d would be of interest.

Corollary 3

 3 Assume that {z k } k∈Z a sequence of i.i.d. random variables, such that E[z 2 0 ] = 1 and P{|z 0 | = 1} < 1. Then there exists d * ∈ [0, 1) such that, for all d ∈ (d * , 1), the FIGARCH(0, d, 0) equation has a unique causal stationary solution satisfying E[|X n | 2p ] < ∞ for all p < 1.

  We conjecture, but could not prove, that L(d) is increasing, so that (15) holds if and only if d > d * (withd * = 0 if E[ξ 0 log(ξ 0 )] = ∞). But this does not prove that the FIGARCH(0, d, 0) does not exist for d ≤ d * .The problem remains open to know if there exist a stationary solution under a mild assumption on z 0 , such as (13) for instance. If a solution exists, say {X n }, then, as seen in the proof of Theorem 1, the sequence {S n } defined in (8) is well defined and Y n =

	that there exists d * ∈ (0, 1) such that L(d) + E[z 2 0 log(z 2 0 )] > 0 (i.e. (15)
	holds) if d > d * . Thus Corollary 2 proves the existence of the corresponding
	FIGARCH(0, d, 0) processes.		2.
	Remark. It easily seen that L(d) ≤ log(d) so that lim d→0 L(d) = -∞, i.e. (15)
	does not hold for small d. 4 Open problems
	Now that a proof of existence of some FIGARCH and related processes is
	obtained under certain conditions, there still remain some open questions. We
	state a few of them here.
	(i) Condition (15) is not necessary for the existence of a stationary causal
	solution, but it implies finiteness of all moments up to 1 of X 2 n (with of
	course E[X 2			), 1]. More-
	over, it is decreasing and convex with respect to p, H(1, d) = 0 and ∂ p H(1, d) =
	L(d). Also, π j (d)/d is a decreasing function of d and lim d→1 π j (d) = 0 for all
	j ≥ 2. Thus, by bounded (and monotone) convergence, for all p ∈ (1/2, 1), it
	holds that lim d→1,d<1 H(p, d) = 0. By convexity of H with respect to p, the S 1/2 n z n is also a stationary causal solution wich satisfies moreover Y 2 n ≤
	following bound holds: X 2 n . But we cannot prove without more assumptions that these solutions
	are equal.	0 ≤ -L(d) ≤	1 -p H(p, d)	.
	Hence lim d→1 L(d) = 0. By assumption, we have E[z 2 0 log(z 2 0 )] > 0. This implies

n ] = ∞).

  The memory properties of the FIGARCH process are of course of great interest. The sequence {X n } is a strictly stationary martingale increment sequence, but E[X 2 n ] = ∞. So does it hold that the partial sum process n -1/2 [nt] k=1 X k converges weakly to the Brownian motion? For p ∈ [1, 2), do the sequences {|X n | p } have distributional long memory in the sense that n -H [nt] k=1 {|X k | p -E[|X k |

	2 n > x) is regularly varying with
	index -1.
	(iii)

p ]} converge to the fractional Brownian motion with Hurst index H for a suitable H > 1/2? (iv) Statistical inference. The FIGARCH

(p, d, q

Acknowledgement

The authors are grateful to Remigijus Leipus for his comments on a preliminary version of this work.