Exponential gaps in the length spectrum

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.


Introduction
The Anosov property of the geodesic flow on a compact manifold with negative curvature provides a great abundance of closed geodesics : the counting function satisfies where h top > 0 denotes the topological entropy of the flow and ℓ(γ) the length of a closed, unoriented geodesic γ. We refer the reader to [Mar69,MS04,PP90] for comprehensive studies of periodic orbits for hyperbolic systems. For generic metrics in negative curvature, distinct closed geodesics have distinct lengths [Abr70] : as a result, the length spectrum L contains exponentially many points in an interval I(T ) of fixed size centered at T → +∞. This suggests naturally to study the distribution of nearby lengths in L in this asymptotic limit : see [Dol98,Ana00,PS01,PS12] and references given there for refinements of the counting estimate (1.1).
The length spectrum is also of particular interest when studying the connexions between the geometry of M and the spectrum of its Laplace-Beltrami operator ∆ g for the metric g : if closed geodesics are isolated and non-degenerate in the sense of Morse, it is well known since [CdV73,Cha74,DG75] that the singular support of the tempered distribution Trace(e i t √ ∆g ) is given by the period spectrum P = {kℓ, ℓ ∈ L , k ∈ N}. Such distributional traces have been used widely since then in various direct and inverse spectral problems.
A box principle used with equation (1.1) shows that in L ∩ I(T ), there are many gaps of size at least e −htopT when T is large. However, nothing excludes a priori some over-exponential clustering in the length spectrum : in this case it would not be possible to control the gaps uniformly from below. Some interesting results concerning this question have been obtained by Dolgopyat and Jakobson in [DJ16]. To precise the problem of estimating the gaps in L from below, we will say that the length spectrum is exponentially separated if there exist C, ν > 0 with the property that (1.2) ∀ℓ, ℓ ′ ∈ L with ℓ = ℓ ′ , |ℓ − ℓ ′ | ≥ C e −ν max(ℓ,ℓ ′ ) .
In [DJ16] it is established that the length spectrum of hyperbolic manifolds is exponentially separated as soon as the fundamental group have algebraic generators. The authors note that it is always the case for finite volume hyperbolic manifolds of dimension n ≥ 3, see [GR70]. They also show that for compact hyperbolic surfaces, (1.2) is true for a dense set in the corresponding Teichmüller space.
In variable negative curvature, Theorem 4.1 in [DJ16] gives a rather different picture : (1.2) is false for a G δ -dense set of metrics for the C k -topology, k > 3. Hence metrics with a length spectrum which is not exponentially separated are topologically generic, and then dense.
In this note, we provide a complementary result : in variable negative curvature, the set of metrics satisfying (1.2) is also dense for the C k -topology, with k ≥ 2. To state our result more precisely, we assume that (M, g 0 ) is a closed manifold of dimension ≥ 2 with negative curvature, and that g 0 is of class C r with r ≥ 2. Write the bounds of the sectional curvature K(g 0 ) as Fix some integer k ∈ [2, r], and for ε 0 > 0 define the set of Riemannian metrics The constant ε 0 will always be fixed and chosen small enough so that K 1 > 0, which is possible as the sectional curvature is a continuous function of the metric in the C 2 -topology. Our main result can be stated as follows: Theorem 1. Let M be a closed Riemannian manifold equipped with a C r metric g 0 with negative curvature, r ≥ 2. Take k ∈ [2, r] and ε 0 > 0 as above. For some arbitrary ε > 0, define Let us point out that if k is fixed with k < r, the metric g given by our proof of Theorem 1 is not C k ′ for r ≥ k ′ > k. This is due to the fact that g is obtained as a limit of C r metrics (g n ) n∈N with diverging C k ′ norm if k ′ > k. Improving the regularity of g without increasing ν k , if possible, would probably require a new approach than the one presented here.
Some fine control over the gaps in the length spectrum is in particular important for two well-known spectral problems : precise error terms for Weyl's laws on a negatively curved manifold [JP07,JPT07], and lower bounds for the asymptotic distribution of resonances of the Laplacian on non-compact manifolds with hyperbolic trapped sets [GZ99,Pet02]. In both cases, one has to study accurately the contributions of potentially many periodic orbits of the geodesic flow to a semiclassical trace formula, which involves the distributional trace we alluded to above. Periodic orbits are subsets of the unit tangent bundle T 1 M, and it turns out that they can be exponentially isolated there : this is a consequence of the expansivity of the geodesic flow, see also Section 2 below. This allows to overcome a lack of control of the gaps in the length spectrum to establish a trace formula by microlocalizing near each closed geodesic in the unit cotangent bundle -see [JPT07]. Such formulae involve sums of the form where A γ,k > 0 is related to the Poincaré map of γ, and here J(T ) ⊂ [0, T ] is some interval. Obtaining lower bounds for the above sum is one of the main difficulties that stand in the way of improved Weyl laws for eigenvalues (or resonances). Such estimates requires in particular the control of the oscillating terms which are directly connected with the distribution of the gaps in the length spectrum.

Separation of orbits in phase space
Let us start this section by gathering some standard facts and notations about the geodesic flow Φ t : T 1 M → T 1 M. In the following, closed geodesics will always be unoriented. The set of closed geodesics for (M, g) with g ∈ M(ε 0 ) will be denoted by C g . If I ⊂ R + is an interval, we will also make use of the sets If ρ ∈ T 1 M and t ∈ R, we write ρ(t) def = Φ t (ρ). By γ, we will denote one of the two possible lifts of γ in T 1 M, namely (γ(t), ±γ(t)) t∈R where t → γ(t) is any parameterization of γ by arc-length.
We equip T 1 M with the Sasaki metric and write d S for the induced distance. A fundamental estimate coming from the analysis of the Green subbundles [Ebe73] together with Rauch's comparison theorem and (1.3) yields to for all ρ, ρ ′ ∈ T 1 M and t ∈ R. For any γ 0 ∈ C g 0 , the curve γ 0 ⊂ M is still a closed path for g : since g ∈ M k (ε 0 ) is negatively curved, there is a unique g-geodesic γ in the same free homotopy class. Hence, there is a well defined, bijective map (2.2) f g 0 →g : Furthermore, the g-length of γ = f g 0 →g (γ 0 ) satisfies Also, by a theorem of Klingenberg, K(g) < 0 implies that the injectivity radius r inj (g) is determined by the shortest closed geodesic, In the following, we fix a number r m > 0 such that r m < (1 + ε 0 ) −1/2 r inj (g 0 ). Finally, if γ ∈ C g , we define the tubular open neighborhood of a lift γ ⊂ T 1 M by The purpose of this section is to prove the following exponential separation result in T 1 M : where β, γ denote any lifts of β, γ in T 1 M.
We begin with a lemma where the lengths of the geodesics are restricted to a fixed interval.
where γ, γ ′ denote any lifts of γ, γ ′ in T 1 M and δ = rm 2κ 0 . Proof. We borrow the method from [JPT07] and argue by contradiction. Let γ, γ ′ ∈ C g (T ) be two distinct geodesics with |ℓ(γ) − ℓ(γ ′ )| < r m /2 and take two lifts Since r m < r inj (g), this means that there is an homotopy by geodesic segments between the closed curves γ ′ and β. But β being a reparameterization of γ, this implies that we would have two distinct closed geodesics in the same free homotopy class.
Proof of Proposition 2. The preceding lemma can be generalized for β, γ ∈ C g (T ) as follows. Before, let us recall a classical fact for hyperbolic flows which is at the basis of the Anosov Closing and Shadowing lemmas. Denote the dynamical ball Suppose that ρ 0 ∈ T 1 M is a periodic point of period τ for the geodesic flow. There exist a constant σ(g) > 0, depending continuously on the metric in the C 2 -topology such that if Σ σ(g) (ρ 0 ) is a transversal section of the flow at ρ 0 with radius σ(g), then Namely, ρ 0 is the unique periodic point of period τ in Σ σ(g) (ρ 0 ) sufficiently close to ρ 0 for the dynamical distance. This follows from a standard fixed point argument using hyperbolicity and a sequence of quasi-linear C 1 approximations of the flow near the orbit ρ 0 (t) for 0 ≤ t ≤ τ , see [KH95, Chapters 6 and 18].
Since we consider g ∈ M k (ε 0 ) with k ≥ 2, we can define to get a constant uniform with respect to g ∈ M k (ε 0 ). Consider now distinct β, γ ∈ C g (T ) and assume for instance that Fix some ǫ 0 > 0 to be chosen sufficiently small later (in function of g 0 , ε 0 ), and take two lifts β, γ ⊂ T 1 M of these closed geodesics. Suppose that there are z ∈ γ, Hence the point Φ t (z) stays exponentially close to Φ t (z ′ ) for t ∈ [0, l ′ ]. Moreover, Let Σ σ (z) be a Poincaré section of the flow of size σ centered at z. Without loss of generality, we can assume that z ′ ∈ Σ σ (z). Equation (2.5) implies that for ǫ 0 small enough depending on g 0 , ε 0 via κ, σ, the point Φ l ′ (z) belongs to a flow box with section Σ σ (z) : there is a unique s ∈ R such that Here C = C(g 0 , ε 0 ) ≥ 1 can be chosen uniformly for g ∈ M k (ε 0 ) since the distance d S varies continuously with g. But now, ζ and z are on the same periodic orbit, so they are periodic with period l and both belong to Σ σ (z). Also, using (2.1) one more time, we have Therefore, if ǫ 0 is small enough, the right hand side of the previous equation is < σ and we have ζ ∈ B T S (z, σ) ∩ Σ σ (z). From the remark above, this forces ζ = z. Up to shrink ǫ 0 further, this finally implies

Almost intersections of closed geodesics
In this section we prove the key tool needed in the proof of Theorem 1. If x ∈ M and r > 0, we denote by B g (x, r) ⊂ M an open ball of radius r centered at x for the metric g.
and moreover, B g (z, e −αT ) ∩ β consists in a unique geodesic segment of β centered at z.
The main idea is as follow : since two closed geodesics β, γ ∈ C g (T ) are separated by ǫ 0 e −2κT when lifted in T 1 M, it means that if β and γ are very close somewhere in M, the angle "between" their tangent vectors must be bounded below. It follows that the two geodesics cannot stay close to each other in M for a long time. Proposition 6 below is a quantitative version of this observation.
3.1. Local divergence of orbits separated in phase space. The next proposition is a simple application of the Toponogov comparison theorem [Kar89] to study the local divergence of two geodesics close from each other in M, but separated when lifted in T 1 M.
Given two geodesics β, γ ∈ C g (T ), the preceding proposition will enable us to estimate the total length of pieces of these geodesics which are close to each other.

Intersections and almost-intersections.
Roughly speaking, finding z in Proposition 4 can be done if we are able to find a sufficiently large piece of β ∈ C g (T ) that "avoids" both all other geodesics in C g (T ), and all the rest of β . This suggests to study the situations where two given closed geodesics intersect, or more generally come close to each other.
Let β, γ ∈ C g (T ) be two closed geodesics. It is essentially well known that their intersection number i(β, γ) ∈ N grows quadratically with T : This is a consequence of the fact that if x and x ′ are two intersection points such that . The estimate (3.6) follows by cutting γ into pieces of length < r inj (g) and applying the above property repeatedly. Intersections of two closed geodesics are special situations where small tubular neighborhoods of the geodesics intersect, and the above remark shows that the topology of the manifold excludes that too many intersections points can be themselves close to each other.
In dimension 3 or more, intersections and self-intersections are marginal, so we would like to generalize (3.6) to "almost-intersections", namely regions in M where two geodesics are close to each other but without necessarily intersecting. Of course points of almost-intersection will not be countable, so we will have to consider small segments.
Before continuing further, for ǫ > 0 let us denote by an open tubular neighborhood of γ ∈ C g of size ǫ. The next proposition allows to control the size of β ∩ θ ǫ γ when β, γ ∈ C g (T ) : Proposition 6. Let β, γ ∈ C g (T ). Fix α ≥ 2κ + h and set There are T 0 , C 3 > 0 depending only on g 0 , ε 0 such that for T > T 0 , if the set β ∩ θ ǫ γ is not empty, it can be covered by a finite number of geodesic segments {J 1 , . . . , J n(β,γ) } ⊂ β with the following properties : Before giving the proof of this proposition, we need some preliminary results. We fix two parameterizations of β, γ by arc-length and define the continuous maps Consider an open connected component U ⊂ R 2 of D −1 β,γ (]0, ǫ[). Such a set U exists since we assumed β ∩ θ ǫ γ = ∅, furthermore, there is (s u , t u ) ∈ U which is a local minimum of D β,γ | U . Assume that T is large enough so that e −αT ≤ r m /2. By the convexity of the distance function in negative curvature, for s, t such that 0 < |t| ≤ r m /2 and 0 < |s| ≤ r m /2, we have It follows that the local minima of D β,γ are isolated, and the total number of local minima of this map is at most is a local minimum of D β,γ and (x, y) = G β,γ (s, t), we say that (x, y) is an almost-intersection of β and γ.
Consider now an almost intersection (x, y) ∈ β × γ, and let us shift the origin of the parameterizations of β and γ so that we can write Proof. Since Proposition 2 ensures that Θ ǫ 0 e −2κT γ ∩ Θ ǫ 0 e −2κT β = ∅ for any lifts γ, β in T 1 M, this implies d S ((x, ±ξ), (y, η)) ≥ ǫ 0 e −2κT . On the other hand, d g (x, y) < ǫ, so we are in position to apply Proposition 5, which gives readily and the same equation holds true by exchanging the roles of x and y. Define C 3 = 2C −1 1 (1 + C 2 ) and take T ≥ T 0 where C 3 e −hT 0 < r m . In this case, we have Therefore, there is a (maximal) non-empty open interval ]t − We can define an interval ]t − y , t + y [ in the same way by permuting the roles of x and y. The open segments with the desired properties are precisely Proof of Proposition 6. Call   (s 1 , t 1 ), . . . , (s n(β,γ) , t n(β,γ) ), n(β, γ) ≤ 4 T r m 2 the local minima of the function D β,γ , and write (x i , y i ) = G β,γ (s i , t i ) the almostintersections identified with these local minima via the parameterizations of the closed geodesics. We just need to check that if (x, y) ∈ β × γ is such that then there is i ∈ [1, n(β, γ)] such that x ∈ J(x i , y i ) and y ∈ J(y i , x i ) where the intervals are given by the preceding lemma. This is clear if ( . U contains a local minimum (s,t) of D β,γ , and since it is arc-connected (as it is locally), there is a continuous path joining (s,t) to (s x , t y ) which is fully contained in U, namely : Let us definẽ Lemma 7 shows precisely that In view of (3.8), the continuity of f and G β,γ • f (0) = (x,ỹ), this implies that and this shows that x ∈ J(x,ỹ) and y ∈ J(ỹ,x). In particular, in each connected The proof of Proposition 6 is completed since there are at most 4 (T /r m ) 2 terms in the above equation, and for all i, |J(x i , y i )| ≤ C 3 e −(α−2κ)T from Lemma 7.

Proof of Proposition 4 : case of distinct geodesics.
In this section, we establish Equation (3.1). Let β ∈ C g (T ) where g ∈ M k (ε 0 ). For ε > 0, let us choose α > 0 such that We then take T 0 > 0 large enough so that Propositions 2 and 6 hold true for T ≥ T 0 . From Proposition 6, we have We now consider all closed geodesics γ = β with β ∈ C g (T ) fixed and γ ∈ C g (T ).
To get a uniform bound on the counting function for closed geodesics for the metric g, note first that (1.1) implies that there is C 0 > 0 depending only on g 0 such that ♯C g 0 (T ) ≤ C 0 T −1 e htopT for, say, T > r m . For γ 0 ∈ C g 0 , let γ = f g 0 →g (γ 0 ). In view of (2.3), Setting we obtain using (3.9) that By construction, β \U β ⊂ V β and U β is a finite union of O(T e hT ) open segments. Therefore, we see by a box principle along β using (3.11) that for T large enough depending only on g 0 , ε 0 and ε, the set V β contains at least one segment I of size We assumed 2 e −αT ≤ r m , so if we choose z ∈ β to be the middle of I, then and this shows (3.1). It remains to establish that z can be chosen such that the ball B g (z, ǫ) also avoids β except on a single geodesic segment of β containing z.

Almost-intersections of a single closed geodesic.
To conclude the proof of Proposition 4, we now indicate how the results of the preceding sections allow us to control almost-intersections of a closed geodesic with itself. As above, we take α ≥ h + 2κ + ε for some ε > 0. Fix some parameterization of β ∈ C g (T ) by arc-length. Let (x, y) ∈ β × β, and call t x , t y times such that x = β(t x ), y = β(t y ). We will say that the couple (x, y) is an almost-intersection of β with itself if d g (x, y) < ǫ, t y = t x and (t x , t y ) is a local minimum of D β,β : (t, t ′ ) → d g (β(t), β(t ′ )). In particular, either (x, y) is a self-intersection of β, or the segment joining x to y is not included in β. Using as before convexity of the distance function in negative curvature, we get that there are at most O rm (T 2 ) such couples of almost-intersections, and arguments identical to those developed in the proof of Proposition 2 show that if (x, y) is an almost-intersection, then For z = β(t z ), define as before In particular, as in Lemma 7, to (x, y) we can associate segments J(x, y) ⊂ I x and J(y, x) ⊂ I y of β centered at x and y respectively, of size O g 0 ,ε 0 (e −(α−2κ)T ) such that ∀z ∈ I x \ J(x, y), d g (z, I y ) ≥ ǫ and symmetrically when interchanging the roles of y and x. Exactly as in Proposition 6, we can then show that the set can be covered by O g 0 ,ε 0 (T 2 ) segments of size O g 0 ,ε 0 (e −(α−2κ)T ), for T ≥ T 0 (g 0 , ε 0 ). These segments can be added up to U β : exactly as previously, we end up by a box principle with the fact that for T large enough depending now on g 0 , ε 0 and ε, there is a segment I ⊂ β such that |I| ≥ 2 e −αT and if z denotes the middle of I, we have This concludes the proof of Proposition 4. We end this section by a straightforward corollary : Corollary 8. Let g ∈ M(ε 0 ), ε > 0 and α ≥ 2κ + h + ε be as above. For T ≥ T 0 (g 0 , ε 0 , ε) and each γ ∈ C g (T ), there is z γ ∈ γ such that and B g (z γ , ǫ/2) ∩ γ consists in a unique geodesic segment centered at z γ . In particular, if γ, γ ′ ∈ C g (T ) are distinct, then B g (z γ , ǫ/2) ∩ B g (z γ ′ , ǫ/2) = ∅.

Proof of Theorem 1
For T > 0 large enough, Corollary 8 of Proposition 4 allow to perturb the metric near a point of a closed geodesic in C g (T ) without changing the length of all the others in C g (T ). Before exploiting further this property to separate the length spectrum, we recall without proof a standard fact for conformal perturbations of a given metric g near a closed geodesic.
Let T 0 (g 0 , ε 0 , ε, k) > 0 be a positive number such that Proposition 4 with the above value of α is satisfied for T ≥ T 0 , and set T n = T 0 + n, n ∈ N.
For ν > 0 to be defined soon below and n ≥ 1, we will say that C gn ( We proceed iteratively : once a metric g n−1 ∈ M k (ε 0 ) (n > 1) is constructed such that C g n−1 ([T 0 , T n−1 ]) is ν-separated, we consider the next interval ]T n−1 , T n ] and build a metric g n from g n−1 such that C gn ([T 0 , T n ]) is ν-separated.
By a box principle, in view of (3.10) there is at least one couple of distinct ℓ − , ℓ + ∈ L g n−1 (]T n−1 , T n ]) such that ℓ + − ℓ − ≥ C −1 0 e −hTn . Setting ℓ i = ℓ(γ i ), we order the points of L g n−1 ∩]T n−1 , T n ] so that Let us define ǫ n = e −αT n+1 /2 and choose z i ∈ γ i according to Corollary 8. In particular, B g n−1 (z i , ǫ n ) does not intersect the open ǫ n −neighborhood of any γ ∈ C g n−1 (T n+1 ) \ γ i , and B g n−1 (z i , ǫ n ) ∩ γ i consists in a single geodesic segment centered at z i . We are exactly in the settings of Lemma 9 : in the ball B g n−1 (z i , ǫ n ), we dilate the metric by a factor (1+ǫ −1 n δ n,i χ n,i ) 2 , where χ n,i plays the role of χ 0 in this lemma. The constants δ n,i are taken such that In this way, the geodesics (γ i ) 1≤i≤µn have their lengths dilated (for 1 ≤ i ≤ m) or contracted (for m + 1 ≤ i ≤ µ n ) tol 1 , . . . ,l µn with Define now To simplify the notations below, we write f n,i def = δ n,i ǫ −1 n χ n,i . By C k , we will denote a positive constant depending only on g 0 , ε 0 , ε, k whose value may change from line to line. Equation (3.10) gives µ n ≤ C 0 e hTn /T n , so we have We deduce from this equation and Lemma 9 that f n,i satisfies where P k is a polynomial of degree 2 k−1 . Using (4.3) and T 0 large enough, we then get P k (f log(1 + f n,i ) C k ≤ C k e −(ν−h−(k+1)α)Tn T n , k ≥ 0.