key: cord-0164005-pt1nslev
authors: Kuster, Benjamin; Weich, Tobias
title: Pollicott-Ruelle resonant states and Betti numbers
date: 2019-03-03
journal: nan
DOI: nan
sha: ac0b8e49e9942d66c51e721a6636fb8edb84a5fd
doc_id: 164005
cord_uid: pt1nslev

Given a closed hyperbolic manifold of dimension $neq 3$ we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers.

Pollicott-Ruelle resonances have been introduced in the 1980's in order to study mixing properties of hyperbolic flows and can nowadays be understood as a discrete spectrum of the generating vector field (see Section 1.2 for a definition and references). Very recently it has been discovered that in certain cases some particular Pollicott-Ruelle resonances have a topological meaning. Let us recall these results:

In [DZ17] Dyatlov and Zworski prove that on a closed oriented surface  of negative curvature the Ruelle zeta function at zero vanishes to the order | ()|, where () is the Euler characteristic of . They prove this result as follows: By previous results on the meromorphic continuation of the Ruelle zeta function [DZ16, GLP13] the order of vanishing of the Ruelle zeta function at zero can be expressed as the alternating sum

is the multiplicity of the resonance zero of the Lie derivative  along the geodesic vector field ∈ Γ ∞ ( ( * )) acting on perpendicular -forms. The latter are those -forms on the unit co-sphere bundle *  that vanish upon contraction with (for the precise definition of the multiplicities, see Sections 1.1 and 1.2). For closed oriented surfaces it is rather easy to see that  ,Λ 0 ( * ⟂ ) (0) =  ,Λ 2 ( * ⟂ ) (0) = 0 () = 2 (), thus the central task is to prove that  , * ⟂ (0) = 1 (). Dyatlov and Zworski achieve this by combining microlocal analysis with Hodge theory [DZ17, Proposition 3.1(2)]. This is a remarkable result also apart from its implications on zeta function questions because it identifies a resonance whose multiplicity has a precise topological meaning.

Let us mention a second result that establishes a connection between Pollicott-Ruelle resonances and topology: Dang and Rivière [DR17c] examine a general Anosov flow = on a closed manifold. The Lie derivative  has a discrete spectrum (the Pollicott-Ruelle spectrum) on certain spaces of anisotropic -currents and it is shown that the exterior derivative acting on generalized eigenspaces of the eigenvalue zero forms a complex which is quasi-isomorphic to the de Rham complex. 1 While this result gives no precise information about the multiplicities of the resonances, it gives lower bounds for them and it holds in very great generality.

As a third result we would like to mention [GHW18a] where the relation between Pollicott-Ruelle and quantum resonances is studied for compact and convex co-compact hyperbolic surfaces. For this correspondence the resonances at negative integers turn out to be exceptional points and it is shown that their multiplicities can be expressed by the Euler characteristic of the hyperbolic surface. The proof uses a Poisson transform to establish a bijection between the resonant states and holomorphic sections of certain line bundles, and the formula for the multiplicities follows from a Riemann-Roch theorem.

In the present article we broaden the picture regarding the topological properties of Pollicott-Ruelle resonant states. To this end, we combine some of the above approaches: In a first step we use a quantum-classical correspondence to find new examples of resonances with topological multiplicities. In particular, we prove 1 We would like to point out that an analogous statement also holds for Morse-Smale flows [DR16, DR17b, DR17a, DR17c] and in these cases the spectral complex defined by the Pollicott-Ruelle resonances is actually isomorphic to the Morse complex. Consequently, the spectral complex of Dang and Rivière can be considered as a generalization of the Morse complex to Anosov flows. Proposition 0.1. For any closed hyperbolic manifold  of dimension + 1 with ≠ 2, one has

Furthermore, the resonance zero has no Jordan block and if ≥ 3, then zero is the unique leading resonance and there is a spectral gap. 2 We prove these statements using the general framework of vector-valued quantum-classical correspondence developed by the authors [KW18] as well as a Poisson transform of Gaillard [Gai86] . 3 Without any further effort these ingredients provide additional examples of resonance multiplicities that involve not only the first but all Betti numbers (see Proposition 2.3) which should be of independent interest. For = 1 the first statement in Proposition 0.1 is the special case of [DZ17, Proposition 3.1(2)] restricted to hyperbolic surfaces. Interestingly = 2 is an exceptional case and the multiplicity is given by  , * ⟂ (0) = 2 1 () (see Remark 2.2). For > 2 the statement can be considered as a generalization of the Dyatlov-Zworski result to higher dimensions at the cost of restricting to manifolds of constant negative curvature.

In a second step we can partially overcome this restriction and prove 

We prove this statement by combining Proposition 0.1, which has been obtained by a quantum-classical correspondence, with the cohomology results of Dang-Riviére as well as some recent advances concerning the perturbation theory of Pollicott-Ruelle resonances for flows [DGRS18, Proposition 6.3] [Bon18] . Note that also in dimension + 1 = 2 our methods give the equality (0.1) only in a neighborhood of ℊ 0 , whereas Dyatlov and Zworski prove the equality in this dimension for all ℊ ∈ ℛ ,<0 . It seems thus reasonable to conjecture that the equality holds for all ℊ ∈ ℛ ,<0 in all dimensions + 1 ≠ 3.

1.1. Geodesic flows on manifolds of negative curvature. Let (, ℊ) be a closed Riemannian manifold of dimension + 1 with negative sectional curvature. Then the geodesic flow on the unit co-sphere bundle *  is an Anosov flow which implies that there is a -invariant Hölder continuous splitting of the tangent bundle ( * )

where 0 = ℝ is the neutral bundle spanned by the geodesic vector field and + , − are the stable and unstable bundles, respectively (see e.g. [Kni02] ). Additionally, there is a smooth contact one-form ∈ Ω 1 ( * ) which is simply the restriction of the Liouville one-form on *  to * . It fulfills = 1, ker( ) = + ⊕ − , is symplectic on ker( ),  = 0, where  denotes the Lie derivative. Note that the last two properties imply that ∧( ) is a nowhere-vanishing flow-invariant volume form which defines the Liouville measure on * . Using the contact one-form we get a splitting of the cotangent bundle into smooth subbundles * ( * ) = ℝ ⊕ * ⟂ , * ⟂ ∶= { ∈ * ( * ) ∶ = 0}.

We will call the smooth sections of * ⟂ perpendicular one-forms and denote their space by Ω 1 ⟂ ( * ). More generally, we introduce for = 0, … the space of perpendicular -forms

). By the Anosov splitting, the bundle * ⟂ can be further split into (1.1) * ⟂ = * + ⊕ * − , where the dual stable and unstable bundles are defined by * ± ( 0 ⊕ ∓ ) = 0. In contrast to the smoothness of * ⟂ , the subbundles * ± are only Hölder continuous unless  is a locally symmetric space of rank one. Remark 1.1 (Complexifications). When addressing spectral questions involving an operator on any of the bundles mentioned so far, or on any subbundle of a tensor power of * ( * ), it is often more useful to work with the complexified bundle. For simplicity of notation we shall not explicitly distinguish in the following between real vector bundles and their complexifications. It will be clear from the context whether we refer to the real or the complexified bundle.

and Ruelle [Rue86] in order to study mixing properties of hyperbolic flows (as mentioned before). In the last years it has been found out that these resonances can also be defined as poles of meromorphically continued resolvents [Liv04, BL07, FS11, DZ16, DG16, BW17] (see also [BKL02, Bal05, BT07, Bal18, BT08, GL06, GL08, Liv05, FRS08] for important related work on hyperbolic diffeomorphisms). We follow [DG16] to introduce the notion of Pollicott-Ruelle resonances on an arbitrary smooth complex vector bundle  → * . A first order differential operator on  is called an admissible lift of the geodesic vector field if ( ) = ( ) + , ∈ C ∞ ( * ), ∈ Γ ∞ (). An example of such an admissible lift is the Lie derivative  with respect to the geodesic vector field on any -invariant subbundle of ⊗ * ( * ) for some ∈ ℕ 0 (taking into account Remark 1.1). In Section 2 we will additionally consider covariant derivatives which are further examples of admissible lifts. After choosing a smooth metric on  one defines the space L 2 ( * , ). Note that by the compactness of  only the norm on this space depends on the choice of the metric but neither does the space nor its topology. One checks that there is a constant > 0 such that + ∶ L 2 ( * , ) → L 2 ( * , ) is invertible for Re( ) > . (1.2)

Res , ( 0 ) = { ∈  ′ ( * , ) ∶ ( + 0 ) ( 0 ) = 0, WF( ) ⊂ * + }. If ( 0 ) = 1 we say that the resonance has no Jordan block. Otherwise, the space of Pollicott-Ruelle resonant states res , ( 0 ) ∶= ker( + 0 ) ∩ Res , ( 0 ) is a proper subspace of Res , ( 0 ).

Note that the resolvent, the Pollicott-Ruelle resonances, and the associated resonant states and multiplicities depend on the Riemannian metric ℊ. In Section 3 we will be interested in their variation under perturbations of ℊ. For this reason we will write , ( , ℊ), Res , ( , ℊ), , ( , ℊ), … in order to emphasize the dependence on ℊ. In the other sections we suppress the Riemannian metric in the notation.

Throughout the whole section we assume that (, ℊ) is a closed hyperbolic 4 manifold of dimension + 1.

Furthermore, the resonance zero has no Jordan block, and if ≥ 3, then zero is the unique leading resonance and there is a spectral gap. 5

The first part of this result will be a central ingredient for Theorem 3.1. We will prove Proposition 2.1 using a quantum-classical correspondence. Such correspondences have recently been developed in various contexts [DFG15, GHW18a, GHW18b, Had18] and we will use the general framework for vector bundles developed by the authors in [KW18] . Additionally we use a Poisson transform due to Gaillard [Gai86] and combining both ingredients allows us to construct an explicit bijection between the Pollicott-Ruelle resonant states in perpendicular one forms and the kernel of the Hodge Laplacian.

Remark 2.2. The dimension + 1 = 3 is an exception where the multiplicity is given by  , * ⟂ (0) = 2 1 (). The deeper reason for this exception is that Gaillard's Poisson transform is not bijective in this case. The exceptional case could also be treated with our methods by a more detailed analysis of Gaillard's Poisson transform. This special case has however been worked out already in [DGRS18, Proposition 7.7] by factorizations of zeta functions, so we refrain from taking on the additional effort.

A crucial role in these quantum-classical correspondences is played by the so-called (generalized) first band resonant states

where  − is the horocycle operator which we will introduce below in (2.9). Roughly speaking, first band resonant states are resonant states that are constant in the unstable directions. In the process of proving Proposition 2.1 we will establish the following result. Proposition 2.3. On any closed hyperbolic manifold  of dimension + 1 and for any = 0, … , , ≠ ∕2,

. We consider this result to be of independent interest because it shows that also the higher Betti numbers appear as multiplicities of Pollicott-Ruelle resonances. Again the statement is obtained by constructing an explicit isomorphism onto the kernel of the Hodge Laplacian.

2.1. Lie theoretic structure theory. Any closed connected 6 hyperbolic manifold  of dimension + 1 can be written as a bi-quotient  = Γ∖ ∕ , where = SO( + 1, 1) 0 , 7 = SO( + 1), and Γ ⊂ a cocompact torsion-free discrete subgroup.  is thus an example of a Riemannian locally symmetric space of rank one. There exists a very efficient Lie-theoretic language to describe the structure of , the co-sphere bundle * , as well as the invariant vector bundles which we introduce in this subsection. For more details we refer the reader to [GHW18b, KW18] and for background information to the textbooks [Kna02, Hel01] .

The Lie algebra of any semisimple Lie group of real rank one with finite center possesses a Cartan involution ∶ → and a Cartan decomposition = ⊕ . There is a maximal compact subgroup ⊂ with Lie algebra and the Cartan decomposition ⊕ is Ad( )-invariant. The tangent bundle (Γ∖ ∕ ) can then be identified with the associated vector bundle Γ∖ × Ad( ) . Note that via the Killing form ∶ × → ℝ and the Cartan involution one can define an Ad( )-invariant inner product ⟨⋅, ⋅⟩ on by ( , ) ↦ − ( , ). The restriction to then allows to define a Riemannian metric on Γ∖ ∕ . For any rank one locally symmetric space 4 I.e., a Riemannian manifold of constant sectional curvature −1. Fixing the curvature at −1 is a common convention. By trivial rescaling arguments all results in this paper involving the resonance 0 remain true if the metric is multiplied by a positive constant. 5 I.e., there exists > 0 such that  acting on * ⟂ has no resonances with real part in the interval (− , ∞) except the resonance zero. 6 If the manifold is not connected, the description given in the following applies to each connected component. 7 Here the subscript 0 indicates the identity component. this is a metric of strictly negative sectional curvature and in the specical case = SO( + 1, 1) 0 the curvature is constant. We will normalize our Killing form such that the sectional curvature is normalized to −1.

We next want to describe the structure of the co-sphere bundle *  and the vector bundles 0∕+∕− . The condition real rank one for means that there is a maximal one-dimensional abelian subalgebra ⊂ . After choosing a notion of positive and negative roots for the abelian subalgebra , the unions of all positive and all negative root spaces provide two additional Lie subalgebras ± ⊂ and one obtains the Iwasawa decompositions on the Lie algebra level = ⊕ ⊕ + = ⊕ ⊕ − . Also on the group level there are two corresponding Iwasawa decompositions = + = − . Here ± ⊂ are the analytic subgroups with Lie algebras ± and ⊂ is the analytic subgroup with Lie algebra . Furthermore, the respective exponential maps provide diffeomorphisms ≅ , ± ≅ ± . For each group element ∈ we now have unique Iwasawa (+) and opposite Iwasawa (−) decompositions

where exp( ± ( )) = ± ( ), exp being the exponential map on . This provides us with maps

In addition, define the group ∶= ( ) = ( ), where acts on by the adjoint action, and let be the Lie algebra of . The groups ± are normalized by and . On the Lie algebra level we have the Bruhat decomposition

Now the co-sphere bundle *  can be identified with Γ∖ ∕ . The Lie group ≅ ℝ acts from the right on Γ∖ ∕ because it commutes with , and this action precisely coincides with the geodesic flow. Furthermore, the tangent bundle of *  can be identified as follows:

There is an analogous identification of * ( * ). In view of these identifications all bundles of interest in Propositions 2.1 and 2.3 are of the type  ∶= Γ∖ × → *  for some finite-dimensional complexrepresentation ( , ). On such bundles there always exists a canonical connection that we denote by

. To describe how ∇ is defined, let us regard a section ∈ Γ ∞ ( ) as a right--equivariant function̄ ∈ C ∞ (Γ∖ , ). Moreover, by (2.6) we regard a vector field ∈ Γ ∞ ( ( * )) as a right--equivariant function ∈ C ∞ (Γ∖ , + ⊕ ⊕ − ), that is,̄ (Γ ) = Ad( −1 )̄ (Γ ). Then ∇ is defined by the covariant derivative

So far all constructions were valid for general rank one locally symmetric spaces and we have not yet used the fact that we work on hyperbolic manifolds. The fact of working with = SO( + 1, 1) 0 has however the following implications that will simplify the analysis in the sequel:

≅ ℝ by the following canonical identification: As mentioned above the geodesic flow can be identified with the right--action on Γ∖ ∕ and the geodesic vector field corresponds to a unique Lie algebra element 0 ∈ . We then identify ≅ ℝ by setting 0= 1. • ± ≅ ℝ are abelian Lie algebras and the Ad( )-action on ± is the defining representation of SO( ) on ℝ .

• With 0 ∈ as above one has for ∈ ± the relation [ 0 , ] = ± . In structure theoretic terms this means that SO( + 1, 1) 0 has only one positive restricted root. Recall that we normalized the Killing form in such a way that the sectional curvature equals −1, which implies that the restricted root has length 1. Given a finite-dimensional complex -representation ( , ) we define the boundary vector bundle

The total space × of   carries the -action

that lifts the -action (2.10) on the base space ∕ . Consequently, we get an induced action on smooth sections:

If we consider a section ∈ Γ ∞ (  ) as a right--equivariant smooth function̄ ∶ → , the action (2.12) corresponds to assigning tō for any ∈ the right--equivariant smooth function ∶ → given by

To describe how the principal series representation of associated to an -representation and a parameter ∈ ℂ acts on smooth sections of   , let us regard a section ∈ Γ ∞ (  ) as a right--equivariant function ∈ C ∞ ( , ). We then set 8

(2.14)

,

This representation extends by continuity to a representation , comp ∶ → End( ′ ( ∕ ,   )). One has the following important relation between first band resonant states and the Γ-invariant distributional sections of the boundary vector bundle with respect to the principal series representation 

Proposition 2.5 provides a powerful way to handle first band resonant states of the covariant derivative ∇ along the geodesic vector field. In Proposition 2.1 and 2.3 we are however interested in resonant states of the Lie derivative. Therefore we have to relate these states: Lemma 2.6. For ∈ {0, 1, 2, …}, suppose that is a subrepresentation of ⊗ (Ad( )| ± ). Then the covariant derivative and the Lie derivative along the geodesic vector field , acting on smooth sections of  , are related by

Consequently, one has for every ∈ ℂ and ∈ ℕ (2.16) Res  , ( ) = Res ∇ , ( ∓ ) and res  , ( ) = res ∇ , ( ∓ ).

Proof. Recall that the geodesic flow on * (Γ∖ ∕ ) = Γ∖ ∕ is given by

Any vector ∈ ± is an eigenvector of the adjoint action:

Considering as a left-Γ-, right--equivariant map̄ ∶ → , let * ∶ → be the left-Γ-, right-equivariant function corresponding to * ∈ Γ ∞ ( ). Then we get with (2.19) for ∈ and 1 , … , ∈ ± : … , ) . For the Lie derivative of we then obtain with the analogous "̄̄"-notation and the product rule This result is due to Gaillard [Gai86] although it requires some work (see Section 2.5.1) to translate his statements into the form stated above that we can apply in our setting: For ≠ ∕2 the Poisson transform , is bijective and thus

∈ Ω (Γ∖ℍ +1 ), Δ = 0, = 0 .

As on compact manifolds any harmonic form is co-closed, the right-hand side is simply the kernel of the Hodge Laplacian and Hodge theory implies that its dimension equals the -th Betti number of Γ∖ℍ +1 = . We thus have shown This finishes the proof of Proposition 2.3.

In his article [Gai86] Gaillard considers the vector-valued Poisson transform for forms on ∕ to which we refer in Theorem 2.7. His notation and conventions are however quite different from ours. In the following we will translate his results into the form stated in Theorem 2.7. Gaillard studies -currents on ∕ which we will denote by  ′ ( ∕ ) ∶= (Ω − ( ∕ )) ′ , and we have the canonical dense embedding Ω ( ∕ ) ⟶  ′ ( ∕ ). As acts by diffeomorphisms on ∕ the pullback action on  ′ ( ∕ ) provides a -representation. Proof. Denote by ⟂ ⊂ the orthogonal complement of in . Then acts via the adjoint action on ⟂ . Note that in our setting ⟂ ≅ ℝ and Ad( )| ⟂ is simply the standard action of SO( ) on ℝ . In the following, we shall write simply Ad( ) instead of Ad( )| ⟂ . Note that there is a canonical identification

Let ∈ and ∶ ↦ − ( ) be the diffeomorphism on ∕ given by the left--action, then the derivative acts on ( ∕ ). In order to prove our lemma we have to determine how acts on × Ad( ) ⟂ under the identification (2.21). We have for [ , ] ∈ ( ∕ )

being the projection onto defined by the opposite Iwasawa decomposition of . We can now proceed by studying for fixed ∈ , ∈ , ∈ ⟂ the element (2.24) pr − Ad( − ( ) − ( ))( ) ∈ ⟂ .

By the orthogonal Bruhat decomposition = ⊕ ⊕ + ⊕ − and the fact that lies in the orthogonal complement of in , we have ⟂ ⊂ + ⊕ − , so we can write = + + − with ± ∈ ± and ± = ∓ . The space ± is Ad( ± )-invariant. Consequently Ad( − ( ) − ( ))( − ) ∈ − , so pr − Ad( − ( ) − ( ))( − ) = 0 by the opposite Iwasawa decomposition. This shows that only + contributes to (2.24). Let us write − ( ) = exp( ) with ∈ − . Then we get

Here we use that = 0 ⊕ + ⊕ − is the root-space decomposition of = ( + 1, 1) and consequently

Furthermore, the map Ad( − ( )) acts on ± by scalar multiplication with ± − ( ) and leaves 0 = ⊕ invariant. The opposite Iwasawa projection pr − maps − to 0 and the space 0 onto . However, the Lie algebra element considered in (2.24) is by construction in ⟂ . We therefore arrive at pr − Ad( − ( ) − ( ))( ) = pr − − ( ) + .

In summary, we have proved

Finally, note that ( ∕ ) ≅ × Ad( ) ⟂ induces for each ∈ {1, 2, …} an isomorphism Λ * ( ∕ ) ≅ × Λ Ad * ( ) Λ ( ⟂ ) * . Under that isomorphism, a -form ∈ Γ ∞ (Λ * ( ∕ )) corresponds to a section ∈ Γ ∞ ( × Λ Ad * ( ) Λ ( ⟂ ) * ), and by our above computations the pullback action ≡ ( −1 ) * of an element ∈ on corresponds to the following action on̂ :

(2.26)

Recalling the definition (2.14) of the principal series representations, and taking into account that the pullback action of on -currents as well as the principal series representations of on distributional sections of Λ * ( ∕ ) are the continuous extensions of the respective actions on smooth -forms, the proof is complete.

For the definition of his Poisson transform Gaillard generalizes his setting to currents with values in complex line bundles → ∕ parametrized by a complex number ∈ ℂ. Let us recall their construction [Gai86, Section 2.2]: It is based on a -invariant function 9

(2.27)

where Gaillard's "application visuelle" ∶ * ( ∕ ) → ∕ , ∈ ∕ , is defined by

A straightforward calculation similar to the proof of Lemma 2.8 shows that The stabilizer subgroup of ∈ ∕ with respect to the left--action on ∕ is − and the action of the stabilizer group on the fiber of over is

If we define the − -representation by ↦ − log( ) ∈ ℂ then we can identify with the associated line bundle × ℂ → ∕( − ) ≅ ∕ . Thus the -action on sections of this homogenous bundle is equivalent to the principle series representation , comp , where denotes the trivial -representation on ℂ. By Lemma 2.8 we know that the pullback action on -currents is equivalent to Using the quantum-classical correspondence once more we shall obtain a simple description of the latter spaces.

To this end, note that the Cartan involution | + ∶ + → − is an equivalence of representations Ad( )| + ∼ Ad( )| − which induces an isomorphism * + ≅ * − that is compatible with the connections on the two bundles. Therefore, Res 1st

Now the restriction of the Riemannian metric ℊ to − × − defines a smooth section of * − ⊗ * − . Lemma 2.9. If ≠ 2, there is a number > 0 such that for all ∈ ℂ with Re ∈ (− , 0] one has

Before proving this lemma let us see how it finishes the proof of Proposition 2.1: All that is left to prove is that if  − = ℊ| − × − for some ∈ Res * − (−1) and ∈ ℂ, then = 0. This is easy: where Δ is the Bochner Laplacian associated to the connection ∇. The eigenvalue − ( + ) + 2 appearing here is a real number iff Im = 0 or Re = − , so for Re > −1 only numbers ∈ (−1, ∞) remain as possible candidates for a non-zero resonance space (2.34). In addition, a Weitzenböck type formula (see [DFG15, Lemma 6.1]) says that the spectrum of Δ acting on Γ ∞ ( 2 0 ( * )) is bounded from below by + 1 which is strictly larger than − ( + ) + 2 for ≥ 2 and ∈ (−1, ∞). Consequently, for such and the right-hand side of (2.34) is the zero space and it follows that Res 1st For Re > −1 and ≥ 3, the eigenvalue appearing here is either imaginary or negative, so the right-hand side of (2.35) is the zero space (because Δ is positive) and res 1st ( ) = 0. Finally, let us turn to the third summand in (2.33). As ∇ ( )| − × − = ( ) ℊ| − × − we see that the distribution has to be a scalar resonant state of a resonance . In the scalar case we can however apply Liverani's result on the spectral gap for contact Anosov flows [Liv04] to see that zero is the unique leading resonance (with resonant states the locally constant functions) and there is a spectral gap > 0, so the proof is finished.

We now address the question how the equality  , * ⟂ (0) = 1 () for constant negative curvature manifolds behaves under perturbations of the Riemannian metric. Therefore, throughout this section, let  be a closed manifold admitting a hyperbolic metric and Γ ∞ (S 2 ( * )) the space of symmetric two-tensors endowed with the Fréchet topology. Let ℛ ,<0 ⊂ Γ ∞ (S 2 ( * )) be the open subset of Riemannian metrics with negative sectional curvature and ℋ  ⊂ ℛ ,<0 the nonempty subset of constant negative curvature metrics. For each ℊ ∈ ℛ ,<0 the geodesic vector field ∈ Γ ∞ ( ( * )) is an Anosov vector field. 11 In order to prove this result we first need a perturbation theory argument. Proof. Dyatlov-Zworski [DZ16] have shown that for any metric ℊ ∈ ℛ ,<0 and 0 > 0 such that ( , 0 ( )) > for all ∈ *  the flat trace

is well defined and has a meromorphic continuation to ℂ (see (4.2) in [DZ16] and discussions below). Furthermore, they prove that the poles of this function coincide with the resonances of  on * ⟂ , and the poles of ( , ℊ) are simple with integer residues given by  , * ⟂ ( , ℊ). In the recent article on Fried's conjecture [DGRS18] the main technical ingredient is the continuous dependence of ( , ℊ) on the Anosov vector field . A slight reformulation is Note that inspecting the proof of [DGRS18, Proposition 6.3] shows that the assumptions 0 ∈  and  being simply connected are not necessary for the statement of [DGRS18, Proposition 6.3] but simply technical assumptions necessary for other parts of [DGRS18] .

With this result the upper semicontinuity follows by a simple residue argument: Choose a metric ℊ 0 ∈ ℛ ,<0 and a resonance 0 for ℊ 0 as well as an annulus  around 0 such that no resonances are contained in  and 0 is the only resonance in the interior of the annulus. Let be a closed path in  winding around 0 , then  , ⟂ ( 0 , ℊ) = (2 ) −1 ∫ ( , ℊ) . If is the open neighborhood of ℊ 0 from Proposition 3.3, consider the continuous map ∋ ℊ ↦ (2 ) −1 ∫ ( , ℊ) . As the residues of ( , ℊ) are the multiplicities of the resonances with respect to ℊ one has

11 Note that the unit co-sphere bundles for two different Riemannian metrics are diffeomorphic. In particular, up to continuous isomorphism, all the spaces of functions and distributions considered below that involve *  do not depend on ℊ. We can therefore essentially treat *  as a smooth manifold that does not depend on ℊ.

where the sum runs over all resonances with respect to ℊ in the interior of the annulus. So (2 ) −1 ∫ ( , ℊ) can only take on integer values and is therefore constant. This proves the upper semicontinuity of ( 0 , ℊ).

A second ingredient to Theorem 3.1 is a lower bound on  , * ( * ) (0):

Lemma 3.4. For any ℊ ∈ ℛ ,<0 ,  , * ( * ) (0) ≥ 1 ( * ) + 0 ( * ).

Proof. Dang-Rivière [DR17c] proved that 0 → Res  ,Λ 0 ( * ( * )) (0, ℊ) → Res  ,Λ 1 ( * ( * )) (0, ℊ) → ⋯ → Res  ,Λ 2 dim −1 ( * ( * )) (0, ℊ) → 0 forms a finite-dimensional complex whose cohomology is isomorphic to the de Rham cohomology of * . As Λ 0 ( * ( * )) is simply the trivial line bundle its resonant states at zero are the locally constant functions and thus (Res  ,Λ 0 ( * ( * )) (0, ℊ)) = 0, which implies dim(Res  ,Λ 1 ( * ( * )) (0, ℊ) ∩ ker( )) = 1 ( * ). Furthermore, there is the contact one-form ∈ Γ ∞ ( * ( * )) fulfilling  = 0. By the wavefront characterization of resonant states (1.2) we know that ∈ Res  ,Λ 1 ( * ( * )) (0, ℊ). But as is a contact one-form ≠ 0. The same holds for with being a locally constant function (thus an element in the 0-th de Rham cohomology) and the statement follows.

We can now prove Theorem 3.1:

Proof. Let ℊ ∈ ℛ ,<0 and the contact one-form. From the fact that = 1 we can uniquely decompose any ∈ Res  , * ( * ) (0, ℊ) into = ⟂ + ( ) where ⟂ = 0 and thus ⟂ is a distributional section of * ⟂ . We have for some ∈ ℕ (3.1) 0 =  =  ⟂ + ( ( )) .

Using Cartan's magic formula for  one checks  ⟂ = 0, so the wavefront characterization of resonant states (1.2) implies ⟂ ∈ Res  , * ⟂ (0, ℊ) and ∈ Res  ,Λ 0 ( * ( * )) (0, ℊ). This implies 

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Acknowledgements. After the appearance of [KW18] , Semyon Dyatlov raised the question whether the vectorvalued quantum-classical correspondence might shed light on generalizations of [DZ17] to higher dimensions. We are grateful to him for proposing this question as well as for several helpful discussions. Furthermore, we thank Colin Guillarmou, Viet Nguyen Dang, and Gabriel Rivière for several discussions concerning their works [DR17c, DGRS18] and helpful comments. This research is partially funded by the ERC grant no. 725967 Inverse Problems and Flows.