High frequency limits for invariant Ruelle densities

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances which also leads to a new description of Patterson-Sullivan distributions.


Introduction
Let X be a smooth Anosov vector field on a compact Riemannian manifold M. Furthermore, to each Ruelle resonance λ 0 we can define a canonical generalized density in the following way: Because the wavefront set of Π λ 0 is precisely known [FS11, Theorem 1.7][DZ16, Proposition 3.3], there is a well defined notion of a trace of Π λ 0 , the so called flat trace Tr and we can define the following continuous linear functional If λ 0 = 0, then the functional T λ 0 is given by the SRB measure 1 , see [BL07]. For a general resonance λ 0 ∈ C these functionals are only distributional densities and not measures. As X commutes with Π λ 0 , they are still invariant under the flow, i.e. XT λ 0 = 0, and we call them invariant Ruelle densities. Note that these invariant Ruelle Date: March 20, 2018. 1 More precisely by an SRB measure, because the eigenvalue λ 0 might be degenerate and the SRB measure not unique.
densities have also an explicit expression in terms of the Ruelle resonant states: In the simplest case of a first order pole of multiplicity one the invariant density is simply the distributional product of a resonant and co-resonant state, where co-resonant states are resonant states for the flow in backward time.
We want to study high frequency limits (also called semiclassical limits) of these invariant Ruelle distributions. For Ruelle resonances the reasonable notion of semiclassical limit is to fix a range in the real part Re(λ) > −C and consider |Im(λ)| → ∞. In this limit there have recently been established several results on the distributions of resonances such as Weyl laws [FS11,DDZ14,FT17a] or band structures [FT13,FT17b].
High-frequency limits. If G/K is a rank one Riemannian symmetric space of noncompact type, Γ ⊂ G a co-compact torsion free subgroup, then the locally symmetric space M := Γ\G/K is a compact Riemannian manifold of strictly negative curvature. Its geodesic flow on the unit sphere bundle M := SM is Anosov. Any compact manifold of constant negative curvature can be realized in this way, but the rank one locally symmetric spaces contain also families with nonconstant sectional curvature. In constant curvature the spectrum is known to obey an exact band structure [DFG15,KW17], i.e. for any Ruelle resonance λ 0 one has either Im(λ 0 ) = 0, or Re(λ 0 ) ∈ −ρ − N 0 , where ρ > 0 is the positive constant that is associated to a Riemmanian symmetric space by taking half the sum of its positive restricted roots. Our first result is the following equidistribution theorem.
Theorem 1. Let M be a compact locally symmetric space of rank one, M := SM be the unit sphere bundle and dµ L the Liouville measure on M. Let r n ∈ R + be such that λ n = −ρ + ir n are the Ruelle resonances with Re(λ n ) = −ρ, Im(λ n ) > 0 for the geodesic flow on M. Then there exists a subsequence (r kn ) n>0 ⊂ (r n ) n>0 , such that • T −ρ+ir kn converges weakly towards dµ L as n → ∞.
• The subsequence is of density one, i.e. To prove this result, we have to show an explicit correspondence between the Ruelle resonant states on Re(λ) = −ρ and the eigenstates of the Laplacian ∆ M . This allows us to reduce the problem to a quantum ergodicity result for the Laplacian and use the Shnirelman-Zelditch-Colin de Verdière theorem [Shn74,Zel87,CdV85]. In fact, we prove several results in the article that are of independent interest.
The quantum-classical correspondence (Theorem 2 and Theorem 3). In [DFG15,GHW17,Had17] it is shown that for geodesic flows on compact (resp. convex cocompact) constant negative curvature manifolds the Ruelle resonances are related to eigenvalues (resp. quantum resonances) for the Laplacian; some results related to that problem were obtained previously in [FF03]. A central ingredient in the proof is to establish an explicit bijection between the Ruelle resonant states in D (SM) that are killed by unstable derivatives and the eigenstates of the Laplacian ∆ M on M, at least for Ruelle resonances that are not in a certain exceptional set. The map from Ruelle resonant states to eigenfunctions of ∆ M is given by the pushforward π 0 * : D (SM) → D (M), where π 0 : SM → M is the projection onto the base. We extend this bijection to the setting of all compact locally symmetric spaces of rank one in Theorem 2 and Theorem 3.
A new description of Patterson-Sullivan distributions (Theorem 4). In [AZ07] Anantharaman and Zelditch introduced Patterson-Sullivan distributions on compact hyperbolic surfaces. Given an eigenfunction of the Laplacian in C ∞ (M), these distributions are distributions in D (SM) which are invariant under the geodesic flow and become equivalent to usual semiclassical lifts such as Wigner distributions in the semiclassical limit. In [HHS12] this construction has been generalized to compact higher rank locally symmetric spaces. Using the quantum classical correspondence we give in Theorem 4 a new description of these Patterson-Sullivan distributions for rank one spaces: given a Laplace eigenfunction ϕ ∈ C ∞ (M) the quantum-classical correspondence allows us to associate a unique Ruelle resonant state v ∈ D (SM) as well as a Ruelle co-resonant state v * ∈ D (SM). The Patterson-Sullivan distribution is then precisely given by the distributional product of v · v * which is well defined by a wavefront condition.
A pairing formula (Theorem 5). If the Ruelle resonance λ 0 associated to a Patterson-Sullivan distribution is simple, then it is easy to check that the Patterson-Sullivan distribution coincides with the invariant Ruelle density. If Rank(Π λ 0 ) > 1, one additionally needs a pairing formula in order to express the invariant Ruelle density in terms of the Patterson-Sullivan distributions. The pairing formula relates pairings of resonant states with co-resonant states to pairings of the associated eigenfunctions of ∆ M . Such pairing formulas have previously been proven in [AZ07] for hyperbolic surfaces and [DFG15] for compact constant negative curvature manifolds using different methods. We extend them to all rank-one cases in Theorem 5. We follow the strategy of [DFG15] but we emphasize that new difficulties appear due to the anisotropy of the Lyapunov exponents given by the fact that the curvature is not constant anymore.
We would like to end the introduction with the following remark: in this article we use the precise correspondence between Ruelle and Laplace resonant states in order to prove the first version of quantum ergodicity for Ruelle resonant states. If however it becomes possible to prove stronger properties like quantum unique ergodicity for the high frequency limits of Ruelle resonant states, then the quantum-classical correspondence would allow to transfer these results to the semiclassical limits of Laplace eigenfunctions.
Acknowledgements: C.G. is supported by ERC consolidator grant IPFLOW. We thank Benjamin Küster for helpful comments concerning the manuscript.

Ruelle resonances
In this section we introduce the spectral theory of Ruelle resonances for Anosov flows as well as the notion of their resonant states and invariant distributions.
Within this section let M be a smooth, compact manifold without boundary, X ∈ C ∞ (M, T M) a smooth vector field that generates an Anosov flow ϕ t .
We want to introduce Ruelle resonances as the discrete spectrum of the differential operator X on suitable Hilbert spaces, as they have been introduced by Liverani [Liv04], Butterley-Liverani [BL07] and Faure-Sjöstrand [FS11] and Dyatlov-Zworski [DZ16]. We will use the microlocal approach from [FS11,DZ16].
Proposition 2.1. There exists a family of Hilbert spaces denoted by H N and parametrized by N > 0. Each H N is an anisotropic Sobolev space that fulfills the relations where H N (M) denotes the ordinary L 2 -based Sobolev space of order N . Consider furthermore the operator X acting on D (M) and define which is a dense subset in H N . Then the operator X : H N → H N is an unbounded closed operator defined on a dense domain satisfying: (1) There is C 0 > 0 such that for any N > 0 the operator (X + λ) : Dom N (X) → H N is Fredholm of index 0 depending analytically on λ in the region {Re(λ) > −N/C 0 }. (2) There is a constant C 1 > 0 such that for Re λ > C 1 the operator (X + λ) : Consequently, the operator −X has discrete spectrum on {Re(λ) > −N/C 0 } of finite algebraic multiplicity. We call λ ∈ C a Ruelle resonance if Res X (λ) := ker H N (X + λ) = 0 for some N > −C 0 · Re(λ).
It can also be shown [FS11, Theorem 1.5] that for each j ≥ 0, ker H N (X + λ) j ⊂ D (M) is independent of the choice of N > −C 0 · Re(λ), so Res X (λ) is well defined and we call it the space of Ruelle resonant states associated to the resonance λ. In general the geometric and algebraic multiplicity of a Ruelle resonance λ need not be equal. Thus we define J(λ) to be the smallest integer such that for all k ≥ J(λ). We call ker H N (X + λ) J(λ) the space of generalized Ruelle resonant states. Spectral theory also provides us with a finite rank spectral projector , which commutes with the geodesic flow, i.e. [X, Π λ 0 ] = 0. Note that this spectral projector coincides with the residue of the meromorphically continued resolvent as defined in the introduction (see [DZ16,Section 4]). For f ∈ C ∞ (M), the multiplication operator by f is continuous on H N and the spectral projector allows to define a distribution ) is called multiplicity of the resonance λ 0 . From the invariance of Π λ 0 under the geodesic flow it directly follows that T λ 0 is flow-invariant as well. Note that from the microlocal description of Π λ 0 in [DZ16] it follows that T λ 0 ∈ D (M) does not depend on the choice of H N and is an intrinsic invariant distribution associated to each Ruelle resonance which we will call invariant Ruelle distribution. Note that if the space of generalized resonant states for λ 0 = 0 is onedimensional, then the invariant Ruelle distribution corresponds to the unique SRBmeasure.
We will not need any detailed knowledge on the construction of the Hilbert space structure of H N . However, we will use the microlocal description of the resonant states using the wavefront set. Therefore, let be the Anosov splitting of the tangent bundle into neutral, stable and unstable bundles. We can introduce the following dual splitting of the cotangent space Lemma 2.2 ([DFG15, Lemma 5.1]). The space of Ruelle resonant states for a resonance λ 0 is given by where WF(u) ⊂ T * M denotes the wave-front set of the distribution u. The generalized resonant states can be characterized similarly: By duality one can define the co-resonant states which we will denote by Res X * (λ 0 ) and if the Anosov flow preserves a smooth volume denisty 2 , they can microlocally be described as Note that given a Ruelle resonance λ 0 , since E * s ∩ E * u = 0, the resonant and coresonant states satisfy conditions on their wavefront sets ensuring that their product is well-defined in D (M) by [Hör03, Theorem 8.2.10]. For u ∈ Res X (λ 0 ), v ∈ Res X * (λ 0 ), the product is a flow-invariant distribution: We will see in Section 5 that the Patterson-Sullivan distributions on compact locally symmetric spaces of rank one can be interpreted as such a product of resonant states. It turns out that in the case of a nondegenerate Ruelle resonance without Jordan block, (i.e. for dim(Res X (λ 0 )) = 1 and J(λ 0 ) = 1) also the invariant Ruelle distribution T λ 0 can be expressed in this way.

Ruelle resonances on rank one locally symmetric spaces
In this section we want to relate the Ruelle resonant states of the so called "first band" on rank one locally symmetric spaces to certain distributional vectors in principle series representations (Proposition 3.5).
3.1. Riemannian symmetric spaces. We first recall some standard notations for Riemannian symmetric spaces. Let G be a noncompact, connected, real, semisimple Lie group of real rank 1 with finite center and K ⊂ G a maximal compact subgroup. We will write G = KAN for an Iwasawa decomposition and let M be the centralizer of A in K in what follows. The Killing form K : g × g → R is a non-degenerate bilinear form and the Cartan involution θ : g → g on the Lie algebra g of G allow to define a natural positive-definite scalar product ·, · g = −K(·, θ·) on g (and thus on g * as well). Moreover, as G/K is of rank 1, i.e. dim R (A) = 1, we have an isomorphism a * C → C by identifying λ with λ(H 0 ), after choosing a suitable element H 0 ∈ a: we choose H 0 to be the uniquely determined element of a which satisfies α 0 (H 0 ) = α 0 , α 0 g =: ||α 0 ||, where α 0 ∈ a * is the unique simple positive restricted root. We shall denote by ρ ∈ a * the half-sum of the positive restricted roots weighted by multiplicity, and let m α 0 := dim g ±α 0 and m 2α 0 := dim g ±2α 0 be the multiplicities of the possible restricted roots. In particular, one gets ρ = 1 2 ||α 0 ||m α 0 + ||α 0 ||m 2α 0 when identifying a * C C. Under the above assumptions G/K is a Riemannian symmetric space of rank 1. More precisely, G/K is a hyperbolic space H n K where K is either R, C, H, O (H denotes the quaternions and O the octonions) and n is its real dimension 3 . To simplify notation, we will write H n = G/K. The Killing form induces a canonical Riemannian metric on H n and with this metric the spaces have negative, but possibly non-constant, curvature. Furthermore, it induces a smooth left G-invariant measure, which we denote by dx. The unit sphere bundle SH n can be identified with G/M and we denote the left G-invariant Liouville measure by dµ L . Using a trivialization SH n ∼ = H n × S n−1 the Liouville measure can be written as dµ L = dx ⊗ dµ S n−1 where dµ S n−1 is the standard Lebesgue measure on the unit sphere. Note that the measures dx and dµ L are intrinsically defined by the Riemannian geometry of H n . For further reference let us mention how one can normalize the bi-invariant Haar measures on the Lie groups G, M, K, A and N in a consistent way: we start by fixing dm by the condition vol(M ) = 1 and in addition set vol(K) = vol(S n−1 ). The adjoint action of K on p gives an identification K/M ∼ = S n−1 and our choice implies that, under this identification, dµ S n−1 = dkM . Next we fix dg such that dgK = dx. With these choices one obtains the identification dgM = dµ L in the following way: the G = N AK decomposition gives a trivialization G/M ∼ = G/K × K/M and using the normalizations from above one checks that dgM = dgK ⊗ dkM = dx ⊗ dµ S n−1 = dµ L . Finally it remains to normalize da: recall that we identify a ∼ = R or respectively a * Let Γ ⊂ G be a torsion-free discrete co-compact subgroup, then M := Γ\G/K is a smooth compact Riemannian locally symmetric space of rank 1. We denote the respective positive Laplacians by ∆ H and ∆ M . Again we have a Lebesgue measure defined by the Riemannian metric as well as the Liouville measure and by slight abuse of notation we also denote them by dx and dµ L . The unit tangent bundle M := SM of M = Γ\H n can be identified with the quotient Γ\G/M . Under this identification the geodesic flow is simply the natural right action of A. It is known to be an Anosov flow (see e.g. [Hil05]), thus all the definitions of Ruelle resonances and resonant states from Section 2 apply. Moreover, the Anosov splitting of T M into neutral, stable, and unstable directions can be expressed explicitly as associated vector bundles Here a is the Lie algebra of A, n + is the Lie algebra of N and n − = θn + is the Lie algebra of N := θN , where θ denotes the Cartan involution on G as well as its derivative on the Lie algebra g of G. The bundles E u and E s are identified with Similarly, the geodesic flow on the cover SH n = G/M also has an Anosov splitting with smooth stable and unstable bundles 3.2. The first band of classical Ruelle resonances. In [DFG15] it is shown for compact real hyperbolic manifolds that the spectrum of Ruelle resonances forms an exact band structure. A particularly important subset of resonances are those that are invariant by the horocyclic flows (i.e. killed by the unstable derivatives). Similarly, we say a co-resonant state u belongs to the first band if for each section U + of E s we have U + u = 0. We write Res 0 X * (λ 0 ) for the first band Ruelle co-resonant states at the resonance λ 0 ∈ C.
Remark 1. The notion first band is justified by the following result of an exact band structure (see [DFG15] for constant negative curvature manifolds and [KW17] for compact locally symmetric spaces of rank one): If λ 0 ∈ C is a Ruelle resonance with Im(λ 0 ) = 0, then Re(λ 0 ) ∈ −ρ − N 0 α 0 , i.e. the resonances with nonvanishing imaginary parts are arranged on vertical lines parallel to the imaginary axis. Furthermore, if λ = −ρ+ir for r = 0 i.e. if λ lies on the first line, then it is shown in [DFG15,KW17], that the resonant states are first band resonances in the sense of Definition 3.1.
We can lift the resonant states to the cover G/M by the quotient map π Γ : G/M → Γ\G/M . By a slight abuse of notation, X will also denote the infinitesimal generator of the geodesic flow on G/M which descends to the infinitesimal generator of the geodesic flow on Γ\G/M via π Γ . The geodesic flow (the flow of X) on M = Γ\G/M and on SH n = G/M will be denoted by ϕ t . The splitting G × M (a + n + + n − ) of the tangent bundle of G/M is G-invariant and descends to the Anosov splitting for Γ\G/M via π Γ . With this notation, for λ ∈ a * C , let Remark 2. In view of the above characterizations of Res 0 X (λ) and Res 0 X * (λ) we obtain the linear isomorphisms 3.3. Points at infinity. Next we want to identify the first band of (co)-resonant states with distributions on the Furstenberg boundary ∂H n of the symmetric space, which is identified with G/P = K/M where P = M AN is the minimal parabolic of G. Let us explain how this boundary can be naturally obtained from the geodesic flow on G/M . For any point y ∈ SH n = G/M we define the limiting points of the geodesic passing through y: B ± (y) := lim t→+∞π 0 (ϕ ±t (y)) ∈ ∂H n ifπ 0 : SH n → H n is the projection onto the base. In terms of Lie groups, the resulting maps are simply the projections where w + represents the trivial and w − the nontrivial element of the Weyl group W = We will refer to B − and B + as the initial respectively the end point map.
To see (3.2) for the initial point map, note that Ad(w − ) interchanges n + and n − and is −id on a. Furthermore, the initial and the endpoint map are invariant under changes in the unstable, resp. the stable direction.
Finally, both maps intertwine the left G-actions on G/M and Lemma 3.2. The maps Now consider the AN -fiber bundle B + : G/M → G/P and note that (3.5) implies that v is a distribution which is constant along the fibers. Thus there is a T + ∈ D (G/P ) such that v = B * + T + . Analogously, we obtain the surjectivity of Q − . The fact that Q ± intertwines the pullback action is a direct consequence of the intertwining property (3.4) of B ± .
The continuity of Q ± follows directly from the continuity of push-forwards of distributions under submersions. For the continuity of the inverse map consider the embedding ι : is a subset of the annihilator of G × M n + in T * (G/M ). Thus the Hörmander condition for pullbacks [Hör03, Thm 8.2.4] implies that ι * : R ± (0) → D (K/M ) is well defined and continuous. As B ± • ι = Id K/M and Q ± is bijective, we find that ι * = Q −1 ± and we deduce the continuity of the inverse map.
Similar isomorphisms can also be defined for nontrivial resonances λ = 0. This requires the so called horocycle bracket defined on H n × ∂H n = G/K × K/M by ·, · : where H : G → a is given by H(kan) = log a. Geometrically, they correspond to Busemann functions. Using this horocycle bracket as well as the initial and endpoint maps, we can define the smooth functions on where ν 0 = α 0 / α 0 . A straightforward calculation using (3.2) and (3.3) shows that they are ±1 eigenfunctions of the geodesic vector field, and they are constant in the stable and unstable directions, respectively: For γ ∈ G, x ∈ H n = G/K and b ∈ ∂H n = K/M we have the following equalities A last ingredient is the compact picture of the spherical principal series.
Definition 3.3. Let µ ∈ a * C and H µ := L 2 (∂H n ) be the Hilbert space of square integrable functions w.r.t. the K-invariant measure on ∂H n = K/M . Then the spherical principal series representation (π cpt µ , H µ ) is the representation of G on H µ given by Here Note that we can express (3.13) via the functions N γ as follows: where (γ −1 ) * is the pullback of distributions with the diffeomorphism obtained by the left G-action on K/M = G/P .
Proposition 3.4. For λ ∈ a * C the initial and end point transforms defined by intertwining the left regular representation on R ± (λ) ⊂ D (SH n ) with the representation π cpt −(λ+ρ) , D (∂H n ) .
Proof. In view of Lemma 3.2 and the properties of Φ ± , the property of Q λ,± being a topological isomorphism is clear. It only remains to verify the intertwining part: for Combining Proposition 3.4 with Remark 2 we arrive at the promised description of the first band of Ruelle resonances.
Proposition 3.5. There are isomorphisms of finite dimensional vector spaces ) denotes the spaces of Γ-invariant distribution vectors in the spherical principal series with spectral parameter µ = −(λ + ρ).
After having described the Ruelle resonances by distributions on the boundary, we now turn to the description of generalized resonant states via boundary distributions.
Proposition 3.6. Let λ ∈ C be a Ruelle resonance of X on M = Γ\G/M . Then the following conditions are equivalent.
(1) There is a Jordan block of first band resonant states of size J, i.e. there are distributions u 0 , . . . , u J−1 ∈ D (M) and some λ such that and Proof. We start with a Jordan basis u k on M = Γ\G/M as in (1) and we lift them to a set of Γ-invariant distributions on the cover SH n = G/M where π Γ : SH n → M = SM is the natural projection. They satisfy the relations (3.16) withũ k replacing u k . Then we use the functions Φ − defined in (3.7) and define the distributions on SH n Using (3.8) as well as the Γ-invariance ofũ k , a straightforward calculation yields Xv i = 0. According to Lemma 3.2 there are unique distributions T k ∈ D (∂H n ) fulfilling v k = Q − T k . The transformation property (3.17) finally follows, as Q − intertwines the pullback actions, from the following calculation for γ ∈ Γ: This proves that (1) implies (2).
The converse follows similarly: Given T 0 , . . . , T J−1 ∈ D (∂H n ) fulfilling (3.17) we can define v k = Q − T k . Next, we can obtainũ l from the v k by (3.19). From (3.17) we conclude thatũ ∈ D (SH n ) are Γ-invariant, thus we obtain distributions u k ∈ D (M). By a straightforward calculation they fulfill (3.18) and U − u k = 0 for all smooth sections 4. Quantum-classical correspondence 4.1. Poisson Transformation. A central role for the relation between classical and quantum resonances is played by the Poisson transform which we now introduce.
As explained in Section 3.2, we can identify a * with C via λ → λ(H 0 ), and we shall do so in what follows, writing λ 2 instead of λ(H 0 ) 2 . Given a spectral parameter µ ∈ a * C we introduce the eigenspace (cf. [vdBS87]) for the positive Laplacian ∆ H n on H n . Note that by elliptic regularity the elements of E µ (H n ) are real analytic. If we define the space of quantum eigenstates of ∆ M on M as taking the lift to the universal coverπ Γ : H n → M we obtain a bijection between eigenfunctions of ∆ M and Γ-invariant elements in E µ (H n ), denoted by which is the Schwartz kernel of the Poisson transform and which defines, using the K-invariant measure db = dµ S n−1 = dkM on K/M , a linear operator Here we use the notation that T (db) is the generalized density associated with the distribution T via the invariant measure db.
In the case of rank 1 symmetric spaces, kernel and image of the Poisson transform have very explicit descriptions. In this paper we mostly restrict our attention to spectral parameters µ for which the Poisson transform is injective. The maximal domains of definition for the Poisson transforms are spaces of hyperfunctions. As we restrict our attention to spaces of distributions we need to introduce spaces of smooth functions with moderate growth in order to describe the image of our Poisson transforms.
For f ∈ C ∞ (H n ) and r ≥ 0 the norm where d H n is the Riemannian distance function on H n = G/K and o = eK is the base point of H n . Then we define The space of eigenfunctions of weak moderate growth (see [vdBS87,Remark 12.5]) can then be defined as and we can equip it with the direct limit topology.
In the following remark we collect the mapping properties of the Poisson transform we will use (cf. [vdBS87]).
which is a topological isomorphism if and only if µ / ∈ Ex.
Note that E * µ (H n ) is invariant under the left regular representation. Moreover, if one considers the compact picture π cpt −µ of the spherical principal series representation of G associated with the spectral parameter µ (cf. [vdBS87]), then D (∂H n ) can be interpreted as the space of distribution vectors of π cpt −µ . As is well-known, the Poisson transform P µ intertwines these two representations.
For later reference we collect some of the spectral properties of ∆ M .
Remark 4. Let Γ ⊂ G be a co-compact discrete subgroup. Then for all µ ∈ a * C the pullback of smooth functions is a bijection where Γ E µ (H n ) denote the space of Γ-invariant elements in E µ (H n ). Consequently Γ E µ (H n ) is finite dimensional and Γ E µ (H n ) = 0 only holds on a discrete set of values for µ ∈ a * C that fulfill ρ 2 − µ 2 ≥ 0. Furthermore, In the same way we obtain the associated pushforward on distributionsπ 0 * on the universal cover, which allows us to state the following useful expressions for the Poisson transformation.
Proof. By a density argument we restrict our attention to the map for smooth functions φ ∈ C ∞ (K/M ). Recall from section 3.1 that the invariant measures on G/K, G/M and K/M are normalized such that This impliesπ For ψ ∈ C ∞ (K/M ) = C ∞ (G/P ) we recall Q µ−ρ,− (ψ) ∈ C ∞ (G/M ) from (3.15) and note that (we use the isomorphism gP → k(g)M identifying G/P with K/M , k(g) being the K element in the KAN -decomposition of g) (the third line corresponds to a change of variable k → k := k(gk ) in ∂H n = K/M ). The equality P µ =π 0 * • Q µ−ρ,+ follows analogously.
For the special case of a real hyperbolic space a description of Jordan blocks is given in [DFG15]. The proof there however relies on the pairing formula and the self adjointness of the Laplacian. Here we have given a different proof, more in the spirit of [GHW17], that also allows the precise description of the spectral value at λ = −ρ which was untractable with the methods in [DFG15].
Let us finally give a rough description of the first band resonant states at the exceptional points. At these points the Poisson transform is not injective anymore, but one has a nontrivial, closed G-invariant subspace ker P µ ⊂ D (∂H n ) ∼ = H −∞ µ as well as a G-invariant subspace Im P µ ⊂ E * µ (H n ). In particular, the existence of the kernel implies that at these exceptional points there could be more Ruelle resonant states in the first band than expected from the Laplace spectrum.
Note that for real and complex hyperbolic spaces, µ ≥ ρ if µ ∈ Ex.

A new description of Patterson-Sullivan distributions
We briefly recall the construction of Patterson-Sullivan distributions from [AZ07, HS09,HHS12]. For µ, µ ∈ a * C we we introduce a weighted Radon transform R µ,µ by Definition 5.1. Let µ, µ ∈ a * C and ϕ µ ∈ E * µ (G/K) ϕ µ ∈ E * µ (G/K). Then the associated Patterson-Sullivan distribution PS ϕµ,ϕ µ ∈ D (G/M ) is the generalized density defined by its evaluation at f ∈ C ∞ c (G/M ): Here [P −1 µ (ϕ µ )](db) and [P −1 µ (ϕ µ )](db ) are the generalized densities on ∂H n = K/M = G/P obtained by the boundary distributions and the invariant measure. Their tensor product is a generalized density on ∂H n × ∂H n and can be restricted to the open subset (∂H n ) 2 ∆ := (∂H n × ∂H n ) \ ∆(∂H n ), where ∆(∂H n ) is the diagonal in ∂H n × ∂H n . Note that G acts transitively on (∂H n ) 2 ∆ with respect to the diagonal action and that (∂H n ) 2 ∆ ∼ = G/M A as a G-homogeneous space.
Proof. We will prove the statement for the corresponding Γ-invariant distributions on G/M . Letφ µ ∈ E µ (G/K) andφ µ ∈ E µ (G/K) be the lifted Laplace eigenfunctions. Then the lift of the left hand side of (5.1) simply becomes PSφ µ.φ µ , while the right hand side becomes Note that with the diffeomorphism Ψ : we can write the G invariant measure dgM as dgM = e 2ρ(H(g)+H(gw)) (Ψ −1 ) * (da db db ).
In fact, in view of G/M A ∼ = (∂H n ) 2 ∆ and the description of the K-invariant measures on G/P ∼ = K/M and G/P ∼ = K/M this follows from and Ψ(gaM ) = (a(g)a, B + (gM ), B − (gM )). Now inserting the expressions for Q λ,± from (3.15) and replacing the measure, we obtain for the right-hand-side . This is exactly the Patterson-Sullivan distribution as defined in Definition 5.1. It has a meromorphic continuation to z ∈ C (see e.g. [Hel84, IV.6]) which is given by
One easily checks that the zeros and poles of the c-function are contained in the real line.
Theorem 5. Let λ ∈ C \ {−ρ − N 0 α 0 } be a Ruelle resonance in the first band and let v ∈ Res 0 X (λ) and v * ∈ Res 0 X * (λ) be some associated resonant/co-resonant states. Then we have 4 This normalization is slightly different from normalizations common in the literature of symmetric spaces, where one has c(ρ) = 1. We have chosen this normalization such that they give simple formulas in our geometric context The product v · v * is well defined by the wavefront set properties of v, v * .
As a direct consequence of Theorem 5 we obtain that for two quantum eigenstates ϕ, ϕ ∈ Eig ∆ M (µ) the normalization of the corresponding Patterson-Sullivan distribution is given by Furthermore, the pairing formula allows to relate the invariant Ruelle distributions (defined in (2.1)) to Patterson-Sullivan distributions: Corollary 6.1. Let r > 0 such that −ρ + ir is a Ruelle resonance of multiplicity m in the first band. Then ρ 2 + r 2 is an eigenvalue of ∆ M with multiplicity m and, for an L 2 -orthonormal basis ϕ 1 , . . . , ϕ m of Eig ∆ M (ir), we have Proof. Via the quantum-classical correspondence (Theorem 2) we define a basis of Ruelle resonant states u l := I − (ir)(ϕ l ) ∈ Res X (−ρ + ir) as well as a basis of coresonant states u * l := c(ir)I + (ir)(ϕ l ) ∈ Res X * (−ρ + ir). Now the pairing formula (Theorem 5) together with the chosen normalization of u * i implies the bilinear pairing formula u * i , u j := SM u i · u * i dµ L = δ ij , which means that the basis u * i is dual to the chosen basis u i . By Theorem 3 we know that there are no Jordan blocks at the spectral parameter −ρ+ir and thus the spectral projector can be written as Π −ρ+ir = m l=1 u l ⊗ u * l . Now for f ∈ C ∞ (M) we obtain which completes the proof.
So far, pairing formulas like Theorem 5 have been shown for compact hyperbolic surfaces [AZ07, Theorem 1.2] and compact real hyperbolic manifolds [DFG15, Lemma 5.10]. We follow the strategy of proof of [DFG15]: first we consider M π 0 * (v)π 0 * (v * ) with respect to the measure which is obtained by restricting the measure dx on G/K to a Γ-fundamental domain. Then construct a coordinate transformation on an open dense subset that formally makes the Harish-Chandra c-function appear as an integral over N (see Lemma 6.2). As we integrate distributions. However, we have to cut out an -neighborhood of the points where the coordinate transform is not defined and consider the limit → 0 (see Lemma 6.3). As in [DFG15] this turns out to be a subtle limit that requires a suitable regularization of divergent integrals. Compared to the constant curvature case there are two major challenges: first, one has to replace the explicit calculations in the hyperboloid model in the construction of the coordinate transformations. Second, one has to deal with the fact that the defining integral of the c-function becomes an integral over a non-commutative group N , adding some anisotropy to the regularization process.
6.1. A suitable coordinate transformation. Let us define the double unit sphere bundle S n−1 × S n−1 → S 2 M → M as the pullback of SM × SM under the diagonal embedding M → M × M and equip this bundle with the measure dx ⊗ dµ S n−1 ⊗ dµ S n−1 for which we will use the slightly shorter notation dx dη + dη − . Define The integral on the right makes sense by the wave-front set properties of v, v * and the fact that E u , E s are transverse to the vertical bundle ker dπ 0 ⊂ T SM. We furthermore define the open dense subset S 2 ∆ M := {(x, η − , η + ) ∈ S 2 M : η − + η + = 0}. Recall that the unstable bundle E u → M is given by an associated vector bundle E u = Γ\G× M n − and, using the exponential map, can be identified with Γ\G × M N . We will denote points in E u by equivalence classes [g, n], where [gm, n] = [g, mnm −1 ] for all m ∈ M . Thus, any M -conjugation invariant function χ ∈ C ∞ c (N ) defines a function in C ∞ c (E u , C), which we also denote by χ. We have: Lemma 6.2. There is a diffeomorphism A : S 2 ∆ M → E u such that for any Mconjugation invariant function χ ∈ C ∞ c (N ), any Ruelle resonance λ ∈ C in the first band and associated resonant/co-resonant states v ∈ Res 0 X (λ), v * ∈ Res 0 X * (λ), we have Proof. (For a geometric interpretation of the appearing constructions see Figure 1. We first give an explicit construction of the diffeomorphism by constructing a left G invariant diffeomorphismÃ : via the initial and endpoint maps B ± : SH n → ∂H n from (3.1) and note that it restricts to a diffeomorphism E : S 2 ∆ H n → H n × (∂H n ) 2 ∆ . Furthermore, recall (see e.g. after identifying ∂H n ∼ = G/P , where w = w − denotes a representative of the nontrivial Weyl group element. Note that the map ψ : is well defined and left G-equivariant. Furthermore, we can construct an inverse using the AN K-decomposition g = a AN K (g)n AN K (g)k AN K (g) Using these three diffeomorphisms we define A : It remains to prove (6.2). We clearly have that the left hand side equals Thus, we have to calculate the pushforward of the distribution A * (vv * ) and the measure A * (dx dη − dη + ).
Let us first consider the pushforward A * v * : as v * is a first band co-resonant state, we know by Proposition 3.4 that v * = Φ λ + B * + w for some distribution w ∈ D (∂H n ). If we write A(x, η − , η + ) = [g, n], then by construction of A we have B + (x, η + ) = B + (gM ) = k KAN (g)M and consequently Since x = gnK, we can use (3.6) and (3.10) to simplify this expression to Analogously, we calculate (A * v)([g, n]) = v(gM ). The latter equality can also be understood geometrically as (x, η − ) and gM lie, by construction, on the same instable manifold and v is constant along the instable leaves, because it is a first band coresonant state.
Let us next consider the transformation of measures. In analogy to the final step in the proof of Proposition 4.3 we use [Hel84,Lemma I.5.19] to establish the formula Moreover, by the Propositions B.2 and B.3 proven in the appendix, we have Putting these three transformations together and simplifying the exponents using (3.6) and (3.10) we obtain a constant c A > 0 such that Because dn is invariant under inversion, this establishes (6.2) up to a multiplicative constant. With the chosen normalizations of the measures, this constant has, however, to be equal to one. This can be seen by passing to a co-compact quotient and integrating a constant function on both sides of the variable transform. This completes the proof of Lemma 6.2.
6.2. Renormalization. The variable transformation from Lemma 6.2 would directly imply the desired pairing formula if we could set χ = 1. However, as v, v * are distributions, this is not allowed and in fact we see for the important case of taking a Ruelle resonance with Re(λ) = −ρ that the integral over N on the right hand side of (6.2) would not converge anymore. We thus have to perform a careful regularization of the appearing quantities.
As a first step we introduce a suitable cutoff function: We take an arbitrary Mconjugation invariant function χ ∈ C ∞ c (N ) which is equal to 1 in a neighborhood of the identity. Then we define for any ε > 0 whereȞ 0 := H 0 / α 0 . As conjugation by exp(Ȟ 0 ) on N is strictly contracting towards the identity, χ ε converges to 1, uniformly on any compact subset of N . As M centralizes A, χ ε is still M -conjugation invariant, so that it defines a function in C ∞ c (Γ\G× M N , C). After a pullback with A this function can be identified with a function in C ∞ c (S 2 ∆ M) ⊂ C ∞ (S 2 M). By abuse of notation we will denote all these instances of the function by χ ε and it will be clear from the context where the function lives. Having defined χ ε we can write Note that by this decomposition the term I c (ε) can now be treated with the transformation from Lemma 6.2. The idea of taking formally χ = 1 would correspond to considering the limit ε → 0. We will see that in general this limit is defined neither for I c (ε) nor for I 0 (ε). However, we can prove the following important asymptotic expansions.
An important tool in the proof of the expansions (6.6) and (6.7) will be the following differential operator on N which satisfies the relation ε∂ ε χ ε = Lχ ε . Note that for U 1 ∈ g −α , U 2 ∈ g −2α we have Thus, the operator L corresponds to a special linear combination of Euler operators on the two different root spaces.
As L commutes with the M -conjugation on N we can lift it to a differential operator L with smooth coefficients on Γ\G × M N in the following way: Note thatL is not just the differential operator that differentiates along the fibers of Γ\G × M N , but it is twisted with a derivative along the geodesic flow on the base space Γ\G/M . Nevertheless, we still have the property ε∂ ε χ ε =Lχ ε , where χ ε is now understood as a function on Γ\G × M N . The twist is crucial for the following lemma.
Lemma 6.4. The differential operator (A −1 ) * L on the open dense set S 2 ∆ M ⊂ S 2 M extends to a first order differential operator on S 2 M with smooth coefficients.
By abuse of notation we will denote this extended operator on S 2 M again byL.
Proof. As for the construction of A, we lift everything to H n . By definition the vector field A −1 * L is smooth on S 2 ∆ H n , so we have to study its behavior near the antidiagonal {η − + η + = 0} ⊂ S 2 H n , and we need some appropriate coordinates for this neighborhood. By the N AK-decomposition we can identify S 2 H n ∼ = G/K × K/M × K/M ∼ = G/K × G/P × G/P . The antidiagonal consists of the points (naK, kP, kwP ), na ∈ N A, k ∈ K, where again w is some representative of the nontrivial Weyl group element. By the Bruhat decomposition (naK, kP, kwn ∆ P ) with na ∈ N A, k ∈ K, n ∆ ∈ N parametrizes an open neighborhood of the antidiagonal and the antidiagonal corresponds to n ∆ = e. We now want to calculate how these coordinates are transformed for n ∆ = e under the diffeomorphism A.
The crucial point is that we need an explicit expression for G −1 . This can also be achieved by the Bruhat decomposition which for real rank one implies that N P ⊂ G is open and dense. Given g ∈ N P we define n N P (g) ∈ N to be the unique element such that gP = n N P (g)P . Then we can write G −1 (g 1 wP, g 2 P ) = g 1 n N P (g −1 1 g 2 )M A. Putting everything together we obtain where Ω : N \ {e} → N \ {e} n → n N P (wn)) .
Using this identity and (6.9) we calculate the transformation of the global vector field: which clearly is smooth in a neighborhood of the diagonal. 5 Using the smooth vector fieldL on S 2 M and the identity ε∂ ε χ ε =Lχ ε integration by part yields is the L 2 adjoint on S 2 M. By virtue of Lemma 6.2 we obtain the alternative expression ε∂ I c (ε) = SM vv * dµ L · N (L * e −2(ρ+λ)H(n) )χ ε (n) dn, where L * = −L − div dn (L) is the L 2 -adjoint on N .
The following lemma implies that L is suitable for a regularization at infinity in n and L is suitable for a regularization near the antidiagonal in S 2 M . In order to formulate it, let us define n := U 1 2 + U 2 where n = exp(U 1 + U 2 ), U 1 ∈ g −α 0 , U 2 ∈ g −2α 0 .
Ruelle resonance in the first band and v, v * are corresponding resonant/ co-resonant states, then (L * − β (λ))Ṽ −1 vv * =Ṽ vv * (6.12) Before we prove Lemma 6.5 let us show how it implies Lemma 6.3: 5 Without the derivative along the geodesics inL there would occur a derivative d dτ |τ =0 nake −τȞ0 Ω(n)e τȞ0 for A −1 * L. But near the diagonal in the limit n → e, Ω(n) → ∞ and this derivative would diverge. This shows the necessity for the precise choice ofL.
With this notation let us first show that (6.12) can in fact be reduced to (6.11). We note that, asL is a smooth differential operator, we can calculate its divergence on S 2 ∆ M. Using (6.5) as well as the fact that dµ L is preserved by the geodesic flow, we calculate div(L) = −2ρ LN N + div dn (L).
Furthermore, using (A * (vv * ))([g, n]) = (vv * )(gM )N −λ (n) as well as the description of L near the antidiagonal (6.10), we obtainL(vv * ) = −λ LN N (vv * ), whencẽ On the other side, for L acting on N , we directly get Using the fact that (log) * dn is the Lebesgue measure on n − and the expression of L as an Euler operator (6.8), we get div dn (L) = 2ρ. For both relations, (6.12) and (6.11), we are thus lead to study the properties of (LN )/N . Using (6.14) we get where Q(n) = 2 + 2c U 1 2 . Putting everything together we get We note that Q N 2 = O( n −1 ) for n → ∞. Thus, we have found β 1 (λ) = 2(ρ +λ) and V 1,λ = −(2ρ +λ) Q N 2 . In order to see that inductively one gets functions V ,λ that decay faster and faster as n → ∞, we calculate All terms on the right hand side are O( n −2 ). Now, using this formula together with (6.15), a straightforward induction gives us the existence of the β (λ) and the V satisfying (6.11) and (6.12), where only integer powers of Q N 2 and N −2 appear: we get β (λ) = 2(ρ +λ + − 1) and V is some homogeneous polynomial of degree in the variable (Q/N 2 , 1/N 2 ) with coefficients polynomial in λ.
The existence of a smooth extension to S 2 M is obvious by the following global argument:L is a smooth vector field thus its divergence is a smooth function, and all V are built out of derivatives of this divergence. The smoothness can, however, also be seen in the coordinates n, a, k, n ∆ around the antidiagonal, which were introduced in the proof of Lemma 6.4. In these coordinates N = e α 0 H(n N P (wn ∆ )) = e α 0 (H(n ∆ )−B(wn ∆ )) where B = log(a N M AN ). Now for n ∆ = exp(V 1 + V 2 ), V 1 ∈ g −α 0 , V 2 ∈ g −2α 0 [Hel78, Thm IX.3.8] gives us and with a completely analogous calculation as above we check the vanishing of the iteratively definedṼ .

Equidistribution for Ruelle resonant states
Let us finally draw the desired conclusions concerning the high frequency limts of invariant Ruelle distributions from the explicit relations between Ruelle resonant states and Patterson Sullivan distributions.
For M = Γ\G/M = SM a number λ = −ρ + µ ∈ C is a Ruelle resonance of the first band if and only if the complex conjugate λ is a Ruelle resonance of the first band as well. This follows from the fact that the generating vector field X commutes with complex conjugation. By this symmetry of the spectrum it is enough to consider first band resonances with Im(λ) ≥ 0, and since we are interested in high frequency limits, we take Im(µ) > 0. We thus denote by λ n = −ρ + ir n ∈ C a sequence of Ruelle resonances in the first band with Re(r n ) > 0 and r n+1 ≥ r n . We do not want to repeat them according to multiplicity but rather work with the multiplicities m n := dim(Res 0 X (λ n )). Now to any subsequence (r kn ) n>0 ⊂ (r n ) n>0 we can associate the sequence of invariant Ruelle distributions T λ kn ∈ D (M) and we study the weak limits of these sequences.
Then we obtain the following reformulation of Theorem 1. Proof. According to Theorem 2 we can associate to each Ruelle resonance λ n = −ρ+ir n a spectral parameter µ n := ir n ∈ a * C of the Laplacian such that dim(Eig ∆ M (µ n )) = dim(Res 0 X (λ n )) = m n . For each of these ∆ M -eigenspaces we choose an orthonormal basis ϕ n,l of real-valued functions, where l = 1, . . . , m n . Let B = {ϕ n,l } be the set of all these basis vectors. Then the quantum ergodicity theorem of Shnirelman-Zelditch-Colin de Verdière [Shn74,CdV85,Zel87] implies the existence of a density one subsequence such that the Wigner distributions W φ,n (see Appendix A for a precise definition) converge towards the Liouville measure. More precisely, we can split B into the disjoint union B = B good ∪ B bad such that for any sequence in B good the Wigner distributions converge towards the Liouville measure and where m bad n := #(B bad ∩ Eig ∆ M (ir n )). In order to obtain the subsequence r n k for the convergence of the invariant Ruelle distributions, we remove from the full sequence (r n ) n>0 all elements for which m bad n /m n ≥ n , where n := (sup k≥n q k ) 1/2 is a decreasing sequence converging to 0. Thus, we obtain a subsequence (r kn ) n≥0 for which we have Let us finally show the convergence of the subsequence of invariant Ruelle distributions: we fix a Ruelle resonance λ kn . Then, using the basis ϕ kn,l and the isomorphism I − (ir kn ), we define a basis u l := I − (µ kn )(ϕ kn,l ) ∈ Res X (λ kn ). Theorem 2 implies that u l is a basis of Res 0 X (λ n ). In addition we define the basis of co-resonant states u * l := 1 PS Γ ϕ kn,l ,ϕ kn,l (1) I + (µ kn )(ϕ kn,l ) ∈ Res X * (λ kn ).
The pairing of co-resonant states and resonant states is simply given by the pairing of distributions with disjoint wavefront sets, which in turn is the product of the distributions paired with 1. We recall the pairing formula of Theorem 5: let λ ∈ a * C and for v ∈ Res 0 X (λ), v * ∈ Res 0 X * (λ). Then one has the identity where c(µ) = N e −(µ+ρ)H(nw) dn is the Harish-Chandra c-function.
Thus, the pairing formula together with the chosen normalization of u * i implies, that u * i [u j ] = δ ij , which means, that the basis u * i is dual to the chosen basis u i . By Theorem 3 we know that there are no Jordan blocks at the spectral parameter λ kn , and thus the spectral projector can be written asΠ kn = Iff ∈ C ∞ c (T M) is an arbitrary compactly supported function such thatf | SM = f , then by [HHS12] T But as we have chosen the subsequence such that m bad kn m kn → 0, we conclude that in the limit n → ∞ only the "good" Wigner distributions contribute and we obtain concluding the proof.
In quantum ergodicity, one is then interested in understanding the weak limits of these generalized densities. They have the following important properties.
Proposition A.2. Let r n > 0 be the positive real numbers, such that Eig ∆ M (ir n ) = 0, where we repeat r n according to the multiplicity (dimension of Eig ∆ M (ir n )), and let ψ n ∈ Eig ∆ M (ir n ) be an associated L 2 -normalized eigenfunction. If there is µ ∈ C −∞ (T * M) as well as a subsequence r n k such that W ψn k → µ weakly, then µ is a positive Radon measure that is supported on S * M and we call it a semiclassical measure. There is a constant c G such that is actually equal to Let d(gK) be the G-invariant measure on G/K normalized by the Killing-metric on G/K. By our choices in Section 3.1 this coincides with the push-forward of the Haar measure on G by the canonical projection pr K : G → G/K. Further, letpr M : G × N → G × M N be the canonical projection with respect to the M -action. We now use Fubinis theorem and arrange the integrals, such that the dn integral is the inner integral. Then the invariance of dn by left multiplication implies that the latter integral equals c 1 c 2 N N A Nh nn a AN K (n −1 an ), n AN K ((n −1 an ) dn da dn dn.
Reordering the integrals such that the da integral is the interior integral and using the invariance of da we can transform our expression to c 1 c 2

N N A Nh
(nn a, (a −1 n −1 a) n AN K (n )) da dn dn dn.
Note that N → N , n → w −1 n w is measure preserving. Therefore we can rewrite this integral as c 1 c 2 N A N Nh (nn a, (a −1 n −1 a) n AN K (wn w −1 )) dn dn da dn.
Moreover we have that the map ν : N → N , n → n AN K (wnw −1 ) has Jacobian equal to one (see Lemma B.4), thus we obtain has Jacobi determinant | det Dν| = 1.
Case 1: Suppose that w = w −1 . Then n = ν(ñ), i.e. we have that ν = ν −1 . As a consequence the Jacobian determinant is 1. Case 2: Suppose that one cannot choose w such that w = w −1 . Then one can choose w of order 4. The calculation above shows that n = n AN K (ww −2ñ w 2 w −1 )ν(w −2ñ w 2 ).
Thus we deduce ν 4 = Id and the Jacobian determinant is again equal to 1.