For a compact Riemannian locally symmetric space M of rank one and an associated vector bundle V τ over the unit cosphere bundle S * M, we give a precise description of those classical (Pollicott-Ruelle) resonant states on V τ that vanish under covariant derivatives in the Anosovunstable directions of the chaotic geodesic flow on S * M. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators D(G, σ) on compatible associated vector bundles W σ over M. As a consequence of this description, we obtain an exact band structure of the Pollicott-Ruelle spectrum. Further, under some mild assumptions on the representations τ and σ defining the bundles V τ and W σ , we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott-Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of W σ . Our methods of proof are based on representation theory and Lie theory.and call the elements of Res 0 (λ) first band resonant states. We call λ a first band resonance 2 if Res 0 (λ) = {0}. Now, let π : S * M → M be the bundle projection and recall that the pushforward of distributions along this submersion is a well-defined continuous operator π * : D ′ (S * M) → D ′ (M). Dyatlov, Faure, and Guillarmou then prove the following remarkable result [DFG15]: If λ is a first band Pollicott-Ruelle resonance, π * restricts towhere µ(λ) = − (λ + ̺) 2 +̺ 2 and ̺ = (dim(M)−1)/2. Furthermore, provided one has λ / ∈ −̺− 1 2 N 0 , the above map is an isomorphism of complex vector spaces.The aim of the present article is to extend these results to the setting of associated vector bundles over arbitrary compact Riemannian locally symmetric spaces of rank one. From now on, suppose that M is such a space. Let us briefly introduce the setting: We can write our manifold as a double quotient M = Γ\G/K, where G is a connected real semisimple Lie group of real rank one with finite center, K ⊂ G a maximally compact subgroup, and Γ ⊂ G a discrete torsion-free cocompact subgroup. The Riemannian metric on M is defined in a canonical way from the non-degenerate Killing form on the Lie algebra g of G. This metric has strictly negative sectional curvature 3 , so 1 In the language of physics, the geodesic flow ϕt : S * M → S * M generated by X describes the classical mechanical movement of a single free point particle on M with constant speed, while ∆ is, up to rescaling, the Schrödinger operator associated to the corresponding quantum mechanical system. 2 The name "first band" comes from the observation due to Dyatlov, Faure, and Guillarmou that the Pollicott-Ruelle spectrum on surfaces of strictly negative constant curvature has an exact structure of bands parallel to the imaginary axis. The first band resonances turn out to be those lying closest to the imaginary axis.3 The case of manifolds with strictly negative constant curvature, studied in [DFG15], ...

Given a closed orientable hyperbolic manifold of dimension = 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 of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers.

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