Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmonic inversion the distribution of the imaginary parts of the resonances is determined. With increasing opening of the systems, a spectral gap is observed for thick as well as for thin repellers and for the latter case it is compared with the known topological pressure bounds. The maxima of the distributions are found to coincide for a large range of the distance to radius parameter with half of the classical escape rate. This confirms theoretical predictions based on rigorous mathematical analysis for the spectral gap and on numerical experiments for the maxima of the distributions. A paradigm for systems with fractal repellers is the ndisk scattering system (see Fig. 1). It was introduced in the 1980s by Ikawa in mathematics [16] and by Gaspard and Rice [17][18][19] and Cvitanović and Eckhardt [20] in physics. It is given by n hard disks with centers forming a regular polygon. The distance between the centers is denoted by R and the disk radius by a; R/a determines the system up to scaling (see Fig. 1).The quantum system is described by the Helmholtz equationThe quantum resonances k n = Re k n + iIm k n , are the complex poles of the scattering matrix. For the three-disk system this scattering matrix is expressed using Bessel functions and that allowed Gaspard and Rice [19] to calculate the quantum resonances numerically. Classically, particle trajectories are given by straight lines reflected by the disks. From periodic trajectories a wide range of classical quantities such as the classical escape rate, the fractal dimension of the repeller and the topological pressure can be calculated using the Ruelle zeta function [18],where the product runs over the primitive periodic orbits, T p are the corresponding period lengths, and Λ p are the stabilities. The topological pressure P (β) is then defined as the largest real pole of ζ β (z). An effective method for its calculation is the cycle expansion [20,21]. The classical escape rate is given by γ cl = −P (1) and the FIG. 1. (color online)A sketch of a three-disk system is shown on the left side, where one fundamental domain is shaded.On the right side a photograph of the experimental cavity without top plate supporting the disk inset and the absorber is presented.

Abstract. For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

Given a holomorphic iterated function scheme with a finite symmetry group G, we show that the associated dynamical zeta function factorizes into symmetry-reduced analytic zeta functions that are parametrized by the unitary irreducible representations of G. We show that this factorization implies a factorization of the Selberg zeta function on symmetric n-funneled surfaces and that the symmetry factorization simplifies the numerical calculations of the resonances by several orders of magnitude. As an application this allows us to provide a detailed study of the spectral gap and we observe for the first time the existence of a macroscopic spectral gap on Schottky surfaces.

We present microwave experiments on the symmetry reduced five-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with a leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to a noninteger value. For the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture for resonances.

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], ...