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Given a non-compact semisimple real Lie group G and an Anosov subgroup Γ, we utilize the correspondence between ℝ-valued additive characters on Levi subgroups L of G and ℝ-affine homogeneous line bundles over G/L to systematically construct families of non-empty domains of proper discontinuity for the Γ-action. If Γ is torsion-free, the analytic dynamical systems on the quotients are Axiom A, and assemble into a single partially hyperbolic multiflow. Each Axiom A system admits global analytic stable/unstable foliations with non-wandering set a single basic set on which the flow is conjugate to Sambarino's refraction flow, establishing that all refraction flows arise in this fashion. Furthermore, the ℝ-valued additive character is regular if and only if the associated Axiom A system admits a compatible pseudo-Riemannian metric and contact structure, which we relate to the Poisson structure on the dual of the Lie algebra of G.

Related project(s):
65Resonances for non-compact locally symmetric spaces

A three-dimensional quasi-Fuchsian Lorentzian manifold M is a globally hyperbolic spacetime diffeomorphic to Σ×(−1,1) for a closed orientable surface Σ of genus ≥2. It is the quotient M=Γ∖ΩΓ of an open set ΩΓ⊂AdS3 by a discrete group Γ of isometries of AdS3 which is a particular example of an Anosov representation of π1(Σ). We first show that the spacelike geodesic flow of M is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for Γ extend meromorphically to ℂ. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian ◻ of M, which is meromorphic on ℂ with poles of finite rank, defining a notion of quantum resonances and quantum resonant states related to the Ruelle resonances and (co-)resonant states by a quantum-classical correspondence. This initiates the spectral study of convex co-compact pseudo-Riemannian locally symmetric spaces.

Related project(s):
65Resonances for non-compact locally symmetric spaces

We prove that the Patterson-Sullivan and Wigner distributions on the unit sphere bundle of a convex-cocompact hyperbolic surface are asymptotically identical. This generalizes results in the compact case by Anantharaman-Zelditch and Hansen-Hilgert-Schröder.

Related project(s):
65Resonances for non-compact locally symmetric spaces

By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Guéritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Related project(s):
65Resonances for non-compact locally symmetric spaces

For negatively curved symmetric spaces it is known that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.

Related project(s):
65Resonances for non-compact locally symmetric spaces

We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.

Related project(s):
65Resonances for non-compact locally symmetric spaces

Related project(s):
65Resonances for non-compact locally symmetric spaces