Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f ₀, f ₁). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.     

Pollicott–Ruelle Resonances for Open Systems

We define Pollicott–Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane.     

Identification of a Connection from Cauchy Data on a Riemann Surface with Boundary

We consider a connection ?^X{\nabla^X} on a complex line bundle over a Riemann surface with boundary M ₀, with connection 1-form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) L : = ?^X*?^X + q{L := \nabla^X{^*\nabla^X} + q} , with q a complex-valued potential, uniquely determines the connection up to gauge isomorphism, and the potential q.