Let (Σ, g) be a closed, oriented, negatively curved surface, and fix pairwise disjoint simple closed geodesics γ ,1 , . . . γ ,r . We give an asymptotic growth as L → +∞ of the number of primitive closed geodesic of length less than L intersecting γ ,j exactly n j times, where n 1 , . . . , n r are fixed nonnegative integers. This is done by introducing a dynamical scattering operator associated to the surface with boundary obtained by cutting Σ along γ ,1 , . . . , γ ,r and by using the theory of Pollicott-Ruelle resonances for open systems.
We provide a meromorphic continuation for Poincaré series counting orthogeodesics of a negatively curved surface with totally geodesic boundary, as well as for Poincaré series counting geodesic arcs linking two points. For the latter series, we show that the value at zero coincides with the inverse of the Euler characteristic of the surface.
In this paper, we compute special values of certain combinatorial zeta functions counting geodesic paths in the (n−1)-skeleton of a triangulation of a n-dimensional manifold. We show that they carry a topological meaning. As such, we recover the first Betti number and L 2 -Betti number of compact manifolds, and the linking number of pairs of null-homologous knots in a 3manifold.The tool to relate the two sides (counting geodesics/topological invariants) are random walks on higher dimensional skeleta of the triangulation. Given a hyperbolic surface Σ, Fried [5] initiated the study of the behavior at the origin of the so-called Ruelle zeta function [10] ζ Σ (s) = γ 1 − e −s (γ) ,
Let D ⊂ R d , d 2, be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let µ j ∈ C, Im µ j > 0 be the resonances of the Laplacian in the exterior of D with Neumann or Dirichlet boundary condition on ∂D. For d odd, u(t) = j e i|t|µj is a distribution in D ′ (R \ {0}) and the Laplace transforms of the leading singularities of u(t) yield the dynamical zeta functions η N , η D for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. Under the non-eclipse condition, for d2 we show that η N and η D admit a meromorphic continuation in the whole complex plane. In the particular case when the boundary ∂D is real analytic, by using a result of Fried [Fri95], we prove that the function η D cannot be entire. Following Ikawa [Ika88], this implies the existence of a strip {z ∈ C : 0 < Im z δ} containing an infinite number of resonances µ j for the Dirichlet problem.
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