Dynamical zeta functions for billiards

Let D⊂ℝ d ,d⩾2, be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let μ j ∈ℂ,Imμ 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|μ j 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 a non-eclipse condition, for every d⩾2 we show that η N and η D admit a meromorphic continuation to the whole complex plane. In the particular case when the boundary ∂D is real analytic, by using a result of Fried [17], we prove that the function η D cannot be entire. Following Ikawa [29], this implies the existence of a strip {z∈ℂ:0<Imz⩽α} containing an infinite number of resonances μ j for the Dirichlet problem. Moreover, for α≫1 we obtain a lower bound for the resonances lying in this strip.