Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Let s₀ < 0 be the abscissa of absolute convergence of the dynamical zeta $Z(s)$ for several disjoint strictly convex compact obstacles Kᵢ\⊂ , i = 1,..., \κ₀,\\: ₀ \≥ 3, and let R\χ(z) = \χ(-\ΔD - z²)⁻¹\χ,\\: \χ \∈ C₀\∞(), be the cut-off of the Dirichlet Laplacian -\ΔD in \Ω = \\ \∪i = 1k₀ Kᵢ. We prove that there exists \σ₁ < s₀ that $Z(s)$ is analytic for \ (s) \≥ \σ₁ and the cut-off R\χ(z) has an analytic continuation for \ (z) < -\σ₁,\\: |\ (z)| \≥ C > 0.