Resonances near the real axis for manifolds with hyperbolic trapped sets

For manifolds Euclidian at infinity, we show that for some compact perturbations of the Laplacian with hyperbolic classical trapped set, there are strips below the real axis where the resonance counting function grows sub-linearly. We also provide an inverse result, showing that the knowledge of the scattering poles can give some information about the Hausdorff dimension of the trapped set when the classical flow satisfies the Axiom-A condition.