Lower resolvent bounds and Lyapunov exponents

We prove a new polynomial lower bound on the scattering resolvent. For that, we construct a quasimode localized on a trajectory $\gamma$ which is trapped in the past, but not in the future. The power in the bound is expressed in terms of the maximal Lyapunov exponent on $\gamma$, and gives the minimal number of derivatives lost in exponential decay of solutions to the wave equation.