We prove resolvent estimates for semiclassical operators such as $-h^2 Delta+V(x)$ in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by $h^{-M}$ in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrodinger propagation and to energy decay of solutions to wave equations.
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