Semiclassical non-concentration near hyperbolic orbits

For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h)=−h²Δg+V(x), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if A is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit, then