  For a large class of semiclassical pseudodifferential operators, including
Schr\"odinger operators, $ P (h) = -h^2 \Delta_g + 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 \[ \| u \| \leq C (\sqrt{\log(1/h)}/ h) \| P
(h)u \| + C \sqrt {\log(1/h)} \| (I - A) u \| . \] This generalizes earlier
estimates of Colin de Verdi\`ere-Parisse \cite{CVP} obtained for a special
case, and of Burq-Zworski \cite{BZ} for real hyperbolic orbits.
