  For a large class of semiclassical operators $P(h)-z$ which includes
Schr\"odinger operators on manifolds with boundary, we construct the Quantum
Monodromy operator $M(z)$ associated to a periodic orbit $\gamma$ of the
classical flow. Using estimates relating $M(z)$ and $P(h)-z$, we prove
semiclassical estimates for small complex perturbations of $P(h) -z$ in the
case $\gamma$ is semi-hyperbolic. As our main application, we give logarithmic
lower bounds on the mass of eigenfunctions away from semi-hyperbolic orbits of
the associated classical flow.
  As a second application of the Monodromy Operator construction, we prove if
$\gamma$ is an elliptic orbit, then $P(h)$ admits quasimodes which are
well-localized near $\gamma$.
