Local Smoothing Estimates near a Trapped Set with Infinitely Many Connected Components

We prove a local smoothing result for the Schr\"odinger equation on a class of surfaces of revolution which have infinitely many trapped geodesics. Our main result is a local smoothing estimate with loss (compared to \cite{ChMe-lsm}) depending on the accumulation rate of the critical points of the profile curve. The proof uses an h-dependent version of semiclassical propagation of singularities, and a result on gluing an h-dependent number of cutoff resolvent estimates.