Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods

We introduce the definition of irreducible quasimodes, which are quasimodes with $h$-wavefront sets living on the smallest invariant sets in phase space. We prove a strong conditional unique continuation estimate for these quasimodes in rotationally invariant neighbourhoods on compact surfaces of revolution. The estimate states that irreducible Laplace quasimodes have $L^2$ mass bounded below by $C_\epsilon \lambda ^{-1 - \epsilon }$ for any $\epsilon >0$ on any open rotationally invariant neighbourhood which meets the semiclassical wavefront set of the quasimode. For an analytic manifold, we conclude the same estimate with a lower bound of $C_\delta \lambda ^{-1 + \delta }$ for some fixed $\delta >0$