2D internal waves in an ergodic setting

We study a model of internal waves under periodic forcing in an effectively 2-dimensional aquarium. When the underlying classical dynamics has sufficiently irrational rotation number, we prove that the solution to the internal waves equation remains bounded in energy space in time. This result is in contrast with the works by Colin de Verdi\`ere--Saint-Raymond and Dyatlov--Wang--Zworski, which gave a description of singular profiles when the underlying dynamics has periodic attractors. Our result is proved by showing the spectral measure near the forcing frequency is very small.