Focal points and sup-norms of eigenfunctions

If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(\lambda )$ saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point $x_0\in M$ for the geodesic flow at which the associated Perron-Frobenius operator $U_{x_0}: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)$hasanontrivialinvariant$L^2$ function. The proof is based on an explict Duistermaat-Guillemin-Safarov pre-trace formula and von Neumann’s ergodic theorem.