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(λ) 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 x0 € M for the geodesic flow at which the associated Perron-Frobenius operator Ux0 : L2(S 0M) → L2(S 0M) has a nontrivial invariant L² function. The proof is based on an explicit Duistermaat-Guillemin-Safarov pre-Trace formula and von Neumann's ergodic theorem.