  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 $U_{x_0}:
L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)$ has a nontrivial invariant $L^2$ function.
The proof is based on an explict Duistermaat-Guillemin-Safarov pre-trace
formula and von Neumann's ergodic theorem.
