About the blowup of quasimodes on Riemannian manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ_λ satisfy ||φ_λ||_∞≤ Cλ^1-n/2 where -Δφ_λ =λ² φ_λ. The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere Sⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat toriℝⁿ/Γ. We say that Sⁿ, but not ℝⁿ/Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the se ℒ_x of geodesic loops at x has positive measure in S^*_xM. We strengthen this result here by showing that such a manifold must have a point where the set ℛ_x of recurrent directions for the geodesic flow through x satisfies |ℛ|_x >0. We also show that if there are no such points, L²-normalized quasimodes have sup-norms that are o(λ^(n-1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L^∞-norms that are Ω(λ^(n-1)/2).