  On any compact Riemannian manifold $(M, g)$ of dimension $n$, the
$L^2$-normalized eigenfunctions ${\phi_{\lambda}}$ satisfy
$||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta
\phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of
all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard
$n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds,
e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, 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 set ${\mathcal L}_x$ of geodesic loops at $x$ has positive
measure in $S^*_x M$. We strengthen this result here by showing that such a
manifold must have a point where the set ${\mathcal R}_x$ of recurrent
directions for the geodesic flow through x satisfies $|{\mathcal R}_x|>0$. We
also show that if there are no such points, $L^2$-normalized quasimodes have
sup-norms that are $o(\lambda^{n-1)/2})$, 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 quasi-modes with
$L^\infty$-norms that are $\Omega(\lambda^{(n-1)/2})$.
