  This article contains a generalization of the authors' results on numbers of
nodal points of eigenfunctions on "good curves" in analytic plane domains
(arXiv:0710.0101). The term `good' means that the $L^2$ norms of restrictions
of eigenfunctions of eigenvalue $\lambda^2$ to the curve are bounded below by
$e^{- C \lambda}$. In this article, the result is generalized to all real
analytic Riemannian manifolds $(M, g)$ of any dimension $m$ without boundary.
Moreover, a similar lower bound is given for the Hausdorff $m-2$ measure of the
intersection of the nodal set with a good real analytic hypersurface.
  Most of the article is devoted to giving a dynamical or geometric control
condition for `goodness' of a hypersurface. The conditions are that the
hypersurface $H$ be asymmetric with respect to geodesics and that the flowout
of the unit vectors with footpoint on $H$ have full measure in $S^*M. $ This
gives a partial answer to a question of Bourgain-Rudnick of characterizing
hypersurfaces $H$ on which a sequence of eigenfunctions vanishes. We show that
under our conditions, a positive density sequence cannot vanish on $H$ or even
have smaller $L^2$ norms than $e^{- C \lambda}$

Analysis of PDEs