Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces

Abstract

We consider the intersections of the complex nodal set $\mathcal{N}_{\lambda_{j}}^{\,\mathbb{C}}$ of the analytic continuation of an eigenfunction of $\Delta$ on a real analytic surface $(M^2, g)$ with the complexification of a geodesic $\gamma$. We prove that if the geodesic flow is ergodic and if $\gamma$ is periodic and satisfies a generic asymmetry condition, then the intersection points $\mathcal{N}_{\lambda_{j}}^{\,\mathbb{C}} \cap \gamma_{x, \xi}^{\mathbb{C}}$ condense along the real geodesic and become uniformly distributed with respect to its arc-length. We prove an analogous result for non-periodic geodesics except that the ‘origin’ $\gamma_{x, \xi}(0)$ is allowed to move with $\lambda_j$.