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We consider the the intersections of the complex nodal set of the analytic continuation of an eigenfunction of the Laplacian on a real analytic surface with the complexification of a geodesic. We prove that if the geodesic flow is ergodic and if the geodesic is periodic and satisfies a generic asymmetry condition, then the intersection points 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' is allowed to move with $\lambda_j$. The proof uses the quantum ergodic restriction theorem due to J. Toth and the author (see also Dyatlov-Zworski).

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