## Journal of Differential Geometry

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

Steve Zelditch

#### 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 $\lambda_j$.

#### Article information

Source
J. Differential Geom., Volume 96, Number 2 (2014), 305-351.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.jdg/1393424920

Digital Object Identifier
doi:10.4310/jdg/1393424920

Mathematical Reviews number (MathSciNet)
MR3178442

Zentralblatt MATH identifier
1303.32002

#### Citation

Zelditch, Steve. Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces. J. Differential Geom. 96 (2014), no. 2, 305--351. doi:10.4310/jdg/1393424920. https://projecteuclid.org/euclid.jdg/1393424920