Journal of Differential Geometry

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

Steve Zelditch

Full-text: Open access

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

Permanent link to this document
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


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