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February 2014 Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces
Steve Zelditch
J. Differential Geom. 96(2): 305-351 (February 2014). DOI: 10.4310/jdg/1393424920

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 $\lambda_j$.

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Steve Zelditch. "Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces." J. Differential Geom. 96 (2) 305 - 351, February 2014. https://doi.org/10.4310/jdg/1393424920

Information

Published: February 2014
First available in Project Euclid: 26 February 2014

zbMATH: 1303.32002
MathSciNet: MR3178442
Digital Object Identifier: 10.4310/jdg/1393424920

Rights: Copyright © 2014 Lehigh University

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Vol.96 • No. 2 • February 2014
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