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**Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces.**
*(English)*
Zbl 1303.32002

This paper is part of a program of which the ultimate objective is to establish that the nodal sets of the eigenfunctions \(\varphi_j\) of the Laplacian on a real-analytic Riemannian surface, with ergodic geodesic flow on the tangent bundle, become equidistributed, in an appropriate sense, as \(j\) tends to infinity.

In this paper, the main result is that “in the complex domain, as the eigenvalue goes to \(\infty\), the intersections of nodal sets with generic periodic and non-periodic geodesics condense along the underlying real geodesic and become uniformly distributed with respect to its arclength,” along a subsequence \((\lambda_{j_k})\) of eigenvalues of density one. The results involve an asymmetry condition spelled out in an appendix, in which material from J. A. Toth and S. Zelditch [Geom. Funct. Anal. 23, No. 2, 715–775 (2013; Zbl 1277.53088)] is recalled. The second result “is somewhat weaker due to the non-compactness of non-periodic geodesics.” The proofs rely on the calculus of Toeplitz operators developed by Boutet de Monvel and Guillemin.

In this paper, the main result is that “in the complex domain, as the eigenvalue goes to \(\infty\), the intersections of nodal sets with generic periodic and non-periodic geodesics condense along the underlying real geodesic and become uniformly distributed with respect to its arclength,” along a subsequence \((\lambda_{j_k})\) of eigenvalues of density one. The results involve an asymmetry condition spelled out in an appendix, in which material from J. A. Toth and S. Zelditch [Geom. Funct. Anal. 23, No. 2, 715–775 (2013; Zbl 1277.53088)] is recalled. The second result “is somewhat weaker due to the non-compactness of non-periodic geodesics.” The proofs rely on the calculus of Toeplitz operators developed by Boutet de Monvel and Guillemin.

Reviewer: Satyanad Kichenassamy (Reims)

### MSC:

32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |

58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |