Semiclassical measures for complex hyperbolic quotients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908508844195840 |
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| author | Athreya, Jayadev Dyatlov, Semyon Miller, Nicholas |
| author_facet | Athreya, Jayadev Dyatlov, Semyon Miller, Nicholas |
| contents | We study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold.
The proof uses the one-dimensional fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] along the fast expanding/contracting directions, in a way similar to the work of Dyatlov-Jézéquel [arXiv:2108.10463] in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06477 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semiclassical measures for complex hyperbolic quotients Athreya, Jayadev Dyatlov, Semyon Miller, Nicholas Analysis of PDEs Dynamical Systems Geometric Topology Spectral Theory We study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold. The proof uses the one-dimensional fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] along the fast expanding/contracting directions, in a way similar to the work of Dyatlov-Jézéquel [arXiv:2108.10463] in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory. |
| title | Semiclassical measures for complex hyperbolic quotients |
| topic | Analysis of PDEs Dynamical Systems Geometric Topology Spectral Theory |
| url | https://arxiv.org/abs/2402.06477 |