{"title":"复双曲商的半经典测度","authors":"Jayadev Athreya, Semyon Dyatlov, Nicholas Miller","doi":"10.1007/s00039-025-00717-x","DOIUrl":null,"url":null,"abstract":"<p>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.</p><p>The proof uses the one-dimensional fractal uncertainty principle of Bourgain–Dyatlov (Ann. Math. (2) 187(3):825–867, 2018) along the fast expanding/contracting directions, in a way similar to the work of Dyatlov–Jézéquel (Ann. Henri Poincaré, 2023) in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Semiclassical Measures for Complex Hyperbolic Quotients\",\"authors\":\"Jayadev Athreya, Semyon Dyatlov, Nicholas Miller\",\"doi\":\"10.1007/s00039-025-00717-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>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.</p><p>The proof uses the one-dimensional fractal uncertainty principle of Bourgain–Dyatlov (Ann. Math. (2) 187(3):825–867, 2018) along the fast expanding/contracting directions, in a way similar to the work of Dyatlov–Jézéquel (Ann. Henri Poincaré, 2023) in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.</p>\",\"PeriodicalId\":12478,\"journal\":{\"name\":\"Geometric and Functional Analysis\",\"volume\":\"27 1\",\"pages\":\"\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2025-08-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Geometric and Functional Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00039-025-00717-x\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometric and Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00039-025-00717-x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Semiclassical Measures for Complex Hyperbolic Quotients
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 (Ann. Math. (2) 187(3):825–867, 2018) along the fast expanding/contracting directions, in a way similar to the work of Dyatlov–Jézéquel (Ann. Henri Poincaré, 2023) in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.
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