Semiclassical Measures for Complex Hyperbolic Quotients
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Author(s)
Athreya, Jayadev
Dyatlov, Semyon
Miller, Nicholas
Date Issued
August 28, 2025
Journal
Geometric and Functional Analysis
Publisher
Springer International Publishing
Citation
Athreya, J., Dyatlov, S. & Miller, N. Semiclassical Measures for Complex Hyperbolic Quotients. Geom. Funct. Anal. 35, 979–1050 (2025).
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-ShareAlike
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DOI of Published Version
https://doi.org/10.1007/s00039-025-00717-x
