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<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, <CitationRef CitationID="CR3">2018</CitationRef>) along the fast expanding/contracting directions, in a way similar to the work of Dyatlov–Jézéquel (Ann. Henri Poincaré, <CitationRef CitationID="CR8">2023</CitationRef>) 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>

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Semiclassical Measures for Complex Hyperbolic Quotients

  • 期刊论文

  • 发表日期：

    2025年8月28日

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  • Jayadev Athreya,
  • Semyon Dyatlov,
  • Nicholas Miller

摘要

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.