Duke Mathematical Journal

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Etienne Le Masson and Tuomas Sahlsten

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We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and Le Masson. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Le Masson, and Lindenstrauss on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalized averaging operators over disks, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

Article information

Duke Math. J., Volume 166, Number 18 (2017), 3425-3460.

Received: 25 April 2017
Revised: 15 May 2017
First available in Project Euclid: 13 October 2017

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Primary: 81Q50: Quantum chaos [See also 37Dxx] 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 11F72: Spectral theory; Selberg trace formula

quantum chaos quantum ergodicity Benjamini–Schramm convergence short geodesics hyperbolic dynamics eigenfunctions of the Laplacian Selberg transform rate of mixing mean ergodic theorem


Le Masson, Etienne; Sahlsten, Tuomas. Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces. Duke Math. J. 166 (2017), no. 18, 3425--3460. doi:10.1215/00127094-2017-0027. https://projecteuclid.org/euclid.dmj/1507860019

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