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1 December 2017 Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces
Etienne Le Masson, Tuomas Sahlsten
Duke Math. J. 166(18): 3425-3460 (1 December 2017). DOI: 10.1215/00127094-2017-0027


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.


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Etienne Le Masson. Tuomas Sahlsten. "Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces." Duke Math. J. 166 (18) 3425 - 3460, 1 December 2017.


Received: 25 April 2017; Revised: 15 May 2017; Published: 1 December 2017
First available in Project Euclid: 13 October 2017

zbMATH: 1384.37035
MathSciNet: MR3732880
Digital Object Identifier: 10.1215/00127094-2017-0027

Primary: 11F72 , 37D40 , 81Q50

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

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 18 • 1 December 2017
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