## Duke Mathematical Journal

Published by Duke University Press since its inception in 1935, the Duke Mathematical Journal is one of the world's leading mathematical journals. DMJ emphasizes the most active and influential areas of current mathematics. Advance publication of articles online is available.

New criteria for ergodicity and nonuniform hyperbolicityVolume 160, Number 3 (2011)
On the Lagrangian structure of transport equations: The Vlasov–Poisson systemVolume 166, Number 18 (2017)
Birkhoff normal form for partial differential equations with tame modulusVolume 135, Number 3 (2006)
$\mathrm{G}_{2}$ -manifolds and associative submanifolds via semi-Fano $3$ -foldsVolume 164, Number 10 (2015)
On the zeros of $\zeta'(s)$ near the critical lineVolume 110, Number 3 (2001)
• ISSN: 0012-7094 (print), 1547-7398 (electronic)
• Publisher: Duke University Press
• Discipline(s): Mathematics
• Full text available in Euclid: 1935--
• Access: By subscription only
• Euclid URL: https://projecteuclid.org/dmj

### Featured bibliometrics

MR Citation Database MCQ (2016): 2.29
JCR (2016) Impact Factor: 2.171
JCR (2016) Five-year Impact Factor: 2.417
JCR (2016) Ranking: 10/310 (Mathematics)
Article Influence (2016): 3.852
Eigenfactor: Duke Mathematical Journal
SJR/SCImago Journal Rank (2016): 4.467

Indexed/Abstracted in: Current Contents: Physical, Chemical and Earth Sciences, IBZ Online, Magazines for Libraries, MathSciNet, Science Citation Index, Science Citation Index Expanded, Scopus, and zbMATH

### Featured article

#### Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Volume 166, Number 18 (2017)
##### Abstract

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.