Acta Mathematica

Volume 220 (2018)

Number 2

Semiclassical measures on hyperbolic surfaces have full support

Pages: 297 – 339



Semyon Dyatlov (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Ma., U.S.A.; and Department of Mathematics, University of California at Berkeley)

Long Jin (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China; and Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated in “Spectral gaps, additive energy, and a fractal uncertainty principle” [Dyatlov, S. & Zahl, J. Geom. Funct. Anal., 26 (2016), 1011–1094] and proved for porous sets in “Spectral gaps without the pressure condition” [Bourgain, J. & Dyatlov, S. Ann. of Math., 187 (2018), 825–867].

Received 26 May 2017

Accepted 29 June 2018

Published 16 August 2018