Vol. 11, No. 6, 2018

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Dolgopyat's method and the fractal uncertainty principle

Semyon Dyatlov and Long Jin

Vol. 11 (2018), No. 6, 1457–1485
Abstract

We show a fractal uncertainty principle with exponent 1 2 δ + ε, ε > 0, for Ahlfors–David regular subsets of of dimension δ (0,1). This is an improvement over the volume bound 1 2 δ, and ε is estimated explicitly in terms of the regularity constant of the set. The proof uses a version of techniques originating in the works of Dolgopyat, Naud, and Stoyanov on spectral radii of transfer operators. Here the group invariance of the set is replaced by its fractal structure. As an application, we quantify the result of Naud on spectral gaps for convex cocompact hyperbolic surfaces and obtain a new spectral gap for open quantum baker maps.

Keywords
resonances, fractal uncertainty principle
Mathematical Subject Classification 2010
Primary: 28A80, 35B34, 81Q50
Milestones
Received: 23 February 2017
Revised: 26 October 2017
Accepted: 12 January 2018
Published: 3 May 2018
Authors
Semyon Dyatlov
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Long Jin
Department of Mathematics
Purdue University
West Lafayette, IN
United States