Dolgopyat’s method and the fractal uncertainty principle

Dyatlov, Semyon; Jin, Long

doi:10.2140/apde.2018.11.1457

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Dolgopyat's method and the fractal uncertainty principle

Semyon Dyatlov and Long Jin

Vol. 11 (2018), No. 6, 1457–1485

DOI: 10.2140/apde.2018.11.1457

Abstract

We show a fractal uncertainty principle with exponent $\frac{1}{2} - δ + ε$ , $ε > 0$ , for Ahlfors–David regular subsets of $ℝ$ of dimension $δ \in (0, 1)$ . This is an improvement over the volume bound $\frac{1}{2} - δ$ , 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

Vol. 11, No. 6, 2018