Fractal uncertainty for transfer operators

Abstract

We show directly that the fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] implies that there exists $\sigma >0$ for which the Selberg zeta function for a convex co-compact hyperbolic surface has only finitely many zeros with $\Re s\ge \frac{1}{2}-\sigma$. That eliminates advanced microlocal techniques of Dyatlov-Zahl [arXiv:1504.06589] though we stress that these techniques are still needed for resolvent bounds and for possible generalizations to the case of non-constant curvature.

25 pages, 5 figures; minor revisions. To appear in IMRN

References in corpus (3)

• Spectral gaps without the pressure condition
• Fourier dimension and spectral gaps for hyperbolic surfaces
• Dolgopyat's method and the fractal uncertainty principle

Cited by in corpus (5)

• An introduction to fractal uncertainty principle
• Spectral gap for obstacle scattering in dimension 2
• Fractal uncertainty principle for discrete Cantor sets with random alphabets
• Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces
• Zeros of dynamical zeta functions for hyperbolic quadradic maps