Improved fractal Weyl bounds matching improved spectral gaps for hyperbolic surfaces and open quantum maps

TL;DR

This paper establishes improved fractal Weyl bounds for hyperbolic surfaces and open quantum maps, matching enhanced spectral gaps through refined determinant estimates and fractal uncertainty principles.

Contribution

It introduces a new determinant function approach that yields sharper fractal Weyl bounds and spectral gap matches for hyperbolic surfaces and quantum maps.

Findings

Matching spectral gaps with improved fractal Weyl bounds

Enhanced resolvent estimates for hyperbolic surfaces

Refined bounds for open quantum baker's maps

Abstract

We prove a new fractal Weyl upper bound for the high-energy distribution of resonances of convex co-compact hyperbolic surfaces which matches the improved spectral gap given by Fourier decay. This improves upon the fractal Weyl bound of Dyatlov which matches the Patterson-Sullivan spectral gap. We also give a new resolvent estimate improving the ones given by Dyatlov-Zahl and Dyatlov. Analogous results are obtained for quantum open baker's maps, improving an estimate of Dyatlov-Jin, where we also give an improved fractal Weyl bound matching a spectral gap given by additive energy estimates. We refine known methods for proving fractal Weyl bounds which reduce the problem to an estimate of a certain determinant function; however, we use a different determinant function which allows us to make sharper estimates by applying the methods of proof of the fractal uncertainty principle in each setting.