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

Travis Cunningham

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.

Improved fractal Weyl bounds matching improved spectral gaps for hyperbolic surfaces and open quantum 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.
Paper Structure (8 sections, 12 theorems, 134 equations, 2 figures)

This paper contains 8 sections, 12 theorems, 134 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Main results.
  3. Further context and overview.
  4. PRELIMINARIES
  5. PROOFS OF THEOREMS 1.1 AND 1.2
  6. Improved fractal Weyl bound matching the improved spectral gap.
  7. Improved resolvent bound in the improved spectral gap.
  8. PROOF OF THEOREM 1.3

Key Result

Theorem 1.1

Let $M$ be a convex co-compact hyperbolic surface with $\delta\in(0,1)$, and let $\beta_{BD}$ be the size of the spectral gap proved in BD17. Then for each $\nu>0$, $\varepsilon>0$, there exists $C>0$ such that

Figures (2)

  • Figure 1: (a) Plot of $m(\nu, \delta)$ in Theorem 1.1 for a $\delta<\frac{1}{2}$. The dashed line is the previous bound of Dya19b. (b) Plot of $c(\nu, \delta)$ in Theorem 1.2, again for a $\delta<\frac{1}{2}$. The solid line is the upper resolvent bound $c=2 \nu$ of DZ16, BD17. The dashed curve is the previous upper bound of Dya19b.
  • Figure 2: Plot of $m(\nu, \delta)$ from Theorem 1.3 when $(a)\; \delta<\frac{1}{2}$ and $\beta>\beta_E$, and (b) $\delta>\frac{1}{2},\left|\delta-\frac{1}{2}\right|$ small so that $\beta_E>\beta$. The dashed line in both is the previous bound of DJ17.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 12 more