Journal of the European Mathematical Society


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Published online first: 2019-02-01
DOI: 10.4171/JEMS/867

Improved fractal Weyl bounds for hyperbolic manifolds (with an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich)

Semyon Dyatlov[1]

(1) Massachusetts Institute of Technology, Cambridge, USA

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $\delta$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-\beta,0]$ is $\mathcal O(R^{m(\beta,\delta)+})$, where the exponent $m(\beta,\delta)=\mathrm {min}(2\delta+2\beta+1-n,\delta)$ changes its behavior at $\beta={n-1\over 2}-{\delta\over 2}$. In the case $\delta<{n-1\over 2}$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\Im\lambda > \delta-{n-1\over 2}\}$. Both results use the fractal uncertainty principle point of view recently introduced in [DyZa]. The appendix presents numerical evidence for the Weyl upper bound.

Keywords: Resonances, hyperbolic quotients, fractal Weyl law

Dyatlov Semyon: Improved fractal Weyl bounds for hyperbolic manifolds (with an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich). J. Eur. Math. Soc. Electronically published on February 1, 2019. doi: 10.4171/JEMS/867 (to appear in print)