We investigate how bounds of resonance counting functions for Schottky surfaces behave under transitions to covering surfaces of finite degree. We consider the classical resonance counting function asking for the number of resonances in large (and growing) disks centered at the origin of $\mathcal C$, as well as the (fractal) resonance counting function asking for the number of resonances in boxes near the axis of the critical exponent. For the former counting function we provide a transfer-operator-based proof that bounding constants can be chosen such that the transformation behavior under transition to covers is as for the Weyl law in the case of surfaces of finite area. For the latter counting function we deduce a bound in terms of the covering degree and the minimal length of a periodic geodesic on the covering surface. This yields an improved fractal Weyl upper bound. In the setting of Schottky surfaces, these estimates refine previous results due to Guillopé–Zworski and Guillopé–Lin–Zworski. When applied to principal congruence covers, these results yield new estimates for the resonance counting functions in the level aspect, which have recently been investigated by Jakobson–Naud. The techniques used in this article are based on the thermodynamic formalism for L-functions (twisted Selberg zeta functions), and twisted transfer operators.

中文翻译：

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肖特基表面盖的共振密度

我们研究了肖特基表面的共振计数函数的边界在过渡到有限度的覆盖表面时的行为。我们考虑经典的共振计数函数，该函数要求以$ \ mathcal C $的原点为中心的大（且正在增长）的磁盘中的共振数，以及（分形）共振计数函数，该函数要求附近的盒子中的共振数关键指数的轴。对于前一个计数函数，我们提供了一个基于转移运算符的证明，可以选择有界常数，以使在转换为覆盖范围时的转换行为与韦尔定律在有限面积曲面的情况下一样。对于后一个计数函数，我们根据覆盖度和覆盖面上的周期性测地线的最小长度得出一个边界。这产生了改进的分形Weyl上限。在Schottky曲面的设置中，由于Guillopé–Zworski和Guillopé–Lin–Zworski的缘故，这些估计值完善了先前的结果。将这些结果应用于主同余覆盖时，这些结果将为级别上的共振计数函数提供新的估计，Jakobson–Naud最近对其进行了研究。本文中使用的技术基于L函数的热力学形式（扭曲的Selberg zeta函数）和扭曲的传递算符。Jakobson-Naud最近对其进行了调查。本文中使用的技术基于L函数的热力学形式（扭曲的Selberg zeta函数）和扭曲的传递算符。Jakobson-Naud最近对其进行了调查。本文中使用的技术基于L函数的热力学形式（扭曲的Selberg zeta函数）和扭曲的传递算符。