In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations G is hyperbolic then the extremal conformal measures and the hyperbolic boundary of G coincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

中文翻译：

---

协紧 Fuchsian 群的正规子群的保形测度

在本文中，我们研究了协紧 Fuchsian 群的正规子群的保形测度。特别是，我们将极值保形测度与合适的 Ruelle 算子的特征测度联系起来。使用适用于 Ruelle 算子设置的 Ancona 定理，我们证明如果甲板变换组G是双曲线的，那么极值保形测量和双曲线边界G重合。然后，我们根据测地线切割序列在紧凑双曲线表面的规则覆盖上的渐近行为来解释这些结果。