In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris 311 (1990), 301–306; Teichmüller spaces and BMOA. Math. Ann. 289 (1991), 613–625] by proving that all convergence-type Fuchsian groups of the first kind fail to have Ruelle’s property. Finally, we give some results about second-kind Fuchsian groups. [-3.2pc]

在本文中，我们研究了 Ruelle 属性的有效性范围。首先，我们证明了每个有限生成的 Fuchsian 群都具有 Ruelle 的性质。我们还证明了满足 Ruelle 属性的无限生成 Fuchsian 群的存在。关于负面结果，我们首先推广 Astala 和 Zinsmeister 的结果 [Mostow 刚性和 Fuchsian 群。CR 数学。学院派。科学。巴黎 311 (1990), 301–306; Teichmüller 空间和 BMOA。数学。安。 289 (1991), 613-625] 通过证明所有第一类收敛型 Fuchsian 群都不具有 Ruelle 的性质。最后，我们给出了关于第二类 Fuchsian 群的一些结果。[-3.2pc]