In this paper, we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp-winding number. We completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalize previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph directed Markov systems.

中文翻译：

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双曲曲面上尖角绕组的多重分形分析

在本文中，我们研究了一个有限生成的自由 Fuchsian 群的极限集关于平均 cusp-winding 数的多重分形分解。我们通过一定的自由能函数完全确定了它的多重分形谱，并证明由具有相同标度指数的极限点组成的集合的豪斯多夫维数与该自由能函数的勒让德变换相吻合。作为副产品，我们将先前获得的关于无限迭代函数系统的多重分形形式的结果推广到无限图有向马尔可夫系统的设置。