The limit set of a Fuchsian group
The limit set of a Fuchsian group
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DOI：
10.1007/bf02392046
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发表时间：
1976-12
期刊：
影响因子：
3.7
通讯作者：
S. Patterson
中科院分区：
文献类型：
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作者：
S. Patterson
在本文中，我们将考虑第二类有限生成的 Fuchsian 群的极限集。特别是，我们将尝试计算极限集的豪斯多夫维数。如果群 G 没有抛物线元素，这实际上是在第 4 节中实现的，而如果 G 具有抛物线元素，我们可以获得第 5 节中讨论的部分结果。这些结果的证明涉及构造极限集上支持的测度，至少在原则上，从中我们可以获得 Hausdorff 维数的下界。该界限与 Beardon [4] 发现的上限相同。我们构造的实际测度被证明与 G\H 上的拉普拉斯算子理论密切相关，因此我们对这两者都获得了新的见解。例如，这使我们能够给出 Beardon 定理的新证明 [4]（参见定理 7.1 的推论）。
In this paper we shall consider the limit set of a finitely generated Fuchsian group of the second kind. In particular, we shall attempt to calculate the Hausdorff dimension of the limit set. If the group, G, has no parabolic elements this is actually achieved in section 4, whereas, if G has parabolic elements we can obtain partial results which are discussed in section 5. The proof of these results involves the construction of a measure supported on the limit set, from which, at least in principle, we can obtain a lower bound for the Hausdorff dimension. This bound is the same as the upper bound found by Beardon [4]. The actual measure which we construct proves to be intimately related to the theory of the Laplace operator on G\H, and consequently we obtain new insights into both of these. This allows us, for example, to give a new proof of a theorem of Beardon [4](see the Corollary to Theorem 7.1).