Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups
Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups
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豪斯多夫维数与共形动力学 I：克莱尼群的强收敛性
DOI：
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发表时间：
1999
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通讯作者：
C. McMullen
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作者：
C. McMullen
研究了Kleinian群序列Γn → Γ的极限集Λn和Λ的Hausdorff维数的性质，其中M = H/Γ是几何有限的.我们证明了如果Γn → Γ是强的，则：（a）Mn = H3/Γn是几何有限的，（B）Λn → Λ是Hausdorff拓扑，（c）H. dim（Λn）→ H. dim（Λ），如果H. dim（Λ）≥ 1。另一方面，我们给出了一些例子，证明了当H。dim（Λ）< 1。连续性可以通过要求偶然抛物线径向收敛来恢复。类似的结果也适用于高维流形。应用程序给出拟fuchsian群及其极限。
This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H/Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3/Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H. dim(Λ) < 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higher-dimensional manifolds. Applications are given to quasifuchsian groups and their limits.