Kleinian群作用的精细horoball计数与共形测度
Refined horoball counting and conformal measure for Kleinian group actions
AI总结：
该研究针对非初等几何有限Kleinian群，推广了Stratmann-Velani的horoball计数结果到局部球内，并应用于共形测度的分形维数计算与连续性分析。
摘要：
抛物不动点在具有至少一个抛物元素的非初等几何有限Kleinian群的极限集中形成可数稠密子集。给定这样的群，可以关联一组标准的两两不相交的horoball，每个horoball在抛物不动点处与边界相切。这样的horoball的直径可以被视为通过相关抛物点逼近极限集中任意点的“逆代价”。Stratmann和Velani的一个结果允许人们计数给定大小的horoball，粗略地说，对于小的$r>0$，存在大约$r^{-δ}$个大小约为$r$的horoball，其中$δ$是群的Poincaré指数。我们研究该结果的局部化，即试图在给定球$B(z,R)$内计数大小约为$r$的horoball。粗略地说，如果$r \lesssim R^2$，那么我们得到Stratmann-Velani结果的类似物（通过$B(z,R)$的Patterson-Sullivan测度归一化）。然而，对于更大的$r$值，计数以微妙的方式依赖于$z$。我们的计数结果有多个应用，特别是对支撑在极限集上的共形测度的几何。例如，我们计算或估计某些$s>δ$的$s$-共形测度的若干“分形维数”，并利用这一点来研究$s$-共形测度在$s=δ$处的连续性。
英文摘要：
Parabolic fixed points form a countable dense subset of the limit set of a non-elementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the `inverse cost' of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small $r>0$ there are $r^{-δ}$ many horoballs of size approximately $r$, where $δ$ is the Poincaré exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately $r$ inside a given ball $B(z,R)$. Roughly speaking, if $r \lesssim R^2$, then we obtain an analogue of the Stratmann-Velani result (normalised by the Patterson-Sullivan measure of $B(z,R)$). However, for larger values of $r$, the count depends in a subtle way on $z$. Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several `fractal dimensions' of certain $s$-conformal measures for $s>δ$ and use this to examine continuity properties of $s$-conformal measures at $s=δ$.
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