论文标题
精致的Horoball计数和kleinian群体行动的共形度量
Refined horoball counting and conformal measure for Kleinian group actions
论文作者
论文摘要
抛物线固定点形成了一个至少一个抛物线元件的非元素几何有限的kleinian组的极限集的可数密度子集。给定这样的组，一个人可以将一组标准的成对分离型霍布尔关联到抛物线固定点的边界。可以将这种horoball的直径视为近似于相关抛物线点设置的限制的任意点的“反费用”。 Stratmann和Velani的结果允许一个人计算给定尺寸的Horoball，并且大致来说，对于小$ r> 0 $，有$ r^{ - δ} $大约$ r $的许多horoballs，其中$Δ$是该组的poincaré指数。我们调查了该结果的本地化，我们试图在给定的球$ b（z，r）$内计算大约$ r $的horoballs。大致说明，如果$ r \ lyssim r^2 $，那么我们获得了Stratmann-Velani结果的类似物（由$ B（Z，R）$的Patterson-Sullivan量度标准化。但是，对于$ r $的较大值，计数以微妙的方式取决于$ z $。我们的计数结果有多种应用，尤其是在极限集中支持的保形度量的几何形状上。例如，我们计算或估计$ s>δ$的某些$ s $符合量度的几个“分形维度”，并使用此操作来检查$ s $ con-grom-on-of-S-符合措施的连续性属性。
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 $s=δ$.
