Harmonicity of Quasiconformal Measures and Poisson Boundaries of Hyperbolic Spaces
Harmonicity of Quasiconformal Measures and Poisson Boundaries of Hyperbolic Spaces
复制标题
双曲空间的拟共形测度与泊松边界的调和性
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
R. Muchnik
中科院分区：
文献类型：
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作者：
C. Connell;R. Muchnik
摘要：我们考虑作用在真（不一定是测地的）δ-双曲空间X上的等距群Γ.对于任意连续的α-拟共形测度ν，将满测度赋给Γ的径向极限集Λ r，我们在Γ上产生一个（非平凡的）测度μ，其中ν是平稳的。这意味着极限集合与ν一起形成μ边界，并且ν关于由μ诱导的随机游走是调和的。作为一个基本的例子，
$$X ={mathbb {H}}^n $$
和Γ是任何几何有限的Kleinian群，其中ν是Γ的Patterson-Sullivan测度。当X是CAT（− 1）空间且Γ是离散的且具有拟凸作用时，我们证明了（Λ r，ν）是μ的Poisson边界.在证明过程中，我们建立了一般度量测度空间上一致正的下连续函数空间的一组连续函数在L~1或L~∞范数下构成正基的充分条件。
Abstract.We consider a group Γ of isometries acting on a proper (not necessarily geodesic) δ -hyperbolic space X. For any continuous α-quasiconformal measure ν on ∂X assigning full measure to Λr, the radial limit set of Γ, we produce a (nontrivial) measure μ on Γ for which ν is stationary. This means that the limit set together with ν forms a μ-boundary and ν is harmonic with respect to the random walk induced by μ. As a basic example, take
$$X = {mathbb{H}}^n$$
and Γ to be any geometrically finite Kleinian group with ν a Patterson-Sullivan measure for Γ. In the case when X is a CAT(−1) space and Γ is discrete with quasiconvex action, we show that (Λr, ν) is the Poisson boundary for μ. In the course of the proofs, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L1 or L∞ norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space.