论文标题
相对双曲线的谐波度量的Hausdorff尺寸
The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups
论文作者
论文摘要
本文研究了相对双曲线群的各个边界上的谐波测量的Hausdorff维度，这些范围与有限的概率度量驱动的随机步行相关。关于弗洛伊德度量标准和快捷方式度量，我们证明了谐波测量的Hausdorff尺寸等于熵的比率和随机行走的漂移。 如果该组是无限限制的，则可以为带有视觉度量的端边界获得相同的维度公式。另外，视觉度量标准的Hausdorff尺寸与单词度量的生长速率一起识别。这些结果是通过对无限端访问组的双倍视觉指标的表征来补充的：末端边界上的视觉指标在且仅当组几乎是免费时才加倍。因此，在末端边界上至少有两个不同的Bi-Hölder类（因此是准对称类别）。
The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks driven by a probability measure with finite first moment. With respect to the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension of the harmonic measure equals the ratio of the entropy and the drift of the random walk. If the group is infinitely-ended, the same dimension formula is obtained for the end boundary endowed with a visual metric. In addition, the Hausdorff dimension of the visual metric is identified with the growth rate of the word metric. These results are complemented by a characterization of doubling visual metrics for accessible infinitely-ended groups : the visual metrics on the end boundary is doubling if and only if the group is virtually free. Consequently, there are at least two different bi-Hölder classes (and thus quasi-symmetric classes) of visual metrics on the end boundary.
