Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets
Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets
复制标题
DOI：
10.1090/s0002-9939-01-05969-x
复制
发表时间：
2001-02
期刊：
影响因子：
--
通讯作者：
Y. Peres;M. Rams;K. Simon;B. Solomyak
中科院分区：
文献类型：
--
作者：
Y. Peres;M. Rams;K. Simon;B. Solomyak
如果紧集 K 是其图像通过保形收缩的有限并集，则它是自共形的。众所周知，如果保形收缩满足“开集条件”（OSC），则 K 具有正维 Hausdorff 测度，其中 s 是 Bowen 压力方程的解。我们证明 OSC、强 OSC 和 s 维 Hausdorff 测度的正性对于保形收缩是等效的；这回答了 R. D. Mauldin 的问题。在自相似情况下，当收缩是线性时，这种等价性由 Schief (1994) 证明，他使用了 Bandt 和 Graf (1992) 的结果，但这些论文中的证明并没有扩展到非线性设置。
A compact set K is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the "open set condition" (OSC), then K has positive sdimensional Hausdorff measure, where s is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the s-dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting.