Hausdorff dimensions of the divergence points of self-similar measures with the open set condition
Hausdorff dimensions of the divergence points of self-similar measures with the open set condition
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DOI：
10.1088/0951-7715/25/1/93
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发表时间：
2011
期刊：
影响因子：
1.7
通讯作者：
Jinjun Li;Min Wu;Ying Xiong
中科院分区：
文献类型：
--
作者：
Jinjun Li;Min Wu;Ying Xiong
设 μ 为具有开集条件的自相似集 K 上支持的自相似测度。对于 x ∊ K，设 A(D(x)) 表示 r ↘ 0 的累加点集合。在本文中，我们证明集合 A(D(x)) 对于任何 x ∊ K 来说要么是单例，要么是闭子区间，并且对于任何闭子区间，都确定了点集 x 的豪斯多夫维数，其中集合 A(D(x)) 等于 I。我们的主要结果解决了由Olsen 和 Winter (2003 J. Lond. Math. Soc. 67 103–22) 积极地概括了 Arbeiter 和 Patzschke (1996 Math. Nachr. 181 5–42) 的经典结果。
Let μ be the self-similar measure supported on the self-similar set K with the open set condition. For x ∊ K, let A(D(x)) denote the set of accumulation points of as r ↘ 0. In this paper, we show that the set A(D(x)) is either a singleton or a closed subinterval of for any x ∊ K, and for any closed subinterval determines the Hausdorff dimension of the set of points x for which the set A(D(x)) equals I. Our main result solves the conjecture posed by Olsen and Winter (2003 J. Lond. Math. Soc. 67 103–22) positively and generalizes the classical result of Arbeiter and Patzschke (1996 Math. Nachr. 181 5–42).