Measures with predetermined regularity and inhomogeneous self-similar sets
Measures with predetermined regularity and inhomogeneous self-similar sets
复制标题
具有预定规律和非齐次自相似集的测量
DOI：
10.4310/arkiv.2017.v55.n1.a8
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发表时间：
2016
影响因子：
0.9
通讯作者：
Juha Lehrback
中科院分区：
文献类型：
--
作者：
A. Kaenmaki;Juha Lehrback
我们证明了如果X是一个满足有限倍性的一致完美完备度量空间，那么就存在一个完全支持的低正则维测度，它尽可能地接近于X的低维。进一步证明了在缩聚开集条件下，非齐次自相似集EC的下维与缩聚集C的下维重合，而EC的共轭维数是相应自相似集E与缩聚集C的共轭维数的最大值。如果C的共轭维数严格小于E的共轭维数，则C的共轭维数小于E的共轭维数。则支撑在EC上的任何测度的上正则维数严格大于EC的下维数。令人惊讶的是，较低规则维度的相应语句失败了。
We show that if X is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC coincides with the lower dimension of the condensation set C, while the Assouad dimension of EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension of E, then the upper regularity dimension of any measure supported on EC is strictly larger than the Assouad dimension of EC . Surprisingly, the corresponding statement for the lower regularity dimension fails.