具有预定正则性的测度与非齐次自相似集
Measures with predetermined regularity and inhomogeneous self-similar sets
摘要：
证明了如果$X$是一个满足有限倍性质的一致完美完全度量空间，则存在一个完全支撑测度，其低正则维数尽可能接近$X$的低维数.进一步证明了在凝聚开集条件下，非齐次自相似集$E_C$的低维与凝聚集$C$的低维一致，而$E_C$的Assouad维数是对应的自相似集$E$和凝聚集$C$的Assouad维数的最大值.如果$C$的Assouad维数严格小于$E$的Assouad维数，则在$E_C$上支持的任何测度的上正则维数严格大于$E_C$的Assouad维数。令人惊讶的是，较低的正则性维度的相应陈述失败了。
英文摘要：
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 $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ 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 $E_C$ is strictly larger than the Assouad dimension of $E_C$. Surprisingly, the corresponding statement for the lower regularity dimension fails.
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