论文标题
一类随机Moran测度的Assouad样维度II-非均匀的Moran集
Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets
论文作者
论文摘要
在本文中，我们确定了在$ \ r^d $中支持的$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ \ r^d $ $ $的$φ$维度的几乎确定的值。本文概括了对均匀摩兰的随机度量进行的早期工作，将\ cite {hm}与不等缩放因子的情况下进行。 $φ$ dimensions是具有（准）Assouad尺寸的中级尺寸，而$θ$ -assouad Spectrum是特殊情况。 $ \dim_φμ$的几乎肯定的值表现出阈值现象，其中一个价值为``大''$φ$（以Quasi-Assouad Dimension为例，作为``大'''''''''''''''''''''''''''''''''我们提供了许多应用程序，包括固定尺度因子的位置和概率均匀分布的位置。基础随机集的几乎确定的$φ$维度也是我们结果的结果。
In this paper, we determine the almost sure values of the $Φ$-dimensions of random measures $μ$ supported on random Moran sets in $\R^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets \cite{HM} to the case of unequal scaling factors. The $Φ$-dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the $θ$-Assouad spectrum being special cases. The almost sure value of $\dim_Φμ$ exhibits a threshold phenomena, with one value for ``large'' $Φ$ (with the quasi-Assouad dimension as an example of a ``large'' dimension) and another for ``small'' $Φ$ (with the Assouad dimension as an example of a ``small'' dimension). We give many applications, including where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure $Φ$ dimension of the underlying random set is also a consequence of our results.
