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arXiv · 2105.13927

Assouad-like dimensions of random Moran measures

Kathryn E. Hare·Franklin Mendivil
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Abstract

In this paper, we determine the almost sure values of the $\Phi $-dimensions of random measures supported on random Moran sets that satisfy a uniform separation condition. The $\Phi $-dimensions are intermediate Assouad-like dimensions, the (quasi-)Assouad dimensions and $\theta $-Assouad spectrum being special cases. Their values depend on the size of $\Phi $, with one size coinciding with the Assouad dimension and the other coinciding with the quasi-Assouad dimension. We give many applications, including to equicontractive self-similar measures and $1$-variable random Moran measures such as Cantor-like measures with probabilities that are uniformly distributed. We can also deduce the $\Phi $-dimensions of the underlying random sets.

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Kathryn E. Hare, Franklin Mendivil. 2021-05-28. Assouad-like dimensions of random Moran measures. https://arxiv.org/abs/2105.13927

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