Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets
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arXiv
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| Format: | Preprint |
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2022
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| author | Hare, Kathryn E. Mendivil, Franklin |
| author_facet | Hare, Kathryn E. Mendivil, Franklin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_14654 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets Hare, Kathryn E. Mendivil, Franklin Classical Analysis and ODEs Probability Primary: 28A80, Secondary 28C15, 60G57 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. |
| title | Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets |
| topic | Classical Analysis and ODEs Probability Primary: 28A80, Secondary 28C15, 60G57 |
| url | https://arxiv.org/abs/2207.14654 |