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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2308.04569 |
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| _version_ | 1866916803854204928 |
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| author | Allaart, Pieter Jones, Taylor |
| author_facet | Allaart, Pieter Jones, Taylor |
| contents | We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree, where $M\geq 2$. The Cantor set $C$ is the attractor of an equicontractive iterated function system $\{f_1,\dots,f_N\}$ that satisfies the open set condition with $(0,1)$ as the open set. For a fixed probability vector $(p_1,\dots,p_N)$, each edge in the infinite $M$-ary tree is independently labeled $i$ with probability $p_i$, for $i=1,2,\dots,N$. Thus, each infinite path in the tree receives a random label sequence of numbers from $\{1,2,\dots,N\}$. We define $F$ to be the (random) set of those points $x\in C$ which have a coding that is equal to the label sequence of some infinite path starting at the root of the tree.
The set $F$ may be viewed as a statistically self-similar set with extreme overlaps, and as such, its Hausdorff and box-counting dimensions coincide. We prove non-trivial upper and lower bounds for this dimension, and obtain the exact dimension in a few special cases. For instance, when $M=N$ and $p_i=1/N$ for each $i$, we show that $F$ is almost surely of full Hausdorff dimension in $C$ but of zero Hausdorff measure in its dimension.
For the case of two maps and a binary tree, we also consider deterministic labelings of the tree where, for a fixed integer $m\geq 2$, every $m$th edge is labeled $1$, and compute the exact Hausdorff dimension of the resulting subset of $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04569 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Random subsets of Cantor sets generated by trees of coin flips Allaart, Pieter Jones, Taylor Probability Classical Analysis and ODEs 28A78 We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree, where $M\geq 2$. The Cantor set $C$ is the attractor of an equicontractive iterated function system $\{f_1,\dots,f_N\}$ that satisfies the open set condition with $(0,1)$ as the open set. For a fixed probability vector $(p_1,\dots,p_N)$, each edge in the infinite $M$-ary tree is independently labeled $i$ with probability $p_i$, for $i=1,2,\dots,N$. Thus, each infinite path in the tree receives a random label sequence of numbers from $\{1,2,\dots,N\}$. We define $F$ to be the (random) set of those points $x\in C$ which have a coding that is equal to the label sequence of some infinite path starting at the root of the tree. The set $F$ may be viewed as a statistically self-similar set with extreme overlaps, and as such, its Hausdorff and box-counting dimensions coincide. We prove non-trivial upper and lower bounds for this dimension, and obtain the exact dimension in a few special cases. For instance, when $M=N$ and $p_i=1/N$ for each $i$, we show that $F$ is almost surely of full Hausdorff dimension in $C$ but of zero Hausdorff measure in its dimension. For the case of two maps and a binary tree, we also consider deterministic labelings of the tree where, for a fixed integer $m\geq 2$, every $m$th edge is labeled $1$, and compute the exact Hausdorff dimension of the resulting subset of $C$. |
| title | Random subsets of Cantor sets generated by trees of coin flips |
| topic | Probability Classical Analysis and ODEs 28A78 |
| url | https://arxiv.org/abs/2308.04569 |