Fractal percolation on statistically self-affine carpets
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916387294806016 |
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| author | Falconer, Kenneth Feng, Tianyi |
| author_facet | Falconer, Kenneth Feng, Tianyi |
| contents | We consider a random self-affine carpet $F$ based on an $n\times m$ subdivision of rectangles and a probability $0<p<1$. Starting by dividing $[0,1]^2$ into an $n\times m$ grid of rectangles and selecting these independently with probability $p$, we then divide the selected rectangles into $n\times m$ subrectangles which are again selected with probability $p$; we continue in this way to obtain a statistically self-affine set $F$. We are particularly interested in topological properties of $F$. We show that the critical value of $p$ above which there is a positive probability that $F$ connects the left and right edges of $[0,1]^2$ is the same as the critical value for $F$ to connect the top and bottom edges of $[0,1]^2$. Once this is established we derive various topological properties of $F$ analogous to those known for self-similar carpets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02829 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractal percolation on statistically self-affine carpets Falconer, Kenneth Feng, Tianyi Metric Geometry 28A80 We consider a random self-affine carpet $F$ based on an $n\times m$ subdivision of rectangles and a probability $0<p<1$. Starting by dividing $[0,1]^2$ into an $n\times m$ grid of rectangles and selecting these independently with probability $p$, we then divide the selected rectangles into $n\times m$ subrectangles which are again selected with probability $p$; we continue in this way to obtain a statistically self-affine set $F$. We are particularly interested in topological properties of $F$. We show that the critical value of $p$ above which there is a positive probability that $F$ connects the left and right edges of $[0,1]^2$ is the same as the critical value for $F$ to connect the top and bottom edges of $[0,1]^2$. Once this is established we derive various topological properties of $F$ analogous to those known for self-similar carpets. |
| title | Fractal percolation on statistically self-affine carpets |
| topic | Metric Geometry 28A80 |
| url | https://arxiv.org/abs/2401.02829 |