Fractal percolation on statistically self-affine carpets

Kenneth John Falconer; Tianyi Feng

doi:10.1090/proc/17076

Fractal percolation on statistically self-affine carpets

Kenneth John Falconer^*, Tianyi Feng^*

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider a random self-affine carpet F based on an n x m subdivision of rectangles and a probability 0<p<1. Starting by dividing [0,1]² into an n x m grid of rectangles and selecting these independently with probability p, we then divide the selected rectangles into n x 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.

Original language	English
Pages (from-to)	1121-1134
Number of pages	14
Journal	Proceedings of the American Mathematical Society
Volume	153
Issue number	3
Early online date	29 Jan 2025
DOIs	https://doi.org/10.1090/proc/17076
Publication status	Published - Mar 2025

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Copyright © 2025 the Authors. This work has been made available online in accordance with the University of St Andrews Open Access policy. This accepted manuscript is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.1090/proc/17076.
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Cite this

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abstract = "We consider a random self-affine carpet F based on an n x m subdivision of rectangles and a probability 02 into an n x m grid of rectangles and selecting these independently with probability p, we then divide the selected rectangles into n x 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.",

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N2 - We consider a random self-affine carpet F based on an n x m subdivision of rectangles and a probability 02 into an n x m grid of rectangles and selecting these independently with probability p, we then divide the selected rectangles into n x 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.

AB - We consider a random self-affine carpet F based on an n x m subdivision of rectangles and a probability 02 into an n x m grid of rectangles and selecting these independently with probability p, we then divide the selected rectangles into n x 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.

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Fractal percolation on statistically self-affine carpets

Abstract

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