I'm wondering. Can any compact subset of $\mathbb{R^2}$ be written as a suitable IFS attractor?
Can someone explain? Thank you for visiting my question.
2026-03-25 17:40:38.1774460438
Can any compact subset of $\mathbb{R^2}$ be written as a suitable YFS attractor?
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In general the answer is no; they prove it not just for the plane, but for any uncountable Polish space. I haven't read the paper, so I'm not sure EXACTLY where the result is:
http://www.acadsci.fi/mathematica/Vol38/vol38pp797-804.pdf
This paper discusses the result, as well as generalizations of the question to infinite IFS's:
https://www.sciencedirect.com/science/article/pii/S0022247X1931008X
There are some positive results; in the paper above, they prove it for certain sets containing clopen copies of the Cantor Set. It's also known for polyhedra in $\mathbb{R}^n$.