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Results tagged with fractals
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Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
5
votes
Complexity of the Mandelbrot set on rationals
You probably cannot practically do 100 iterations, at least not in the current universe. Let
$$ f_c(z) = z^2 + c $$
and write $f_c^{\circ n}(z)$ for the $n$'th iterate of $f_c$. You want to start wit …
- 47.4k
8
votes
Accepted
A question about Julia set for quadratic family
I expect that you will find the answer (and a lot more) in the following paper, since it's easy to figure out which quadratic polynomials (if any) commute with one another:
Commuting polynomials and …
- 47.4k
16
votes
Julia sets using other fields
As noted, $\mathbb{Q}_p$ and its finite extensions are locally compact, but the Julia set is often empty. Indeed, in this case that the Fatou set is always non-empty, another difference from the compa …
- 47.4k