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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).
24
votes
Accepted
Parametrization of the boundary of the Mandelbrot set
Lasse's answer expanded: Let $\psi$ be the map of the exterior of the unit disk onto the exterior of the Mandelbrot set, with Laurent series
$$
\psi(w) = w + \sum_{n=0}^\infty b_n w^{-n} =
w - \frac{1 …
- 41.1k
17
votes
Fractal questions: Weierstraß-Mandelbrot
My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals?
Of course this depends on your definition of fractal. … Added:
Kiesswetter function, two figures from Classics on Fractals
Figure 18.2 (source: Wayback Machine)
Figure 18.3 (source: Wayback Machine) …
- 41.1k
11
votes
Accepted
Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?
Topological dimension (say, covering dimension) $\dim_\mathrm{T}$ and Hausdorff dimension $\dim_\mathrm{H}$ both make sense for metric spaces. Benoit Mandelbrot defined $A$ to be a fractal iff $\dim_ …
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10
votes
Fractals of dimension zero
One example: the set of
Liouville numbers has Hausdorff dimension zero.
In number theory, a Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, the …
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9
votes
Accepted
Why in the Sierpiński Triangle is this set being used as the example for the OSC and not a m...
I guess that illustration relates to the paper
Bandt, Christoph; Nguyen Viet Hung; Rao, Hui, On the open set condition for self-similar fractals, Proc. Am. Math. Soc. 134, No. 5, 1369-1374 (2006). …
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8
votes
Dimensions of self-affine sets
Falconer, "The Hausdorff dimension of self-affine fractals" Math. Proc. Cambr. Phil. Soc. 103 (1988) 339-350
[2] G. A. …
- 41.1k
7
votes
Accepted
Does finite Hausdorff dimension imply finite packing dimension?
A construction used (repeatedy) in the paper
Edgar, G. A., Centered densities and fractal measures, New York J. Math. 13, 33-87 (2007). ZBL1112.28004.
For more information, see that paper.
We constru …
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7
votes
How can we not know the $s$-measure of the Sierpiński triangle?
The latest I could find is
Móra, Péter, Estimate of the Hausdorff measure of the Sierpinski triangle, Fractals 17, No. 2, 137-148 (2009). ZBL1178.28007. …
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6
votes
Accepted
Hausdorff dimension: subset of $\mathbb{R}^n$ vs. boundary of this subset
"Smaller" in the sense of $\le$ ... If $S$ is closed and has Hausdorff dimension $< n$, then $S$ has empty interior, so (as noted by Joel) $S$ is its own boundary, and thus we have equality for the tw …
- 41.1k
6
votes
How to define a differential form on a fractal?
There is an example of Hassler Whitney that is a lot of fun.
Whitney, Hassler, "A function not constant on a connected set of critical points."
Duke Math. J. 1 (1935), no. 4, 514–517.
He con …
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5
votes
Measure 0 sets on the line with Hausdorff dimension 1
Perhaps Hausdorff's original paper? He uses gauge functions other than powers of x. And constructs Cantor sets corresponding to them. For example if you take $x |\log x|$ then you get a set of Haus …
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4
votes
Hausdorff dimension for invariant measure?
Hausdorff dimension of a measure is studied, yes.
The mathematical texts should treat this.
Falconer, Fractal Geometry 2nd edition p. 209
Edgar, Integral, Probability, and Fractal Measures p. 123
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4
votes
Hausdorff dimension of inverse images.
Indeed, more is true: (1) The topological dimension is $\ge d-1$. And (2) the Hausdorff dimension is $\ge$ the topological dimension.
For (1) note that $C$ is a closed set that separates $\mathbb …
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4
votes
Self-similarity of a dendrite fractal
OK, as hinted in my comment. Here is the fractal $J$:
Now choose a branch of the squareroot so that $\sqrt{w-i}$ is continuous on this set. Here is the image of $J$ under the map $\sqrt{w-i}$ …
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4
votes
Accepted
Angles and proportions occurring in L-system fractals
In your first example, I think the ratio is
$$
\arctan\frac{\sqrt{127+96\sqrt{2}\;}}{7} \approx 66.643774 \text{ degrees}
$$
There is no reason to think this is a rational number of degrees. And: the …
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