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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).
2
votes
Fractal sets and dimensions
Suppose we construct sets $A,B,C$ with
$A \subseteq [0,1]$, $\dim_\mathrm{H}(A) = 1/5$ and $\dim_\mathrm{P}(A) = 4/5$,
$B \subseteq [2,3]$, $\dim_\mathrm{H}(B) = 1/5$ and $\dim_\mathrm{P}(B) = 2/5$,
$ …
- 41.1k
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). …
- 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 …
- 41.1k
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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1
vote
Accepted
Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$
Take $s \ge 1$. We will use $(x+y)^s \le x^s+y^s$ for positive $x,y$.
Let $C=2^{1+s/2}(1+c)$. I claim: $\mathcal H^s(G) \le C$.
Let $N \in \mathbb N$. Let $\eta > 0$ be so small that
$N2^s\et …
- 41.1k
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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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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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. …
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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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3
votes
Reference for the iterated function system of the Koch snowflake
Aidan Burns, "78.13 Fractal tilings", Mathematical Gazette 78 (1994), 193–196
This article describes two remarkable tilings. The first is the Koch snowflake which will only tile the plane if tile …
- 41.1k
2
votes
Iterated function system on the plane
partial solution
Something like this should work. We are given $r_1, r_2, r_3 > 0$ with $r_1^2+r_2^2+r_3^2<1$. Let $s$ be such that $r_1^s+r_2^s+r_3^s = 1$; this $s$ is the similarity dimension …
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3
votes
Fourier decay rate of Cantor measures
I would start looking here:
MR1785620 (2001m:42020)
Peres, Yuval; Schlag, Wilhelm; Solomyak, Boris,
Sixty years of Bernoulli convolutions.
Fractal geometry and stochastics, II (Greifswald/Koserow, 19 …
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2
votes
Accepted
Fractals as solution to optimization problem?
standard book reference:
B. Mandelbrot, THE FRACTAL GEOMETRY OF NATURE
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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 $\sqrt{w-i}$ …
- 41.1k
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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