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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
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 …
- 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). …
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1
vote
A calculus question related to quantization dimension
My response to the answer by fedja, Jun 5, 2011. This should be a comment, but won't fit.
It didn't work. Taking values of $A,B,\epsilon$ that satisfy your conditions, then tracing back through u …
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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 …
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2
votes
Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
Chaos and Fractals: New Frontiers of Science, by
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 14 …
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0
votes
Minimum number of contractions needed to obtain a particular invariant set
Perhaps you can consult some of the literature on "finite type condition" for IFSs.
That's if you are willing to allow overlap in the IFS.
MR1488232 (98i:28010)
MR1825981 (2002c:28010)
MR2304331 (2 …
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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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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
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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4
votes
Accepted
Determining a lower bound on the Hausdorff dimension of a set
Upper bound for the Hausdorff dimension is often easy, from the definition.
Lower bound can be harder. One method can be used if you have a measure on your set. Even better, a measure that naturall …
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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 …
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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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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 $\dim_\mathrm{P}(B) = 2/5$,
$ …
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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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