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Results tagged with fractals
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user 21437
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).
3
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Iterated function system on the plane
Corrected:
Let $R = \max \{r_1+r_2, r_2+r_3, r_3+r_1 \}$. This can be done when $R \leq 1$.
Take 3 non-collinear points $p_1,p_2,p_3$ on the plane and take the usual self-similar maps $f_i(x)=r_i(x …
- 153
2
votes
1
answer
847
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Hausdorff dimension of a subset of Cantor set
What is the Hausdorff dimension of the subset
$$F := \{ x = \sum^\infty_{n=1} \frac{2 x_n}{3^n} \in [0,1] : x_n \in \{ 0 , 1 \} , x_n = 1 \Rightarrow x_{n+1}=0 \}$$
of the Cantor set? Is it known alre …
- 153
0
votes
1
answer
467
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Littlewood-Paley theory and norm estimation
In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1.
It is claimed that Lemma 2 is a …
- 153