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Results tagged with fractals
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user 48839
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).
4
votes
Accepted
Average size of the Fourier--Stieltjes transform of the fractal measures
No, we cannot have $\int_{-T}^T |\widehat{\mu}|^2 \lesssim T^{1-s-\epsilon}$. This would imply that
$$
I_{s+\epsilon/2}(\mu) = \int d\mu(x)\int d\mu(y) |x-y|^{-s-\epsilon/2} = c \int |t|^{s+\epsilon/2 …
- 24.2k
20
votes
Accepted
A set whose Hausdorff dimension gradually changes?
I assume you want a set $A\subseteq [0,1]$ such that $\dim (A\cap [0,x])=x$ for all $x$. We can define $A_1$ by taking the union of a (Borel) subset of dimension $0$ of $[0,1/2]$ with a subset of dime …
- 24.2k
4
votes
Accepted
Random Cantor sets on the unit interval
Like the classical Cantor set, your set has a natural representation $C=\bigcap C_n$, where $C_1\supseteq C_2\supseteq C_3\supseteq\ldots$, and each $C_n$ is a disjoint union of $2^{n-X_n}$ intervals …
- 24.2k