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 ...

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For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?

The fractal dimension of the 3D Apollonian packing is computed in this paper. In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension ...
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Decay rate of measures on Cantor set

I've read that Kahane and Salem show that if $\mu$ is any measure supported on the ternary Cantor set, then $\hat{\mu}(\xi) \not\to 0$ as $|\xi| \to \infty$, however I have been unable to find a ...
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Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets

Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
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Picture of the set of discontinuity of degree 2 rational Julia sets

Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the ...
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A fractal object at origin but nowhere else: derived from Brownain motion

Hi, Please consider this object: Start with a realization of Brownian motion in 2D, which I'll denote by rho(t) where -infinity < t < +infinity. Next, lets smooth rho. There are various ...
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Self-similarity of Riemann's “non-differentiable” function

I hope it doesn't seem inappropriate for me to raise on MO an unanswered question from MSE, indeed a question actually posed there by someone other than myself. I want to ask the following: 1) ...
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Complexity of a variant of the Mandelbrot set decision problem?

This is a modified version of a question posted on StackExchange TCS. Mandelbrot set is defined using the complex equation $P_c (z)=z^2 +c$ where $c$ is a complex number. Let us define ...
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Classification of Self similar sets

I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are ...