Questions tagged [fractals]

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).

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Who proved that the Mandelbrot set's Julia sets are locally connected?

I'd be greatly interested in a reference to the respective article. Was it Douady? Julia? Hubbard? Fatou? Bonus question: Who gave the proof that can be found in the Orsay notes? EDIT: The question ...
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The Koch snow flake, Holder exponents of conformal mappings

The Koch snow flake $K$ is a domain of $\mathbb{C}$, complex plane. Though, I do not state the precise definition, you can see the picture in wikipedia Koch snow flake. The Koch snow flake $K$ is a ...
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Does fractallity depend on the Riemannian metric?

Edit: According to comment of Andre Henriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...
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Hausdorff dimension and von Neumann dimension

There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type ${\rm II_1}$ ...
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Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ [closed]

Let $f:[0,1] \to \mathbb{R}, G = graph(f)$. If $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $0 = x_0< \ldots < x_m = 1$ then $H^s(G) < \infty$ What technique can I ...
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Pointless characterization relating between a fractal and its code space

Given an hyperbolic IFS $(X,\{f_i:i=1,\ldots,N\})$ and denoting its code space by $\Sigma_N = \{1,\ldots,N\}^{\mathbb{N}}$ and the generated fractal set by $\mathcal{A}$. There is a continuous ...
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Set with modified lower box counting dimension strictly less than Hausdorff dimension

Please, can someone give (as simple as possible) example of the set for which modified lower box counting dimension is strictly smaller than Hausdorff dimension?
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Complexity of the Mandelbrot set on rationals

Given two rationals $a,b \in \mathbb{Q}$, call $c = a + ib$, i.e., the complex number represented by these two rationals. A point $c$ is contained within the Mandelbrot set $M$ if the following ...
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Relationship between the Hurst exponent and the alpha parameter

I have a question about the relationship between the Hurst exponent $H$ and the $\alpha$ parameter in the autocorrelation function when long memory is present. As we know in this case the decay of the ...
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Literature on the total variation of fractal graphs/fractal Brownian motion?

I know that for standard Brownian motion, the total variation sampled at intervals of length $\Delta$ converges to $V(\Delta) = C \Delta^{-1/2}$ for some constant $C$. I wish to use this fact to study ...
If a function $f : \mathbf{R} \to \mathbf{R}$ is $\mathscr{C}^{0,\alpha}$ for every $0 < \alpha < 1$ then its graph has Hausdorff dimension $1$. I would like to see an example of such a ...