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Results tagged with fractals
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user 46214
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
votes
Fractal dimension of scaling limits of discrete structures
Martin Barlow And S. James Taylor published a number of papers on this topic, for example:
Fractional dimension of sets in discrete spaces, Journal of Physics A: Mathematical and General, Volume 22 …
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6
votes
A question about Julia set for quadratic family
Joe's answer and the cited paper are, indeed, quite nice. I think the basic idea is fairly self-contained and probably easier than the reference.
Two complex polynomials $f$ and $g$ of the same degre …
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4
votes
Integral over the Cantor's set Hausdorff dimension
First, I don't think you want to integrate with respect to Hausdorff measure, whose exact value is quite difficult to compute. Rather, you want to integrate with respect to a uniform measure on $C$ or …
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4
votes
Accepted
Dimension of the graph of a function $\varphi : \mathbb R^2 \to \mathbb R$
I don't believe it necessary that the dimension of the graph of $\varphi$ be larger than 2. An example is provided by examining Poisson's integral formula for the upper half plane:
$$
u(x,y) = \frac …
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6
votes
Dimensions of self-affine sets
As the previous two answers correctly indicate, there is no simple, closed form expression for the dimension of a self-affine set. Here is a concrete example illustrating why we might expect difficul …
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6
votes
Accepted
On the boundary of the twindragon
The link that you refer to does not describe the boundary as the attractor of a simple IFS, rather it describes a collection of portions of the boundary as the invariant list of a digraph IFS - or dir …
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27
votes
Why are the Julia sets so simple? (quadratic family)
Julia sets are all very closely related to self-similar sets - each one can be thought of as the invariant set of something like an iterated function system. Specifically, the Julia set of $f(z)=z^2 + …
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10
votes
What is the limit of the sequence of iterated cosines?
You should have a look at Bob Devaney's notes entitled Complex Exponential Dynamics, which you can download from his web page:
http://math.bu.edu/people/bob/papers.html
While that paper focuses on the …
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-2
votes
Hausdorff dimension and sigma finiteness
I believe the Takakgi function satisfies this property. According to The Takagi Function: A Survey, the Takagi function $T$ satisfies
$$T(x+h)-T(x) = O(h\log(1/|h|) \: \text{ as } \: h\to0$$
and this …
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9
votes
Accepted
What is the Hausdorff dimension of this fractal?
Strictly speaking, the graph is not self-similar. It is (nearly) self-affine. More precisely, if $b>1$ is an integer and $c>1$ is real, then the graph $G$ of $f$ over the interval $[0,b]$ agrees wit …
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2
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Is there a two-dimensional unimodal function with fractal level sets
I think this is essentially equivalent to Martin Hairer's comment but expressed a bit differently.
Let $F$ be any continuous, positive valued function defined on $[0,2\pi]$ with the property that $F(0 …
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