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Results tagged with fractals
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user 5701
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).
7
votes
Accepted
Hausdorff dimension of a subset of Cantor set
The comments by Andreas and Anton give you the answer already to your specific question. Let me give a more general answer, since your question is very representative of a whole class of examples.
T …
- 8,903
8
votes
Hausdorff dimension for invariant measure?
The way it's usually done is as follows:
$$
\dim_H \mu = \inf \{ \dim_H Z \mid \mu(Z) = 1 \}.
$$
You can also study box dimension of measures, but there you take an infimum over all sets $Z$ with $\mu …
- 8,903
9
votes
Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?
To the best of my knowledge there is no universally agreed upon precise definition of the word "fractal", so it's not clear to me exactly what would or would not constitute an example of a fractal tha …
- 8,903
3
votes
Hausdorff dimension of non-recurrent walks
Anthony's answer settles the matter, but I'll say a few words about relevant terminology and references that are too long to fit in the comment box. (And go a bit beyond what you actually asked, but …
- 8,903
12
votes
Accepted
Hausdorff dimension of graphs .
The graph of any Lipschitz function $f\colon [a,b]\to\mathbb{R}$ has Hausdorff dimension $1$ (this follows since Hausdorff dimension is invariant under bi-Lipschitz mappings). Your example of $f(x) = …
- 8,903