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Results tagged with fractals
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user 172802
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).
1
vote
Dimension of the graph of a function $\varphi : \mathbb R^2 \to \mathbb R$
This answer is not complete (I am not sure that the function below is smooth).
It seems $\dim_{\mathbb{H}} G(\varphi)$ may be exactly $2$.
Let $f:\mathbb{R}\to\mathbb{R}$ be any continuous function, a …
- 10.6k
10
votes
Accepted
Relationship between doubling constant of a metric space and of a metric measure space
Apart from the obvious counterexample of the measure being $0$, if $(X,d,m)$ is doubling in the sense of metric measure spaces it will be doubling in the sense of metric spaces.
Consider a ball $B(x,r …
- 10.6k