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Results tagged with fractals
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user 8131
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).
10
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Box dimension of the set of Pisot numbers?
The set $\mathcal P$ of all Pisot numbers is known to be closed (Salem). Its limit points $\mathcal P'$ are also known in $(1,2)$ (Talmoudi). The smallest element of $\mathcal P'$ is the golden ratio …
- 2,135
8
votes
Fractals of dimension zero
If by a fractal you mean a self-similar set and if the corresponding iterated function system (IFS) satisfies the Open Set Condition (OSC), then the answer is no.
The Moran-Hutchinson formula gives …
- 2,135
4
votes
Measure 0 sets on the line with Hausdorff dimension 1
It is a very common phenomenon in ergodic theory when the set of points which do not satisfy the Birkhoff ergodic theorem (i.e., a set of zero measure) has full Hausdorff dimension.
See, for instance …
- 2,135
7
votes
A point set of power series with coefficients in {-1, 1}. Connected or not?
What you are referring to is called the connectedness locus. The most recent important paper on the subject is probably this paper by Calegari, Koch and Walker.
In particular, they show that the con …
- 2,135