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Results tagged with fractals
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user 11009
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).
6
votes
Iterated function system on the plane
Given the other (incorrect) answers, I think it is worth to give a detailed proof of the following partial negative answer: if the maps $f_i$ are only allowed to be homotheties, then there is no such …
- 4,660
3
votes
Accepted
Precise density estimates for Cantor sets
Upper densities
In the following, I freely use the well known fact that $s_\lambda$-Hausdorff measure gives mass $2^{-k}$ to all intervals that make up the stage $k$ in the construction of $C_\lambda …
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20
votes
Accepted
Is there some known way to create the Mandelbrot set (the boundary), with an iterated functi...
I know very little about the Mandelbrot set or complex dynamics, but there are several features which strongly suggest that its boundary is not the attractor of an iterated function system, at least a …
- 4,660
20
votes
Accepted
Hausdorff dimension for invariant measure?
There are a wide variety of notions of dimension of a measure. Your basic intuition is completely correct: for a dynamical system, the dimension of a natural invariant measure provides more relevant i …
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8
votes
Accepted
Dimensions of self-affine sets
Falconer's classical theorem from "The Hausdorff dimension of self-affine fractals" says that if $\alpha<1/2$, then for almost every choice of translations $v_1,v_2$, the Hausdorff and box counting dimensions …
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1
vote
Isometrically-invariant measures and dilation of the Cantor set
This is too long for a comment. The following measure (defined on Borel sets) might be a counterexample to Question 1: let $\mathcal{I}_N$ be the collection of all left-closed, right-open intervals of …
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12
votes
Accepted
Arithmetic products of Cantor sets.
The results my paper with M. Hochman yield the following result on the products of self-similar sets:
Let $A=\bigcup_{i=1}^m r_i A + t_i$, $B=\bigcup_{i=1}^n s_i B+u_i$ be two self-similar sets on …
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4
votes
Accepted
Measures of full Hausdorff dimension for self-affine sets
If $\beta^{-1}$ is Pisot, then there is an ergodic measure of maximal dimension. This is a special case of the rather difficult Theorem 2.15 in the paper Dimension Theory of iterated function systems …
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5
votes
local behavior of a finite Borel measure
I think in order to prove the sharp result you need to use a covering theorem. The best option seems to be Vitali's covering theorem for Radon measures, which says the following: let $\mu$ be a Radon …
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5
votes
Fourier decay rate of Cantor measures
Edit in response to the editing of the second question:
The central Cantor sets are never Salem (for any $\theta$). Here is a way to see this. If $A\subset \mathbb{R}$ is a Salem set, then
$$
\dim(A+ …
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