Questions tagged [fractals]
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 systems, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems (see Lorenz attractor).
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Limsup of Cantor sets
Let $C_a$ be the middle-$a$ Cantor set, obtained by starting from $[0, 1]$ and removing the open middle $a$ proportion of every surviving interval at each of countably many stages.
Fix $\alpha \in (0, ...
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Min-max tree topology of sequences
let a contracted sequence $x_1,x_2,\dots$ of values be characterized by $x_i\ne x_{i+1}$, i.e. a sequence after adjacent duplicates have been replaced by a single value.
Then, given an infinite ...
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If $a^3+b^3=1$, is there a set that can be divided into two similar copies of itself, scaled by $a$ and $b$ respectively?
I have previously asked this question on MSE, receiving $14$ upvotes and no definitive answers. As such, I am reproducing this question here.
Let $a$ and $b$ be positive reals.
If $a+b=1$, any line ...
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Transcendental numbers in Cantor set
How can we find irrational or even transcendental numbers $x$ and $y$ in the middle-third Cantor set such that $x^2 + y^2 = 1$? I came across the following interesting paper on this topic: https://...
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Self-similar measure of the boundary of the ball
A finite family of contracting similarity maps $\{S_1, \ldots, S_m\}$, with $m\ge 2,$ is called an iterated function system (IFS). A self-similar set is the attractor of the IFS $\{S_1, \ldots, S_m\}$,...
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Does the airplane Julia set contain true circles?
The well known "airplane" Julia set looks like it contains a true circle. To be precise, let $c$ be the real root of $x^3+2x^2+x+1=0$. i.e., $c\approx -1.75$. The Julia set of $z^2+c$ is the ...
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Reference Request: accessible points of Wada domain boundaries in $\mathbb R^d$
Say that a compact $K\subset\mathbb R^d$ has the Wada property if there are disjoint, connected open sets $V_1,V_2,\ldots,V_n\subset\mathbb R^d$, $n\geq3$, such that $K=\partial V_1=\partial V_2=\...
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Non-fractal algebras
Yesterday, the following question came to my mind:
We say that a unital $C^*$-algebra is a non-fractal algebra if $\mathrm{sp}(a)$ is not a fractal set for all $a\in A$. Equivalently the ...
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continuous, strictly increasing univariate real function with derivative 0 almost everywhere
Are there actually a strictly increasing continuous function from $\mathbb{R}$ to $\mathbb{R}$ with derivative of 0 almost everywhere ?
I tried to build one with three real sequences $a_n$, $b_n$ and $...
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The measure of an everywhere nailed set
Let $K$ be a compact subset of the Euclidean plane.
Assume that for every point $k \in K$, there exists a point $x$ such that the open interval $]k, x[$ lies in the complement of $K$.
Is it true that ...
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IFS hull of a finite set of points
Let $S = \{ s_1, \, \ldots, \, s_m \}$ be a finite set of points in $\mathbb{R}^n$. Suppose that $\mathcal{T} = \{ T_1, \, \ldots, \, T_k \}$ is a family of contractions on $\mathbb{R}^n$ such that $s ...
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Diffeomorphic mappings between attractors of IFS's
Suppose we have two IFS's $\{ f_1, \, \ldots, \, f_m\}$ and $\{ g_1, \, \ldots, \, g_m\}$, each of them being a family of contractions from $\mathbb{R}^n$ to itself. Let $K_f$ and $K_g$ be their ...
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When do solutions to $a^d+b^d=1$ correspond to self-similar sets?
Suppose $S$ is a set of Hausdorff dimension $d$, which admits a dissection into two pieces $S_a,S_b$. If the ratios of similarity between those pieces and the original set are $a$ and $b$ respectively,...
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Upper Minkowski dimension of a sum of planar curves
Assume that two continuous parametric planar curves a(t) and b(t) have respective upper Minkowski dimensions A and B. Is it true that their sum, say c(t)=(a+b)(t), is a continuous parametric curve ...
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Does every compact metric space admit a finite contracting family of maps?
Let $X$ be a metric space. A collection $\mathcal F$ of maps $X\to X$ (not necessarily continuous) is called contracting if
$$\forall \varepsilon>0\ \exists n\in\mathbb N\ \forall f_1,\dots,f_n\in\...
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