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).
273 questions
3 votes
1 answer
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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=\...
- 221
3 votes
1 answer
597 views
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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0 votes
2 answers
162 views
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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5 votes
1 answer
306 views
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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0 votes
0 answers
39 views
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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1 vote
0 answers
54 views
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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1 vote
0 answers
136 views
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,...
1 vote
1 answer
115 views
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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4 votes
0 answers
226 views
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 $$\forall \varepsilon>0\ \exists n\in\mathbb N\ \forall f_1,\dots,f_n\in\...
- 1,100
2 votes
0 answers
168 views
Fractal subsets of $\ell^p$ spaces
Let $\ell^p$ be the standard real Banach space of all sequence $(a_n)$ such that $\sum |a_n|^p$ is a convergent series. Is there a subset $A$ of some $\ell^p \cap \ell^q$ for two $p,q\geq 1$ such ...
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5 votes
1 answer
217 views
Is the simple closed curve a topological fractal with two witnessing maps?
Let $\mathbb S$ be the unit circle in the plane. Do there exist two continuous maps $f_1, f_2:\mathbb S\to \mathbb S$ such that $\mathbb S=f_1(\mathbb S)\cup f_2(\mathbb S)$ and $\forall \varepsilon&...
- 1,100
7 votes
2 answers
342 views
$C^1$ function whose critical values are the middle thirds Cantor set
I have seen it claimed on multiple posts here that it is possible to construct a $C^1$ function $f:[0,1]\rightarrow \mathbb{R}$ whose critical values contain the ternary Cantor set defined by the ...
- 173
4 votes
0 answers
143 views
What is $3/2$-dimensional Hausdorff measure of the graph of Brownian motion?
It is well-known, and well-documented, that the Hausdorff dimension of the graph of regular $1$-dimensional Brownian motion is $3/2$ (almost surely). See for example Theorem 4.29 in "Brownian ...
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8 votes
1 answer
283 views
An algebraic condition possibly related with the Hausdorff measure on $\mathbb{R}$
This is my first time to ask a question here. Please tell me if I can improve it. I would like to introduce the following definition inspired from a measure theory exercise. Definition. A subset $K$ ...
3 votes
1 answer
201 views
Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?
Write $g$ as the inverse of $f$. Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that $$ \displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0 $$ as $r\to0$? If you like,...
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1 vote
0 answers
90 views
A functional equation coming from a distribution function
Currently, I am working on a random series as follows. Let $\{Y_k\}$ be a sequence of i.i.d. Bernoulli random variables with expectation $p$. Then we define $$ S = \sum_{k=1}^\infty \prod_{\ell=1}^k 2^...
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5 votes
2 answers
512 views
What is the limit of the sequence of iterated cosines?
I asked this question on MSE here. Define $f_1(z) = \cos(z)$, $f_{n+1}= \cos(f_n (z)) $, The question is: Does $\lim\limits_{n \to \infty}f_n(z)$ exist for certain $z \in \mathbb{C}$? And what is ...
- 697
3 votes
2 answers
206 views
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
I am currently wondering whether it is realistically possible to choose the topic "Fractals and fractal dimensions" for a seminar aimed at undergraduate students in the 2nd semester, with ...
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5 votes
0 answers
522 views
How to define $\mathbb{R}^\frac{1}{2}$?
The Cayley-Dickson construction generates higher-dimensional hyper complex numbers from lower-dimensional ones, producing algebras of dimension $2^n$. I want to generate an algebra of dimension $2^{-1}...
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3 votes
1 answer
139 views
Intersection of IID fractal sets
Let $A, B \subset \mathbb R$ be IID random closed subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\...
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0 votes
1 answer
208 views
Theories for "fuzzy" distributions
When calculating the probability density function for the quotients of adjacent values in an empirical time series, the image of the PDF looked like this: It seems to resemble a lognormal ...
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6 votes
2 answers
436 views
Existence of an $\alpha$-Hölder continuous function whose graph has positive Hausdorff measure of maximal dimension
It is standard that if $f:[0,1] \rightarrow \mathbb{R}$ is $\alpha$-Hölder continuous, then its graph has Hausdorff dimension at most $2-\alpha$. My naive expectation was that "most" graphs ...
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5 votes
1 answer
321 views
Are singular functions dense in the space of Hölder continuous functions?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e. For every positive $\alpha < 1$, is the set of ...
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11 votes
1 answer
1k views
Can the topologist's sine curve be realized as a Julia set?
Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with $$ T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup\big(\{...
- 3,659
13 votes
2 answers
656 views
Convergence of the sequence $s_{n+1}=s_n^2-s_{n-1}^2$
$s_{n+1}=s_n^2-s_{n-1}^2$, $s_0=\sqrt{x}$, $s_1=x$ This sequence seems simple, but is pretty confusing. If you try it with integers, you might think that it always diverges to infinity, but if you try ...
- 467
2 votes
0 answers
193 views
Random matrix with power law decay in eigenvalues
What positive semi-definite random matrices have (roughly) $n^{-\alpha}$ for $n^{th}$ singular value? The power law decay need not be exact. I want to find random matrix ensembles that naturally ...
- 451
3 votes
2 answers
199 views
Dimension of the graph of a function $\varphi : \mathbb R^2 \to \mathbb R$
Let $\varphi : \mathbb{R}^2 \to \mathbb{R}$ be a continuous function, and let $G(\varphi)$ be the graph of $\varphi$. Denote $R:=\{(x,0) \in \mathbb{R}^2 | x \in \mathbb{R}\}$ as the real line in $\...
- 661
0 votes
0 answers
55 views
Reference on multifractal complex measures?
This is a cross-post of this physicsSE post; I am also posting it here since this question lies at the boundary of both physics and math. I am learning about multifractal formalism recently. It seems ...
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1 vote
0 answers
63 views
Deterministic multifractal measure with quadratic singular spectrum?
For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$ ...
- 775
0 votes
1 answer
149 views
Fractal sets and dimensions
Can we construct two sets $E$ and $F$ meeting the following criteria $\dim_H(E) = \dim_H(F) = \dim_H(E ∩ F)$ $\dim_P(E), \dim_P(F)$, and $\dim_P(E ∩ F)$ are distinct? Here $\dim_H$ denotes the ...
- 39
3 votes
1 answer
764 views
What is the Lebesgue covering dimension of this topological space?
Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal. Take the induced topology defined by the Lorentzian metric called Alexandrov topology. This topology matches ...
- 658
0 votes
1 answer
134 views
Box dimension and graph of Hölder function
In Kamont "ON THE FRACTIONAL ANISOTROPIC WIENER FIELD" (found here : https://www.math.uni.wroc.pl/~pms/files/16.1/Article/16.1.6.pdf), on page 96, it is claimed that, if a function $f:I^{d}\...
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2 votes
1 answer
253 views
Macroscopic sets - a notion of largeness for Lebesgue null sets
Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
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3 votes
1 answer
866 views
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Is $$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$ where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime? Context: This question came out as a result in ...
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7 votes
2 answers
700 views
How to define a fractal from the lexicographic sorting on the prime factorization of natural numbers?
Consider on the natural number the lexicographic ordering on the prime factorization: If $m = p_1^{a_1}\cdots p_r^{a_r},n = q_1^{b_1}\cdots q_s^{b_s}$ then we define: $$m \vartriangleleft n :\iff [(...
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3 votes
1 answer
295 views
Average size of the Fourier--Stieltjes transform of the fractal measures
For $0<\theta<1/2$ define $\mu_\theta$ to be the uniform (self-similar) measure on the Cantor set obtained from the dissection pattern $(1-2\theta,\theta)$. For example, when $\theta=1/3$ the $\...
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3 votes
1 answer
196 views
Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency
let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$ I'm studying fractal geometry and ...
1 vote
0 answers
87 views
Cardinality of intersections of lines with irregular 1-sets in the plane
From Falconer's book (The geometry of fractal sets), Lemma 3.2 says that the intersection of irregular 1-sets with straight lines is of zero $H^1$ measure. What do we know about the cardinality of ...
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1 vote
0 answers
158 views
Fractal dimension of a self-similar tree
Consider a binary tree constructed as the following. Given a node with a some value $x$, I construct two children nodes each having value $l(x)$ and $r(x)$ respectively. I repeat the same on the ...
- 451
8 votes
1 answer
328 views
Rolling a sphere on a fractal curve
Given a rectifiable curve C : [0, 1] → R2 in the plane, it makes sense to roll a unit sphere S2 on the plane along that curve and ask what is its net rotation in SO(3). I wonder if this also makes ...
- 3,545
7 votes
1 answer
212 views
Plane curve with continuously increasing Hausdorff dimension
In a recent paper, we required the following fact. Proposition 1. There exists a simple closed curve $\gamma\subset\mathbb{C}$ with the following property. If $\phi$ is a biholomorphic map, defined on ...
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1 vote
1 answer
180 views
The graph of the fractal sets
Could you please provide me with the graph of the fractal set produced by the given following IFS? Consider an IFS $\{\phi_i, i=1,...,9\}$ on ${X}=[0, \infty) \times[0,1]$ defined as follows $$ \phi_i(...
- 39
1 vote
1 answer
237 views
Will this "tree" cover all rational numbers in a range?
Question I am making a tree using the following two functions: $$f(x)=\frac{x}{r},\quad g(x)=\frac{x+b}{r}$$ where $1<r<2$ and $0<b$ are rationals. Everything is a real number here. The ...
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8 votes
1 answer
822 views
Does the family of fat Cantor sets contain a measurable rectangle?
Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard Smith-Volterra Cantor set of parameter $t$. ...
- 9,438
4 votes
0 answers
116 views
Counting fractals modulo "shared complements"
Previously asked at MSE: Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to tweak that if it would lead to a more ...
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11 votes
0 answers
676 views
Does the intersection of middle third and middle half Cantor sets contain an irrational number?
Let $C_\frac{1}{3}$ be the middle third Cantor set, that is, the set of real numbers in the interval $[0,1]$ which can be written in base $3$ using only digits $0$ and $2$. Likewise let $C_\frac{1}{2}$...
- 3,238
1 vote
1 answer
211 views
Is there a two-dimensional unimodal function with fractal level sets
Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$, such that for all $c\in \mathbb{R}$ both sets $$ f_{<c}~=~ f^{-1}\left( (-\...
- 1,726
6 votes
1 answer
755 views
Why in the Sierpiński Triangle is this set being used as the example for the OSC and not a more "natural"?
The Open Set Condition is fulfilled by an Iterated Function System in the plane $\{\mathbb{R}^2, \phi_1, \phi_2, \dots, \phi_m \}$ if there exists a nonempty open set $V$ such that $\bigcup \phi_{i}(V)...
- 165
5 votes
0 answers
196 views
Naïve definition of a measure on a fractal
This question was previously posted on MSE. Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$. One option would be to use ...
- 345
6 votes
2 answers
1k views
Is there an L-system for aperiodic tilings of the plane with the "hat" monotile?
Most aperiodic tilings of the plane, except possibly for spiral tilings like the Voderberg tiling, exhibit a fractal pattern of self-similarity. This is no exception for the recently discovered "...
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