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 system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
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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 $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$
...
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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 ...
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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 ...
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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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Is there a bi-Hölder Weierstrass-type embedding of the circle into some Euclidean space?
We say that $\Phi\colon S^1\to \mathbb{R}^d$ is an $\alpha$-bi-Hölder embedding if there are constants $c_1,c_2>0$, such that
$$c_1\leq \frac{\|\Phi(x)-\Phi(y)\|}{d(x,y)^\alpha}\leq c_2,$$
where $d$...
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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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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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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:
$$m \vartriangleleft n :\iff [(...
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Average size of the Fourier--Stieltjes transform of the fractal measures
For $0<\theta<1/2$ define $\mu_\theta$ to be the uniform (self-similar) measure on the Cantor set obtained from the dissection pattern $(1-2\theta,\theta)$. For example, when $\theta=1/3$ the $\...
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Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency
let $$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\} $$
I'm studying fractal geometry and ...
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Is an asymptotic expansion of the heat kernel known on any space of fractal dimension?
Is an asymptotic expansion of the heat kernel known on any space of fractal dimension? I was able to find a 2012 paper saying it was not (https://www.math.tugraz.at/~grabner/Publications/...
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Is there a fractional derivative that preserves the composition of the one-parameter Mittag-Leffler function with $x\mapsto x^{\alpha}$?
Let $\alpha\in (0,1)$. The Riemann-Liouville fractional derivative of order $\alpha$ is defined by
$$ \sideset{_0^R}{}{D^{\alpha}f(t)}
=\frac{1}{\Gamma{(1-\alpha)}}
\frac{d}{dt}\left(\int_{0}^{t}
\...
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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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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 ...
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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 ...
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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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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
$$
\phi_i(...
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Will this "tree" cover all rational numbers in a range?
Question
I am making a tree using the following two functions:
$$f(x)=\frac{x}{r},\quad g(x)=\frac{x+b}{r}$$
where $1<r<2$ and $0<b$ are rationals. Everything is a real number here.
The ...
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Does the family of fat Cantor sets contain a measurable rectangle?
Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard Smith-Volterra Cantor set of parameter $t$.
...
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Counting fractals modulo "shared complements"
Previously asked at MSE:
Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to tweak that if it would lead to a more ...
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Does the intersection of middle third and middle half Cantor sets contain an irrational number?
Let $C_\frac{1}{3}$ be the middle third Cantor set, that is, the set of real numbers in the interval $[0,1]$ which can be written in base $3$ using only digits $0$ and $2$.
Likewise let $C_\frac{1}{2}$...
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Is there a two-dimensional unimodal function with fractal level sets
Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$,
such that for all $c\in \mathbb{R}$ both sets
$$ f_{<c}~=~ f^{-1}\left( (-\...
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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)...
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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 ...
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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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Sets of Hausdorff measures zero
Let $H^g$ be the Hausdorff measure with respect to gauge function $g$.
I need to construct an example of a set E for which:
$H^g(E)=0$ for $g(r)=r^s$, $s>0$
and
$0<H^g(E)<+\infty$ for $g(r)=2^...
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On convexity of special fractals in the plane
Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$.
For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the ...
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Does anybody know this paperfolding curve?
In experiments with paperfolding curves, I've constructed an interesting example I cannot find anywhere else.
It is constructed like the “terdragon", where every time the strip is folded to the ...
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Fundamental domain of Möbius transformations
In the book Indra's pearls, Möbius transformations are used to construct Kleinian fractals, which are limit sets of a free group generated by two Möbius transformations $a$ and $b$.
In the process of ...
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Continuous nowhere differentiability and constructive mathematics
In some constructive systems, every function from $\mathbb{R}\to\mathbb{R}$ is continuous (roughly speaking from the classical fact that computable functions are continuous). More weakly, in Bishop's ...
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Seeking for references - Bowen Formula and a link between dimension theory and thermodynamic formalism
I'm needing references - preferably published papers and books - about this subject. I'm relatively new to the state of the art of fractal geometry and am way too inexperienced to seek for myself at ...
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Crazy conjecture about Bernoulli umbra and reference request
For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions.
Yet, it still remains mistery what ...
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Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
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Construction of a homogeneous Moran set
Fix a positive integer $N\ge 2$, for $n \in \mathbb{N}$, denote
$$\Sigma=\{0,1,\dots,N-1\},\\
\Sigma^n=\{(\omega_1,\dots,\omega_n):\omega\in\Sigma, i=1,\dots,n\}.$$
Let $p>2$ be a positive integer. ...
6
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1
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Does finite Hausdorff dimension imply finite packing dimension?
In other words, does there exist a metric space $(E,\rho)$ with finite Hausdorff dimension but infinite packing dimension?
Here are my thoughts:
I know that it is generally hard to relate Hausdorff ...
3
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How well do Gauss-Legendre quadrature methods fare on "fractal" functions?
The context
I'm making your tipical Mandelbrot set viewer, and I have a function $f: ℂ → ℕ$ that counts how many iterations of
$$
z_0 = 0 \\
z_{i+1} = z_i^2 + c
$$
it takes for a particular point $c$ ...
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Relationship between doubling constant of a metric space and of a metric measure space
Let $(X,d,m)$ be a metric measure space. We say that it is doubling in the sense of metric spaces if for every:
$x\in X$ and every $r>0$ there exists some (metric) doubling constant $C_d\geq 0$ ...
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Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
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Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games
A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have
$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$
The set of badly ...
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1
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A question about box dimension and Hölder condition
It is known that if a real continuous function $f(x)$ satisfies a local $\alpha$-Hölder condition on a closed interval $[a,b]$, the box dimension of the graph of $f(x)$ on $[a,b]$ will be not greater ...
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Local dimension of measures
For a Borel prob measure $\mu$ in $\mathbb{R}^n$, define the local dimension of $\mu$ at $x$ by
$$
{\rm dim}_*(\mu, x)=\liminf_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}, {\rm dim}^*(\mu, x)=\limsup_{r\...
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Quantifierisation of maps
I will rewrite my question using Matt F. suggestion.
Consider the logical structure $L = (\mathbb{R}, +, *, 1, 0, =)$ and a function $f:\mathbb{R}\to\mathbb{R}$.
Consider the map $Q:2^\mathbb{R}→2^\...
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Is speaking about a fraction of the Mandelbrot's set meaningful?
Sorry if my question is vague, as I have very little background with fractals and measure theory. My question is inspired by a tweet, where a light shone onto the mandelbrot set, and certain rays were ...
4
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0
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"Snowflaked" Hausdorff metric
Let $(X,d_X)$ be a compact metric space and let $Comp(X)$ be the set of closed subsets of $X$ with the Hausdorff metric:
$$
D(A,B)\overset{\text{def}}{=} \, \max\left\{\sup_{b\in B}\,d_{A}(b),\sup_{a\...
1
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1
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Is there a description of the points of the Cantor set on which the Cantor function is differentiable?
Let $C$ be the usual ternary cantor set, and $f$ the Cantor function, or Devil’s staircase associated to it. We know that $f$ is differentiable a.e., and on every point of the complement $C^c$, the ...
2
votes
1
answer
158
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Hemispherical space filling hilbert curve
First question here, sorry for any posting infractions.
I need to create/find/buy a hemispherical space-filling Hilbert(or similar) curve.
something similar to Cube hilbert
but only filling a ...
6
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0
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560
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Is this result on the set of differentiability of the distance function to the fat Cantor set of any interest?
Quick summary:
Consider the fat Cantor set $C$ of parameter $r$ for arbitrary $0 < r < \frac{1}{3}$, and the distance function to $C$, i.e. $D: [0, 1] \to \mathbb R$ given by $D(x) =\text{dist}(...
2
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0
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57
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Limit set for IFS has either empty interior or dense interior
Let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R^n$ be contracting affine maps. By the theory of iterated function systems, there is a unique minimal compact $K\subseteq\mathbb R^n$ such that $K=f_1(K)\cup\...
16
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3
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573
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How to plot this fractal
I'm a graphic designer and my client has asked me to use this fractal in a design that I'm working on. As you can see, it's not a very good copy, so I'm trying to see if I can generate a high-...
1
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0
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274
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Minkowski (box-counting) dimension of generalized Cantor set
I'm trying to solve this problem.
For $0<\alpha, \beta<1,$ let $K_{\alpha, \beta}$ be the Cantor set obtained as an intersection of the following nested compact sets. $K_{\alpha, \beta}^{0}=[0,...