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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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 ...
Bastam Tajik's user avatar
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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}\...
BabaUtah's user avatar
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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 $d$...
Roope Anttila's user avatar
2 votes
1 answer
175 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 ...
Nate River's user avatar
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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 ...
mathoverflowUser's user avatar
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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 [(...
mathoverflowUser's user avatar
3 votes
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201 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 $\...
Subhajit Jana's user avatar
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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 ...
Simple Conjugate's user avatar
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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/...
Idempotent's user avatar
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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} \...
Medo's user avatar
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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 ...
ru0xffian's user avatar
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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 ...
CWC's user avatar
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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 ...
Daniel Asimov's user avatar
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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 ...
Lasse Rempe's user avatar
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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(...
B-S's user avatar
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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 ...
CWC's user avatar
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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 $t$. ...
Nate River's user avatar
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4 votes
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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 ...
Noah Schweber's user avatar
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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 $C_\frac{1}{2}$...
Dmitrii Korshunov's user avatar
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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( (-\...
Karl Fabian's user avatar
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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)...
anchova's user avatar
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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 ...
Matheus Manzatto's user avatar
6 votes
2 answers
583 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 "...
Wolfgang's user avatar
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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^...
B-S's user avatar
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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 ...
gigi's user avatar
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7 votes
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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 ...
PiotrP's user avatar
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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 ...
p6majo's user avatar
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7 votes
2 answers
482 views

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 ...
Bruno Le Floch's user avatar
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1 answer
68 views

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 ...
anchova's user avatar
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1 vote
0 answers
100 views

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 ...
Anixx's user avatar
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6 votes
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263 views

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 ...
Sidharth Ghoshal's user avatar
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0 answers
30 views

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. ...
Dingding Yu's user avatar
6 votes
1 answer
368 views

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 ...
Peter Koepernik's user avatar
3 votes
0 answers
63 views

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 $c$ ...
Eduard Gomez's user avatar
6 votes
1 answer
482 views

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 $C_d\geq 0$ ...
Carlos_Petterson's user avatar
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1 answer
170 views

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 ...
ABIM's user avatar
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6 votes
1 answer
454 views

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 ...
No One's user avatar
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2 votes
1 answer
235 views

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 ...
Watheophy's user avatar
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3 votes
1 answer
328 views

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\...
user119197's user avatar
2 votes
0 answers
127 views

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^\...
A.Skutin's user avatar
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0 answers
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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 ...
Ron Shvartsman's user avatar
4 votes
0 answers
106 views

"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\...
TomCat's user avatar
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1 vote
1 answer
516 views

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 ...
Nate River's user avatar
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2 votes
1 answer
156 views

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 ...
FlashDD's user avatar
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6 votes
0 answers
555 views

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}(...
Nate River's user avatar
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2 votes
0 answers
57 views

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\...
chronondecay's user avatar
16 votes
3 answers
569 views

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-...
Circle B's user avatar
  • 263
1 vote
0 answers
273 views

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,...
Loli's user avatar
  • 111
1 vote
1 answer
311 views

Can the Mandelbrot set be designed through inequalities?

Many years ago I found an inequality that directly controlled whether a point $\displaystyle c$ belongs or does not belong to the Mandelbrot set. Roughly, it was something like this: If $\displaystyle ...
Γιώργος Πλούσος's user avatar
1 vote
1 answer
124 views

Box counting dimension and Besov spaces on $\mathbb R^2$

I found a lemma in this paper of Constantin and Wu, stated with no proof: Lemma 3.2. Let $b=\chi_{D}$ be the characteristic function of a bounded domain $D\subset\mathbb R^2$ whose boundary has box-...
Calvin Khor's user avatar

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