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).

70 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
11 votes
0 answers
314 views

Hausdorff dimension and von Neumann dimension

There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type ${\rm II_1}$ ...
Sebastien Palcoux's user avatar
9 votes
0 answers
470 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 $C_\frac{1}{2}$...
Dmitrii Korshunov's user avatar
9 votes
0 answers
179 views

Is each Peano continuum a topological fractal?

Problem. Is each Peano continuum a topological fractal? A compact Hausdorff space $X$ is a topological fractal if $X=\bigcup_{i=1}^n f_i(X)$ for some continuous maps $f_1,\dots,f_n:X\to X$ such that ...
Lviv Scottish Book's user avatar
9 votes
0 answers
238 views

Decay rate of measures on Cantor set

I've read that Kahane and Salem show that if $\mu$ is any measure supported on the ternary Cantor set, then $\hat{\mu}(\xi) \not\to 0$ as $|\xi| \to \infty$, however I have been unable to find a ...
user43622's user avatar
8 votes
0 answers
306 views

Fractal homological algebra

The usual definition of a chain complex requires for the indices to be integer numbers. However, taking inspiration from the theory of Hausdorff dimension, one can think of 'fractal' chain complexes (...
gm01's user avatar
  • 325
7 votes
0 answers
130 views

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
  • 71
7 votes
0 answers
205 views

Has this self-similar sequence the ratio $(\sqrt2+1)^2$?

This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows: $a_n$ is the smallest number such that $s_n:=...
Wolfgang's user avatar
  • 13.2k
7 votes
0 answers
506 views

Can we define the Mandelbrot set in terms of the generating function of the Catalan numbers?

For each complex number $c$, define $P_{0}(c)=0$ and $P_{n+1}(c) = (P_{n}(c))^{2} + c$ . The Mandelbrot set is the set of complex numbers c for which $|P_{n}(c)|$ stays bounded as $n\rightarrow \...
Andrius Kulikauskas's user avatar
7 votes
0 answers
422 views

Does the Mandelbrot set have infinite VC dimension?

Define a binary classifier for points in the complex plane, whose parameter $\theta$ is an isometry of $\mathbb{C}$, and which classifies $z \in \mathbb{C}$ based on whether or not $\theta(z)$ is in ...
Peter Schmidt-Nielsen's user avatar
7 votes
1 answer
224 views

Hausdorff dimension and sigma finiteness

If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$. I would like to see an example of such a ...
Longyearbyen's user avatar
6 votes
0 answers
268 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
6 votes
0 answers
560 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
  • 4,802
6 votes
0 answers
1k views

What is known about the area of the symmetric Pythagorean tree?

What is known about the area of the symmetric Pythagorean tree? (Closed unit square as base, area of enclosed triangles not included.) The problem in calculating the area is that squares start to ...
Gerard's user avatar
  • 195
5 votes
0 answers
212 views

Is Domineering on any finite approximation of the Sierpinski Carpet always a second-player win?

The game of Domineering can be played on any board consisting of some subset of $\mathbb{Z} \times \mathbb{Z}$. In particular, consider the boards $K_n$ generated by iterating the following inductive ...
Noam Zeilberger's user avatar
5 votes
0 answers
806 views

Fractal covering of a plane with complex-base numeral systems - is periodicity necessary?

Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$: $$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{...
Jarek Duda's user avatar
4 votes
0 answers
84 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 ...
Noah Schweber's user avatar
4 votes
0 answers
147 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 ...
Matheus Manzatto's user avatar
4 votes
0 answers
108 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
  • 93
4 votes
0 answers
141 views

Can this number be interpreted as a fractal dimension?

Under Goldbach's conjecture, let's denote for a large enough integer $n$ by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\}$ and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n))$. Let's ...
Sylvain JULIEN's user avatar
4 votes
0 answers
196 views

Fourier coeffients of Cantor measure

For $0<\theta<\frac{1}{2}$, denote by $\mu_\theta$ the uniform Cantor measure with dissection ratio $\theta$. It is not hard to show that the Fourier–Stieltjes transform of $\mu_\theta$ is $$ \...
user119197's user avatar
4 votes
0 answers
151 views

Is the limit of classical Laver tables connected anywhere?

Let $A_{n}=(\{1,\dots,2^{n}\},*_{n})$ be the $n$-th classical Laver table. Then $*_{n}$ is the unique operation on $\{1,\dots,2^{n}\}$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\...
Joseph Van Name's user avatar
4 votes
0 answers
170 views

Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets

Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
Bill Mance's user avatar
4 votes
0 answers
304 views

Picture of the set of discontinuity of degree 2 rational Julia sets

Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the ...
Albert's user avatar
  • 377
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
3 votes
0 answers
296 views

Covering lemmas in Hochman's ''On self-similar sets with overlaps and inverse theorems for entropy''

I am confused about the covering lemmas in the captioned work and really hope to get some ideas here. Firstly it is lemma 3.7. (Image of Lemma 3.7) (for convenience here is the lemma of this lemma (...
ericc's user avatar
  • 41
3 votes
0 answers
141 views

Is the Mandelbrot set weakly self-similar?

A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...
Sebastien Palcoux's user avatar
3 votes
0 answers
144 views

Causal fermion systems fromm fractal geometry

Okay, first off- I apologise if this is a stupid question. I'm mainly a very young physics guy, but this has primarily math basis. I'm trying to build a theory that is, long story short, some ...
Ringo Hendrix's user avatar
3 votes
0 answers
145 views

Diffeomorphisms preserving "nice" smooth functions

Let $\mathbb{R}^2\supset D=\{(x,y)\in\mathbb{R}^2|x^2+y^2<1\}$ be the open unit disc, and $U\subset\mathbb{R}^2$ be the interior of Koch's snowflake, as constructed in Falconer's book Fractal ...
A. S.'s user avatar
  • 51
3 votes
0 answers
95 views

Dimension of a graph

Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one? If not, what extra condition do we ...
Nikita Sidorov's user avatar
3 votes
0 answers
200 views

Product Fractals

Here is a theorem found in the Falconer's book on fractal geometry: Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$ $$ \dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...
user avatar
2 votes
1 answer
62 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 ...
Simple Conjugate's user avatar
2 votes
0 answers
46 views

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
  • 676
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
  • 319
2 votes
0 answers
90 views

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
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
2 votes
0 answers
93 views

Hausdorff dimension between $(1,2)$

Is there a number $c \in (1,2)$ for which there exist some interesting geometric property/properties which hold for every set of Hausdorff dimension in $(1,c)$ and does not hold for any set of ...
truebaran's user avatar
  • 9,140
2 votes
0 answers
47 views

Numerically Evaluate the limit of the solution of a functional equation

I need to evaluate the limit of $f(x)$ as $x\to0$, where the function $f$ solves the following equation: $$ f(x)=\left\{ \begin{array}{ll} g(x) & \text{if } x\geq \frac{1}{2};\\ \frac{1}{2} f(\...
guigux's user avatar
  • 607
2 votes
0 answers
102 views

Evaluating this limit in Fourier analysis

In my research work related to Fourier transform of the standard Cantor measure, I came across the following elementary problem: For $k\geq 1$, let $$ S_k=\sum_{m=0}^{3^k-1}\,\,\,\prod_{j=1}^\infty \...
Thomas Yang's user avatar
2 votes
0 answers
50 views

Quasilinear elliptic problem on fractal domain

Consider the following quasilinear elliptic equation $$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$ on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|_{\...
user avatar
2 votes
0 answers
78 views

Intersections of Sierpinski carpets with lines

Let $S$ be the Sierpinski carpet contained in the square $[0,1]^2$. For Lebesgue almost every $a\in [0,1]^2$ and every $\theta\in\mathbb Q$ the intersection of the line $L_{a,\theta}$ with equation $y-...
DavidHume's user avatar
  • 741
2 votes
0 answers
96 views

On the convergence problem of box counting for the Rössler attractor

So 5 month ago i posted this Question: Rössler attractor, Convergence of box counting to estimate the fractal dimension. Since then i have assumed, that the rate of convergence of $n(ɛ,n)$ (sum ...
Chopin's user avatar
  • 61
2 votes
0 answers
458 views

Rössler attractor, Convergence of box counting to estimate the fractal dimension

At the moment I want to estimate the fractal dimension of the Rössler attractor. I have written a program, which is counting the boxes hitted N(ɛ) (with ɛ := side length of boxes) by a trajectory on ...
Chopin's user avatar
  • 61
2 votes
0 answers
343 views

Relationship between the Hurst exponent and the alpha parameter

I have a question about the relationship between the Hurst exponent $H$ and the $\alpha$ parameter in the autocorrelation function when long memory is present. As we know in this case the decay of the ...
ggajardo's user avatar
2 votes
0 answers
105 views

Quantitative estimates on space filling curves

To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed ...
random_shape's user avatar
2 votes
0 answers
43 views

What is a precise definition of a twisted fibration of one fern over another?

I am curious if there is a notion of a "twisted fibration" of fractals. Since there are many classes of fractals, I'll ask specifically about L-systems. How can we precisely define the twisted ...
Miko Himmel's user avatar
2 votes
0 answers
84 views

Critical set of a self similar structure

A self similar structure is a triple $L=(K, S, \{F_i\})$ where $K$ is a compact metric space, $S$ is a finite set and for every $i\in S$ the functions $F_i:K\rightarrow K $ are injective and ...
YTS's user avatar
  • 141
2 votes
0 answers
166 views

A fractal object at origin but nowhere else: derived from Brownain motion

Hi, Please consider this object: Start with a realization of Brownian motion in 2D, which I'll denote by rho(t) where -infinity < t < +infinity. Next, lets smooth rho. There are various ...
chris in physics's user avatar
2 votes
0 answers
667 views

Self-similarity of Riemann's "non-differentiable" function

I hope it doesn't seem inappropriate for me to raise on MO an unanswered question from MSE, indeed a question actually posed there by someone other than myself. I want to ask the following: 1) ...
Kerry's user avatar
  • 779
2 votes
0 answers
313 views

Complexity of a variant of the Mandelbrot set decision problem?

This is a modified version of a question posted on StackExchange TCS. Mandelbrot set is defined using the complex equation $P_c (z)=z^2 +c$ where $c$ is a complex number. Let us define $M=${$(c,k,r)...
Mohammad Al-Turkistany's user avatar
1 vote
0 answers
27 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 $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$ ...
MikeG's user avatar
  • 665