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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14
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3answers
399 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-...
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0answers
31 views

Hausdorff dimension of cookie-cutter sets using topological pressure

I'm reading the proof of Theorem 5.3 from "Techniques in Fractal Geometry" by Kenneth Falconer - the theorem wich gives the Hausdorff dimension of a cookie-cutter set using topological ...
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0answers
79 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,...
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1answer
203 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 ...
1
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1answer
56 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-...
10
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1answer
246 views

Convex Julia sets

Consider the classical Julia set $J_f$ associated with $f(z)=z^2+c$. Since $J_c$ is completely invariant, we know that $f^{-1}(J_f) \subseteq J_f$. Now, let $H_f$ be the convex hull of $J_f$. Is it ...
1
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1answer
79 views

Entropy spectrum is not concave

Let $T:[0, 1]\rightarrow [0, 1]$ be map such that $T(x)=4x(1-x)$. For any $\alpha \in \mathbb{R}$, we define the level set as follows $$F(\alpha)=\{x\in [0,1]: \lim_{n\rightarrow \infty}\frac{1}{n}\...
3
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0answers
99 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 ...
12
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2answers
400 views

Algorithm for computing external angles for the Mandelbrot set

Let $M$ be the Mandelbrot set: there exists a unique series $$ \psi(z) := z + \sum_{m=0}^{+\infty} b_m z^{-m} = z - \frac{1}{2} + \frac{1}{8} z^{-1} - \frac{1}{4} z^{-2} + \cdots $$ which defines a ...
16
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3answers
2k views

How to add two numbers from a group theoretic perspective?

It is known that adding two numbers and looking at the carrying operation has a link with cocycles in group theory. (https://www.jstor.org/stable/3072368?origin=crossref) When we add two numbers by ...
3
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2answers
867 views

A set whose Hausdorff dimension gradually changes?

Can there be a set whose Hausdorff dimension gradually changes? For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ...
2
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0answers
78 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 ...
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0answers
171 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 ...
5
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0answers
84 views

How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?

Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask: Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
2
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1answer
161 views

Set operations over iterated function systems

An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically, $$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$ is an IFS if each $...
22
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1answer
657 views

How can we not know the $s$-measure of the Sierpiński triangle?

I'm preparing a presentation that would enable high-school level students to grasp that the (self-similarity) dimension of an object needs not be an integer. The first example we look at is the ...
5
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2answers
532 views

Who proved that the Mandelbrot set's Julia sets are locally connected?

I'd be greatly interested in a reference to the respective article. Was it Douady? Julia? Hubbard? Fatou? Bonus question: Who gave the proof that can be found in the Orsay notes? EDIT: The question ...
2
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0answers
36 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(\...
5
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1answer
183 views

Wavefront set of characteristic function of rough set

It is a standard exercise to show that if $X\subseteq\mathbb{R}^n$ has smooth boundary, then the characteristic function $1_X$ has wavefront set $$\{(x,\xi)\in\partial X\times\mathbb{R}^n\setminus\{0\}...
11
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1answer
584 views

The Koch snow flake, Holder exponents of conformal mappings

The Koch snow flake $K$ is a domain of $\mathbb{C}$, complex plane. Though, I do not state the precise definition, you can see the picture in wikipedia Koch snow flake. The Koch snow flake $K$ is a ...
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1answer
141 views

Does fractallity depend on the Riemannian metric?

Edit: According to comment of Andre Henriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...
10
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0answers
213 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 ${\rm II_1}$ ...
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1answer
68 views

Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ [closed]

Let $f:[0,1] \to \mathbb{R}, G = graph(f)$. If $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $0 = x_0< \ldots < x_m = 1 $ then $H^s(G) < \infty$ What technique can I use to ...
3
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0answers
252 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 (...
3
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0answers
94 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 $$ \...
1
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1answer
108 views

Pointless characterization relating between a fractal and its code space

Given an hyperbolic IFS $(X,\{f_i:i=1,\ldots,N\})$ and denoting its code space by $\Sigma_N = \{1,\ldots,N\}^{\mathbb{N}}$ and the generated fractal set by $\mathcal{A}$. There is a continuous and ...
2
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0answers
97 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 \...
2
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1answer
87 views

Box dimension as the critical value of the fractal content

Let $M \subseteq \mathbb{R}^n$ be bounded and $N_{\epsilon}(M)$ the minimum number of 'squares' of side $\epsilon$ with center in M necessary to cover $M$. The box dimension of M is then defined as $\...
1
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1answer
154 views

Formal justification of the Chaos game in the Sierpinski triangle

I want to justify why the Chaos game works to produce Sierpinski triangle. I use a theorem taken from Massopust Interpolation and Approximation with Splines and Fractals. Suppose that $(X,d)$ is a ...
6
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2answers
181 views

Box dimension of the graph of an increasing function

This Hausdorff dimension of the graph of an increasing function shows that: Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...
2
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1answer
121 views

Failure of Falconer distance problem in one dimension

I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question: For a compact set $E\...
2
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0answers
41 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|_{\...
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0answers
64 views

Set with modified lower box counting dimension strictly less than Hausdorff dimension

Please, can someone give (as simple as possible) example of the set for which modified lower box counting dimension is strictly smaller than Hausdorff dimension?
7
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0answers
171 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:=...
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0answers
96 views

Why do weak and L metric topology for measures coincide?

Let $(X,d)$ be a complete metric space, $$M^1 := \{\mu: \mu \mbox{ is a Borel regular measures having bounded support and } \mu(X) = 1\},$$ and $$BC(X) := \{f : f:X\rightarrow \mathbb{R} \mbox{ is ...
3
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0answers
96 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 $y \in F$, $\...
3
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1answer
195 views

Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$. There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
5
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1answer
268 views

continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only. Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
7
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0answers
114 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 ...
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0answers
110 views

How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
13
votes
1answer
406 views

Limit of homeomorphisms from square to square

Let $\square=[0,1]\times[0,1]$ be the unit square and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary. Assume $f$ is a limit of homeomorphisms $\square\to \...
5
votes
1answer
122 views

Is the function Point -> Julia set “injective”?

Consider the functions $f_c(z) := z^2 + c$ for $c \in \mathbb C$. For each such function, we may form the associated Julia set. My question: If $c, c' \in \mathbb C$ produce in this way the same Julia ...
0
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1answer
112 views

Laplacians and Renormalization

Why is the renormalization factor of the Dirichlet form for the graph approximations of the Sierpinski gasket (5/3)^m?
29
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1answer
675 views

Running most of the time in a connected set

Let $P$ be a compact connected set in the plane and $x,y\in P$. Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small? ...
2
votes
0answers
55 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-...
8
votes
1answer
155 views

Examples of probability measures with `fake' decay

To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ ...
6
votes
1answer
604 views

Hausdorff dimension of the graph of an increasing function

Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Let $\Gamma_f$ denote its graph. What can be said about the Hausdorff dimension of $\Gamma_f$? In ...
2
votes
0answers
79 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 ...
4
votes
1answer
219 views

algebraic structure of fractals

I am wondering whether there exists an algebraic structure(group or modules etc) possess some kind of self-similarity, (sub-group or sub-module have an identical structure with itself) and "...
1
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0answers
90 views

density of fractal measures

Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...

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