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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5
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2answers
433 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 ...
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35 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(\...
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62 views

What is the geometric or dynamic meaning of a global attractor with an infinite fractal dimension?

In Efendiev-Ôtani's article: Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium (Ann.I. H. Poincaré, AN 28,2011), is obtained that the fractal dimension ...
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1answer
116 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\}...
10
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1answer
377 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
135 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 ...
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182 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
64 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 ...
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18 views

Bounds on the Hausdorff dimension for an IFS satisfying the OSC

Let $([a,b],F)$ be an IFS where $F$ satisfies the open set condition for an open interval $G$. Furthermore, assume that each $f_i \in F$ satisfies: $$\forall x,x' \in \stackrel{-}{G}.q_i |x-x'| \le |...
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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 (...
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81 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 $$ \...
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1answer
107 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 ...
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93 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
81 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 $\...
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1answer
126 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 ...
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2answers
163 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 = ...
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1answer
97 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\...
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33 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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60 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?
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158 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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86 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 ...
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85 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 $y \in F$, $\...
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1answer
187 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 ...
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1answer
225 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 ...
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107 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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99 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
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1answer
374 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 \...
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1answer
114 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 ...
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1answer
110 views

Laplacians and Renormalization

Why is the renormalization factor of the Dirichlet form for the graph approximations of the Sierpinski gasket (5/3)^m?
28
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1answer
631 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
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53 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
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1answer
142 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
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1answer
450 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
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0answers
73 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
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1answer
164 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 "...
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84 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 ...
3
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1answer
102 views

Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational maps (or not)

Is it known that local connectivity of the Mandelbrot set (MLC) is sufficient prove the density of hyperbolic conjecture of qudratic family. I wondered is it known that the MLC is not enough (or ...
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0answers
721 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{...
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1answer
340 views

Cardinality of fractal without CH? [closed]

This appears unanswered on reddit. What are possible choices for the cardinality of plane or line fractal without assuming CH? Are there fractals for which the answer is not easy? (Sieprinski ...
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0answers
104 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 ...
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0answers
105 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 ...
6
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1answer
106 views

Equi-Hölder embeddings of compact metric spaces of finite packing dimension into $\ell_2$

Problem. Does a compact metric space of finite packing dimension admit an equi-Hölder embedding into a Hilbert space? A map $f:X\to Y$ between metric spaces $(X,d_X)$, $(Y,d_Y)$ is called equi-Hölder ...
2
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0answers
186 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 ...
7
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0answers
251 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 (...
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107 views

Hausdorff dimension of $X\times X$

I am thinking of the following question: Let $X\subseteq \mathbb R$. Is it true that $$ \mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)? $$ My thoughts: We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(...
7
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2answers
433 views

Complexity of the Mandelbrot set on rationals

Given two rationals $a,b \in \mathbb{Q}$, call $c = a + ib$, i.e., the complex number represented by these two rationals. A point $c$ is contained within the Mandelbrot set $M$ if the following ...
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148 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 ...
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1answer
99 views

Literature on the total variation of fractal graphs/fractal Brownian motion?

I know that for standard Brownian motion, the total variation sampled at intervals of length $\Delta$ converges to $V(\Delta) = C \Delta^{-1/2}$ for some constant $C$. I wish to use this fact to study ...
14
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2answers
918 views

♢ ⧫ ⬠: the fourth kind of Penrose tiling?

It’s known that Penrose tilings have several implementations that are mutually locally derivable; but the sources (such as en.wikipedia) list no more than three essentially different variants. There ...
6
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1answer
171 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 ...