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 ...

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60 views

Usable Change-of-Variables Formula for Hausdorff Measure

Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
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0answers
105 views

Can infinitely many orbifolds be “added up” to form a fractal space?

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it. Intuitively, an orbifold, from what I understand, is a "...
5
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2answers
137 views

Integral over the Cantor's set Hausdorff dimension

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...
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7answers
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Unexpected Occurences of the Sierpinski Triangle

The probably most wellknown occurance of the Sierpinski Triangle is as the odd entries of the Pascal triangle Some month ago however, there was an article about mathematical models of sandpiles along ...
3
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147 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+\...
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54 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 ...
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0answers
43 views

Fundamental system of neighborhoods. Self similar set

I have been reading the text Analysis on Fractals of Jun Kigami. There is a theorem about the fundamental system of neighborhoods of a point in a self similar set. It is stated as follows Let $\...
6
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139 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 ...
27
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3answers
816 views

A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set $$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$ What can be said about the set $M$ ...
16
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666 views

Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let $\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals. Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{...
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66 views

Quantification of the extent of periodicity in a time series using fractal analyses

I need metrics to quantify and compare the extent of periodicity between any two given time series, considering the time series were "almost periodic". By "almost periodic" I mean: if I were to take ...
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55 views

What results exist for functions with regionally fluctuant fractal dimension?

I'm interested in functions that have a varying fractal dimension at different scales and/or regions. Has this been investigated in detail? I'd be interested in results and references in this area of ...
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3answers
581 views

Self-Similar Graphs

Many fractals can be generated using and infinite sequence of graphs. For example, Sierpinski's Gasket could be generated by the following sequence of graphs. Many definitions of fractal dimensions ...
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1answer
185 views

Lightning strike fractal formula [closed]

I need to generate random gold ore channels for a game, I was thinking they would look kinda like lightning strikes. Anyone know any good fractals (recursive functions) that looks like it? Or ...
10
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1answer
359 views

Are there any exact results for Hausdorff Measure?

The computation of the Hausdorff measure is extremely difficult due to the infinum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult ...
0
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0answers
62 views

Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series? I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...
12
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2answers
625 views

What is the Hausdorff dimension of this fractal?

Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= \sum_{i=h}^\...
3
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1answer
71 views

Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension

Call a Borel set $A \subseteq [0,1]$ good if $$0 < \dim(A) \leq \overline{\dim_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim_\text{M}}(A)$ is ...
7
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1answer
285 views

Fractal dimension of scaling limits of discrete structures

Let $S$ be the set of positive integers whose base-three expansion contains only the digits 0 and 2. The discrete set $S$ in a sense has (negative) fractal dimension $(\log 1/2)/(\log 3)$, since if ...
11
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2answers
523 views

On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e., $$ \mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}. $$ Then, as is well known, $\mathcal T$ has a non-...
6
votes
1answer
282 views

Precise density estimates for Cantor sets

Let $C_\lambda$ be the classical Cantor set associated to a real number $0<\lambda<\frac{1}{2}$, as defined for example in the book of K. J. Falconer The geometry of fractal sets. I recall ...
2
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2answers
227 views

Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?

For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ : Is it possible to find a part of the parameter plane, scanned with a given limited precision (...
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1answer
160 views

What is “graph-directed iterated function”?

Im translating an article about Rauzy fractal and I ran into this sentence: ...
6
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1answer
187 views

Spirals in Apollonian circle-packings

Given mutually (externally) tangent circles $C_1,C_2,C_3$, let $C_n$ be the unique circle externally tangent to $C_{n-1}$, $C_{n-2}$, and $C_{n-3}$ for $n \geq 4$. Let $P_{\infty}$ be the point toward ...
3
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1answer
240 views

“Nice” functions on infinite-dimensional space of germs of continuous functions at a point

Consider set of all germs of continuous functions at some point. Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
10
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2answers
385 views

Nim and the Sierpinski Gasket

(I discovered this in high school, sent it off to a journal, never heard back, and forgot about it. I've never found anyone else who appeared to know about it; the combinatorial game theorists I've ...
6
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1answer
217 views

Isotropy of Apollonian disk-packing

Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...
3
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1answer
148 views

boundary density of the Von Koch flake

Given a measurable set $K\subset \mathbb{R}^d$ we consider the occupation ratio $$f_r(x)=vol(K\cap B(x,r))/r^d$$ and especially the asymptotics when $r\to 0$. When $K$ has a fractal boundary and $x$ ...
7
votes
1answer
296 views

For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?

The fractal dimension of the 3D Apollonian packing is computed in this paper. In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (...
7
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2answers
197 views

Angles and proportions occurring in L-system fractals

This is about properties of certain fractals defined by Lindenmayer systems, a.k.a. L-systems. Unlike “classical” fractals like Julia sets or the Mandelbrot set (the name “set” says it all), these ...
11
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286 views

Cantor set intersecting a geometric sequence

I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
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294 views

Proof of im/possibility of constructing any fractal by iterated function systems? [closed]

Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal(object of any non-integer dimension) can be constructed by ...
12
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3answers
506 views

Dimensions of self-affine sets

Let $A$ be a $2\times 2$ matrix which we assume to be contracting, i.e., the exists $\alpha\in(0,1)$ such that $$ \|A {\mathbf x}\|_2\le \alpha\|{\mathbf x}\|_2,\quad \forall {\mathbf x}\in\mathbb R^...
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3answers
504 views

Precise location of the Mandelbrot Bulb Attachment to the main Cardioid

Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...
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1answer
216 views

notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals? the motivation for this question is: fractals are very difficult mathematical objects to work with, and many problems/questions ...
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1answer
326 views

Singular distributions: Applications and Instances

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...
5
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1answer
369 views

Smallest positive zero of Weierstrass nowhere differentiable function

Consider the Weierstrass nowhere differentiable function $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos(4^n \pi x)$. It seems that the smallest positive zero of $f(x)$ occurs at $x=\frac{1}{5}$, but I ...
2
votes
1answer
448 views

Level sets of a Weierstrass nowhere-differentiable function

Can anyone describe level sets of a Weierstrass nowhere-differentiable function? For example, let $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos( 4^n \pi x)$. For some $c \in (-2,2)$, what is known ...
25
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2answers
1k views

Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group generated by the permutation $$ a: \ (m,n) \ \mapsto \ (m-n,m) $$ of order $6$ and the involutions $$ b: \ (m,n) \ \...
6
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1answer
189 views

Escape Time of Fractional Brownian Motion

Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$. Is the expected time known ...
0
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1answer
89 views

Constructing measures with support in a given set

I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...
3
votes
1answer
135 views

How is the Fractal Dimension of a Parametric Curve Related to the Fractal Dimensions of its Coordinate Functions?

Question: Suppose the fractal dimension $1\le d_c\le2$ of a planar parametric curve $c(t) := (x(t),y(t))$ is given; can any nontrivial estimates for the fractal dimensions $d_x$ of $x(t)$ and $d_y$ of ...
3
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1answer
389 views

Existence and Properties of 3D Curves with unusual 2D $(\kappa(s),\tau(s))$ Trajectories

This question is inspired by Surface in 3D that realizes all pairs of principal curvatures While one can imagine, that a 3D surface could exist, that realizes all pairs of principal curvatures, ...
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159 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 ...
3
votes
1answer
194 views

Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set : $$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$ Question : If $K(f)$ ...
6
votes
2answers
336 views

function that is the average of affine transformations of itself

Consider the function $f : \mathbb{R} \to [-1,1]$ with $$ f(x) = \begin{cases} -1 & x \le -1 \\ +1 & x \ge +1 \\ \frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\...
7
votes
1answer
3k views

Relationship between fractal dimension and Hurst exponent

For many basic random processes (like fractional Brownian motion) Hurst exponent "H" and fractal dimensions ( Hausdorf = Minkoswki typically) "D" are realated by simple formula D = 2-H. I want to ...
8
votes
2answers
206 views

Isometrically-invariant measures and dilation of the Cantor set

Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. ...
7
votes
2answers
327 views

What one really can do with fractals built from L-systems?

For any L-system one can naturally associate a fractal. Why these fractals are (mathematically) useful apart that they are a source of nice pictures?
5
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1answer
156 views

Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...