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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How to define a differential form on a fractal?

It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g. the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one ...
25
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2answers
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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) \ ...
21
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6answers
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How might M.C. Escher have designed his patterns?

I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it ...
15
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6answers
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Parametrization of the boundary of the Mandelbrot set

Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question. The ...
13
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1answer
727 views

Relation between math and piano music

What, if any, is the relation between Cantor's function and Ligeti studio: Devil's Staircase?
13
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533 views

Julia sets using other fields

I hope I am forgiven for my noob question. But, does it make sense to think of Julia sets using other fields? More precisely I would like to think of fields in which closed and bounded isn't ...
13
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2answers
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Does “Algebraic numbers coloured by degree” form a fractal?

This picture from Wikipedia's article on Algebraic numbers shows a visualization of Algebraic numbers coloured by degree. I'm wondering if this is a fractal?
12
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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 ...
12
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Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?

Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system (IFS)? Julia sets can be formed by iterating the two functions $z \mapsto \pm \sqrt{z-c},$ and ...
12
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2answers
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Area of the boundary of the Mandelbrot set ?

My second question about Shishikura's result : Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper 1. In a sense, could we consider it ...
11
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1answer
859 views

Analysis of the boundary of the Mandelbrot set

Motivation: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite ...
11
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4answers
888 views

Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
11
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2answers
458 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)= ...
11
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1answer
263 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 ...
10
votes
2answers
457 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 ...
10
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1answer
259 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 ...
9
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5answers
453 views

Iterated function system on the plane

Let $r_1, r_2, r_3$ be three nonnegative real numbers with $r_1^2+r_2^2+r_3^2 <1$. Can you find three similitudes $f_1,f_2,f_3$ on $\mathbb{R}^2$ with similarity ratios $r_1,r_2,r_3$ resp. and a ...
9
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0answers
135 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 ...
8
votes
2answers
945 views

sequences with a fractal dimension

This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...
8
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2answers
700 views

Fractal Tiling of Rhombic Dodecahedra

Hello, this is my first question on Math Overflow... Rhombic dodecahedra can be tiled in 3-space, leaving no gaps. This tiling corresponds to the close-packing of spheres. Consider a "nucleus" ...
7
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2answers
761 views

Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?

Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...
7
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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?
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510 views

local behavior of a finite Borel measure

Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I ...
7
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1answer
180 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 ...
7
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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
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2answers
598 views

Shortest Paths on fractals

How can one find shortest paths between 2 specified points on fractals, or (since I'm pretty sure this is quite complicated) make useful generalizations about them? Since the above question is broad, ...
7
votes
1answer
271 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
votes
2answers
309 views

Minimum number of contractions needed to obtain a particular invariant set

Consider the Koch curve $G \subseteq \mathbb{R}^2$. Clearly $G$ is the invariant set (IS) of the iterated function system (IFS) $\lbrace \phi_1, \phi_2, \phi_3, \phi_4 \rbrace$. Where (not wanting to ...
6
votes
4answers
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Measure 0 sets on the line with Hausdorff dimension 1

I use $\dim_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if ...
6
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1answer
323 views

Arithmetic products of Cantor sets.

Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product $AB=\lbrace ab|a\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are ...
6
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293 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 ...
6
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1answer
151 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 ...
6
votes
1answer
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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 ...
6
votes
2answers
591 views

Symmetries of the Julia sets for $z^2+c$

The julia set seems to have symmetries roughly corresponding to translation, rotation and scaling. In the following image You can see the horizontal translation, which leaves the extremal left and ...
6
votes
1answer
198 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 ...
6
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1answer
200 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 ...
6
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1answer
122 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 ...
6
votes
1answer
640 views

Is Gouvêa-Mazur's “Infinite Fern” a fractal?

[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that I do not know what the right definition of a fractal in the following question should be. But a nice answer might contain such a ...
5
votes
5answers
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Hausdorff dimension for invariant measure?

A fractal set has a Hausdorff dimension. In some cases, we may generate a fractal by iterating $f,$ and let the fractal be the set of starting points $x$ such that $|f^{\circ n}(x)|$ is bounded as ...
5
votes
1answer
278 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 ...
5
votes
2answers
289 views

Measures of full Hausdorff dimension for self-affine sets

Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set ...
5
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1answer
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Self-similar matrices? [closed]

Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?
5
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1answer
159 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 ...
5
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2answers
570 views

A calculus question related to quantization dimension

Going through some old papers, I came up with a simple-looking problem I thought about 5 years ago or so. MO wants motivation ... Associated to a probability measure on a metric space is something ...
4
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4answers
558 views

Lecture on Fractals for Middle School Students

I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject. I'll show some fractal images and a few ...
4
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3answers
363 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 ...
4
votes
3answers
729 views

Estimating the fractal dimension of a point cloud

I have finite set of geolocation point data, and I'd like to estimate the fractal dimension. I know there are several ways to do this, and some of them give different numbers. What is the most ...
4
votes
4answers
602 views

Determining a lower bound on the Hausdorff dimension of a set

Does anyone know of a good method for finding a lower bound of the Hausdorff dimension of a set $G$? The only method I could find is to find an $\alpha$-Hölder function $f \colon G \to H$ then ...
4
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1answer
136 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 ...
4
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2answers
364 views

Algebraicity of the “outer” boundary of the Mandelbrot set

Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as $$ t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu ...