**2**

votes

**1**answer

101 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

**1**answer

243 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 ...

**6**

votes

**1**answer

105 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**

votes

**1**answer

247 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 ...

**-4**

votes

**1**answer

226 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 ...

**9**

votes

**3**answers

261 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 ...

**4**

votes

**3**answers

301 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 ...

**-1**

votes

**1**answer

132 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 ...

**0**

votes

**1**answer

138 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**

votes

**1**answer

198 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

**1**answer

208 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**

votes

**2**answers

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) \ ...

**5**

votes

**1**answer

135 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**

votes

**1**answer

82 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

**1**answer

103 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**

votes

**1**answer

258 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, ...

**9**

votes

**0**answers

118 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 ...

**4**

votes

**1**answer

171 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

**2**answers

259 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 ...

**5**

votes

**1**answer

953 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 ...

**7**

votes

**2**answers

151 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 $aB = \{ ax : x \in B \}$. ...

**5**

votes

**2**answers

206 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?

**4**

votes

**1**answer

129 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 ...

**3**

votes

**0**answers

78 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 ...

**13**

votes

**1**answer

567 views

### Relation between math and piano music

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

**9**

votes

**5**answers

436 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 ...

**3**

votes

**1**answer

148 views

### Correspondence between fractal sets and trees

In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not ...

**10**

votes

**1**answer

708 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 ...

**6**

votes

**1**answer

621 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 ...

**6**

votes

**1**answer

272 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 ...

**0**

votes

**1**answer

244 views

### Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1.
It is claimed that Lemma 2 is ...

**3**

votes

**3**answers

620 views

### L-systems and Sierpinski Triangle

I was just shocked when I saw these consecutive outcomes of an L-system converging to the Sierpinski triangle (shown in the picture below).
I'm interested to know how could one arrange the rules of ...

**0**

votes

**1**answer

327 views

### Sierpinski Triangle and the Chaos Game

The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...

**4**

votes

**4**answers

510 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**

votes

**2**answers

345 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 ...

**1**

vote

**1**answer

151 views

### Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain?

I have a finite set of points, and plot the graph log(N) vs. log(e). I see a polygonal chain (the final slope, starting at some size of e, is zero, of course). If the set represents some physical ...

**0**

votes

**1**answer

470 views

### What are integration on fractal? [closed]

Who can explain the proof of the formula (2.12) given here: J. Phys. A: Math. Gen. 20 (1987) 3861-3875. Printed in the UK ...

**8**

votes

**2**answers

826 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 ...

**5**

votes

**2**answers

267 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 ...

**2**

votes

**2**answers

317 views

### Visualizing the l-adic fractal in the partition function p(n)

This page http://www.aimath.org/news/partition/ and this youtube lecture http://www.youtube.com/watch?v=aj4FozCSg8g speak of a fractal in the values of the partition function p(n).
"We prove that ...

**2**

votes

**1**answer

272 views

### Can we extend a continuous function with keeping Hausdorff dimension?

Let X be a compact subset of R^d, and K be a compact subset of X, such that Dim_H(X)=Dim_H(K). Let F be a continuous function on K, Can we extend F from K to X, with keeping the continuous and the ...

**8**

votes

**2**answers

563 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" ...

**3**

votes

**2**answers

466 views

### Approximating fractal curves

Is there a known algorithm for approximating a fractal curve, say as specified by some iterative procedure e.g. a Koch snowflake, in terms of $f^{-1}(0)$ for some "simple" function $f$?
Specifically, ...

**11**

votes

**4**answers

726 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 ...

**2**

votes

**4**answers

839 views

### Fractals as solution to optimization problem?

What's the scientific reason for fractals being present in nature at such a large scale? Is it perhaps the solution of an optimization problem? For example, would the fractal based shape of certain ...

**2**

votes

**0**answers

153 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 ...

**2**

votes

**1**answer

397 views

### Hausdorff dimension of a subset of Cantor set

What is the Hausdorff dimension of the subset
$$F := \{ x = \sum^\infty_{n=1} \frac{2 x_n}{3^n} \in [0,1] : x_n \in \{ 0 , 1 \} , x_n = 1 \Rightarrow x_{n+1}=0 \}$$
of the Cantor set? Is it known ...

**1**

vote

**1**answer

317 views

### existence of fractal [closed]

I have a question about fractals;
Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F_\alpha$, such that $Dimension(F_\alpha)=\alpha$?
If yes, do we have any method to construct such ...

**2**

votes

**0**answers

451 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) ...

**3**

votes

**0**answers

223 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 ...