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-1
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1answer
96 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
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1answer
76 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 ...
1
vote
1answer
121 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 ...
5
votes
1answer
147 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
396 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 ...
23
votes
2answers
922 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
1answer
104 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 ...
3
votes
1answer
95 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 ...
28
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5answers
2k views

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 ...
0
votes
1answer
75 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 ...
10
votes
1answer
650 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 ...
1
vote
2answers
257 views

Terrain Generation: Infinite 2D space filled with Diffusion-limited aggregation clusters?

Disclaimer: I don't have a deep understanding of fractals or any higher math, I'm just personally interested in it, so please excuse me if I'm using wrong terms or if I'm being inaccurate. Making ...
3
votes
1answer
200 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
0answers
107 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
167 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 $K(f)$ ...
5
votes
2answers
235 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
1answer
550 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
2answers
135 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 $aB = \{ ax : x \in B \}$. ...
6
votes
2answers
188 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
1answer
124 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
0answers
71 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
1answer
482 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
5answers
413 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
5answers
815 views

Reference for the iterated function system of the Koch snowflake

Let $KS$ be the Koch snowflake. This fractal has an iterated function system (IFS) of the form $$ KS = \bigcup_{0 \leq k \leq 6} f_k(KS) $$ with $$ f_0(z)=\frac{1}{\sqrt{3}} e^{i\pi/2} z $$ and for $0 ...
6
votes
1answer
611 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 ...
3
votes
1answer
141 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 ...
6
votes
1answer
238 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
1answer
232 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
3answers
499 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
1answer
289 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
4answers
466 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
2answers
332 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
1answer
134 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
1answer
433 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 ...
13
votes
2answers
1k views

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?
5
votes
2answers
247 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 ...
13
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3answers
507 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 ...
2
votes
2answers
285 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 ...
5
votes
2answers
526 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 ...
8
votes
2answers
522 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
2answers
446 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, ...
2
votes
1answer
251 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 ...
0
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2answers
806 views

What are the fractal parameters?

There are so many fractal which are not uniquely characterize by some fractal parameters like Fractal dimension, Succolarity, Lacunarity, Morphological entropy. Can you suggest some fractal parameters ...
11
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4answers
662 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
4answers
772 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 ...
0
votes
1answer
395 views

Is the notion of fractional dimension compatible with considering a dimension a set of n-tuples?

Hi there, I am trying to teach a friend about higher-dimensions and I have explained them in the following two manners: A higher dimension, e.g. the 56th is the set of all 56-tuples. A fractional ...
2
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0answers
444 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) ...
2
votes
0answers
152 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
1answer
373 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 ...
21
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6answers
4k views

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 ...