The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
2answers
231 views

Hausdorff dimension of inverse images.

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a continuous function. Let $t \in (\inf(f), \sup(f))$ and define $C = f^{-1} (t)$. Is it true that the Hausdorff dimension of C is $\geq d -1$? If no how does ...
2
votes
4answers
754 views

Fractal questions: Weierstraß-Mandelbrot

Hi, Coming from a specific field in algebraic geometry I am a total noob in Fractal Theory and I'd like to learn it a bit. I hope I am tolerated for my maybe-trivial questions. I just read about the ...
0
votes
1answer
2k views

PhD at 30 vs 33 [closed]

I'm almost 28. I have two bachelors degrees, all from the UK. One in Computing and the other in Electrical & Electronics Engineering. My area of interest is nonlinear dynamics/chaos and complex ...
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 ...
5
votes
5answers
982 views

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 ...
2
votes
1answer
414 views

Mandelbrot and “log-derivative”

I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent") \lim_{\epsilon -> 0} log(f(x+\epsilon) - f(x))/log(\epsilon). To simplify, lets assume that f is ...
3
votes
2answers
292 views

Hausdorff dimension of non-recurrent walks

Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is. Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving ...
21
votes
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 ...
10
votes
6answers
2k views

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 ...
7
votes
2answers
492 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 ...
1
vote
2answers
307 views

Fractional Gaussian noise in higher dimensions

I'm having difficulty imagining (or explaining to myself) how the partial sums a two dimensional Gaussian noise can produce a surface. According to equation (20) of the paper, "On two-dimensional ...
2
votes
1answer
399 views

Attractive Basins and Loops in Julia Sets

I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with ...
-1
votes
5answers
784 views

finding cutting edge papers and books

Hi all, what are the best strategies to find cutting edge papers and books on a field of mathematics? .. Example: 2-3 years ago I had to analyze a time series. I found a paper and showed that to ...
1
vote
1answer
649 views

Need help understanding Mandelbrot and Van Ness Fractional Brownian Motion

I need help understanding the Mandelbot and Van Ness' definition of Fractional Brownian motion $ B_H( t , \omega ) - B_H( 0 , \omega ) = \frac{1}{\Gamma(H + \frac{1}{2})} \left\( \int_{-\infty}^0 ...
2
votes
0answers
225 views

Complexity of a variant of the Mandelbrot set decision problem?

This is a modified version of a question posted on StackExchange TCS. Mandelbrot set is defined using the complex equation $P_c (z)=z^2 +c$ where $c$ is a complex number. Let us define ...
1
vote
2answers
346 views

Topologizing a free product G*H of discrete groups?

My -possibly flawed- mental picture of free products of groups certainly comes from the special case usually performed to illustrate the construction that proves the Banach-Tarski paradox. Thus I'm ...
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 ...
1
vote
1answer
185 views

Hausdorff dimension of higher powers of the Mandebrot set ?

My third question about Shishikura's result : Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper1. The Mandelbrot set is defined by ...
9
votes
2answers
1k views

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 ...
1
vote
1answer
252 views

Hausdorff dimension of subsets of the Mandelbot set.

Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper, but I can't figure out one thing : can we say all open subsets of this boundary has ...
6
votes
4answers
816 views

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 ...
2
votes
3answers
531 views

Hausdorff dimension: subset of $\mathbb{R}^n$ vs. boundary of this subset

Let $n$ be a positive integer. Let $S \subseteq \mathbb{R}^n$. Is the Hausdorff dimension of the boundary of $S$ always smaller than the Hausdorff dimension of $S$? I have not found anything ...
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 ...
5
votes
2answers
563 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 ...
0
votes
3answers
485 views

Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?

Ie, is there a way to probe it for regions of depth that involves a function, the domain of which is the Mandelbrot set itself, or a part of that set?
7
votes
2answers
288 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 ...
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?
28
votes
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
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 ...
7
votes
2answers
571 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, ...
2
votes
1answer
306 views

Is there an nontrivial function whose 'period paralellograms' are Gosper Islands?

The Gosper island tiles the plane, so I'm curious if a nontrivial elliptic? function exists which would have a 'period gosper-island' instead of a period parallelogram. In this case, I'm using ...
7
votes
2answers
738 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 ...
3
votes
2answers
358 views

Systematization of deterministic and stochastic integrals

With this question I try to build up a systematization of different kinds of integrals. The following table differentiates between deterministic and stochastic integrals, the summation processes ...
4
votes
4answers
532 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 ...
5
votes
1answer
984 views

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?