0
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1answer
32 views

Reference - formal characterization and analysis of Koch curve

I am studying the Koch curve but most resources I have seen do not describe the Koch curve formally and are similar to the Wikipedia page on the subject. For example, I have looked at books like ...
0
votes
0answers
14 views

How do you detect when an iterated function converges / diverges and calculate limit accurate-enough?

See this post for a background on what I'm doing. So how many iterations $N$ of $P_c(z)$ does it take so that if $f(c) = P_c^N(z)$, and $g(c) = $ infinite iterations of $P_c(z)$, then $h \circ ...
5
votes
1answer
45 views

Reference for a Cantor set in the plane formed from series of roots of unity

This is a long shot, but I'm looking for a particular article that I once read, and I'm trying to find it again. It deals with a certain Cantor set in the plane. The set could be written as something ...
5
votes
0answers
102 views

Fractal dimension of Gaussian white noise is infinite?

I read in this paper that the fractal dimension of Gaussian white noise is infinite. The paper does not prove it nor give a reference to support it. I failed to find a reference from online searching. ...
7
votes
1answer
356 views

Self Study of Fractals

I am looking for a book to self-study fractals with a certain criteria. I have checked out Getting Aquainted with Fractals. Note that Getting Aquainted with Fractals does not include ...
8
votes
1answer
324 views

Regular open set whose boundary has nonzero volume.

I found this question quite interesting, but its answers were disappointingly non-geometric. I'd be interested to know whether there exists a geometric example. To be precise about what I mean by a ...
2
votes
1answer
85 views

Notation Clarification of Koch Curve

I am having trouble making sense of the notation used to describe the Koch Curve in the book Getting Aquanted with Fractals. The link will take you to a preview of the book which describes the ...
5
votes
0answers
333 views

The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.

The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric, $$ d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \} ...
3
votes
1answer
128 views

Is the ball measure of non-compactness a Lipschitz map?

Let $(M,d)$ be a metric space and let $H(M)$ denote the set of closed and bounded subset in $M$. Then $(H(M),d_H)$ is a metric space where $d_H$ denotes the Hausdorff distance. Let $\chi$ be the ...
0
votes
1answer
215 views

Relation between Hausdorff metric of and Hausdorff measure of non-compactness

Let $(X,d)$ be a metric space and let $$K(X)=\lbrace Y\subset X\colon Y\text{ is non-empty and compact}\rbrace.$$ Endow $K$ with the Hausdorff metric (which is the natural metric on this space, see ) ...