2
votes
0answers
76 views

Points in a general Cantor set

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
1
vote
2answers
37 views

Does there exist a Lipschitz map from the unit interval onto the unit square?

It is well-known that continuous space-filling curves exist. But can they be Lipschitz? Specifically, is there a Lipschitz map from [0,1] onto [0,1]x[0,1]?
3
votes
1answer
54 views

Can we construct a Koch curve with similarity dimension $s\in[1,2]$?

We can make a Koch curve $K$ with similarity dimension $s\in \mathbb Q \cap [1,2]$ by writing $s=\frac{p}{q}$, and constructing such a generator that by scaling with the factor of $2^q$, we'd find ...
0
votes
1answer
23 views

Minkowski content of a Cantor-like fractal

Let $K_0 = [0,1]$. Split $K_0$ into 4 intervals and remove the middle half. This gives $K_1 = [0,1/4] \cup [3/4, 1]$ and so on and set $K = \cap K_i$. I computed the upper and lower Minkowski content ...
2
votes
3answers
150 views

What is the topological dimension of the Peano curve?

The Hausdorff dimension of the Peano curve is know to be two. And I assume it to be a fractal since it's on the List of fractals by Hausdorff dimension. Moreover: According to Falconer, one of the ...
2
votes
1answer
50 views

Why do we require a finite number of subsets for self-similarity?

Here is how my text defines self-similarity: We call $M \subset \mathbb R^d$ self-similar if there are $T_1, \ldots, T_m \subsetneqq M$ and similarity maps $\alpha_1, \ldots, \alpha_m$ such that ...
6
votes
1answer
269 views

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
1
vote
0answers
24 views

Intersections of fractal sets with connected sets

Let $\beta \geq \alpha > 0$. Let $A\subset\mathbb R^n$ be a measurable set with Hausdorff dimension $\alpha$. Must there exist a closed connected set $B$ with Hausdorff dimension $\leq \beta$ ...
2
votes
0answers
71 views

What is the name of this metric: Why is $(\mathcal{M}, L)$ complete

I am reading section 4 of this article about invariant measures: http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf Let $(X,d)$ a complete metric space, ...
2
votes
1answer
54 views

Countability of “center” points of line segments in complement of Cantor set

So, start with the set [0,1] of the real line. Remove the middle third, and keep removing the middle thirds of the remaining line segments as usual when making the Cantor set. Each time you remove a ...
27
votes
5answers
721 views

Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
3
votes
0answers
82 views

Lipschitz continuity for an iterated function system

Let $(M,d_M)$ and $(N,d_N)$ be metric and $$ CB(M)=\{\mbox{all closed bounded subsets of }M\}. $$ Let $f: M\to N$ be a Lipschitz map with Lipschitz constant $L$. Define a map $$ F:(CB(M),\rho)\to ...
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
217 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 ) ...
4
votes
2answers
342 views

Is the number of circles in the Apollonian gasket countable?

Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar? What about ...