22 votes
Accepted
How to add two numbers from a group theoretic perspective?
I think the point is that, forgetting the final carry, the group of $n$-digit binary words is isomorphic to $C_{2^n}$. In the simplest case, the group of 2-digit binary words is isomorphic to $C_4$, ...
- 3,228
20 votes
Accepted
A set whose Hausdorff dimension gradually changes?
I assume you want a set $A\subseteq [0,1]$ such that $\dim (A\cap [0,x])=x$ for all $x$. We can define $A_1$ by taking the union of a (Borel) subset of dimension $0$ of $[0,1/2]$ with a subset of ...
18 votes
Accepted
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
The sum is less than $1$. First of all, Mathematica and SAGE independently tell me that $$\sum_{n=1}^{10000} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}=0.950344\dots.\tag{1}\label{1}$$ We estimate the tail sum ...
- 115k
18 votes
Accepted
Can the topologist's sine curve be realized as a Julia set?
The answer is negative. Since every neighborhood of a point on the Julia set is mapped onto the whole Julia set by some iterate of the rational function, it follows that if a small piece of the Julia ...
- 96.8k
14 votes
What is known about the area of the symmetric Pythagorean tree?
The area is the rational number 12823413011547414368862997525616691741041579688920794331363953564934456759066858494476606822552437442098640979 / ...
- 241
13 votes
Accepted
Hausdorff dimension of the graph of an increasing function
Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$. Proof. Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph ...
- 28.7k
12 votes
Accepted
Does the family of fat Cantor sets contain a measurable rectangle?
No. The intuition here is that sets that are missing a lot of "diagonal strips" cannot contain large product sets (even if the strips are very narrow). For technical reasons it is ...
- 120k
10 votes
Cardinality of fractal without CH?
Let's define that a subset $F$ of the plane (or other suitable space) is a fractal, if there is a nontrivial group $G$ of homeomorphisms of the space, at least some of which are not isometries, such ...
- 247k
10 votes
Who proved that the Mandelbrot set's Julia sets are locally connected?
Nobody. This is the principal unsolved problem in the area, which is called MLC (That the Mandelbrot set is locally connected). Two Fields medals were awarded for partial progress in this problem. ...
- 96.8k
10 votes
Convex Julia sets
Edited: The previously found sufficient condition is indeed necessary, but even better, it is satisfied by all polynomials of degree at least two. Thus the conjecture is true: Theorem: Let $p$ be a ...
- 3,759
10 votes
Accepted
How to plot this fractal
The source info (War in the Age of Intelligent Machines) identifies the fractal as a Julia set, iterates of $z\mapsto z^2+z_0$. It has evidently been distorted (warped) to give it a 3D appearance. I ...
- 201k
10 votes
Accepted
Relationship between doubling constant of a metric space and of a metric measure space
Apart from the obvious counterexample of the measure being $0$, if $(X,d,m)$ is doubling in the sense of metric measure spaces it will be doubling in the sense of metric spaces. Consider a ball $B(x,r)...
- 13.1k
10 votes
Accepted
What is the limit of the sequence of iterated cosines?
You should have a look at Bob Devaney's notes entitled Complex Exponential Dynamics, which you can download from his web page: http://math.bu.edu/people/bob/papers.html While that paper focuses on the ...
- 2,148
9 votes
Accepted
Running most of the time in a connected set
The answer to this question is positive. A required path $\gamma$ can be constructed inductively using the following Lemma. For any continuum $P\subset\mathbb R^2$, distinct points $x,y\in P$, and $\...
- 44.3k
9 votes
Accepted
Continuous nowhere differentiability and constructive mathematics
The usual proofs are either constructive or can be made constructive fairly easily, sometimes by a slight weakening of the theorem. For example, let us read through this note by Brent Nelson. (Please ...
- 51.4k
9 votes
Accepted
Why in the Sierpiński Triangle is this set being used as the example for the OSC and not a more "natural"?
I guess that illustration relates to the paper Bandt, Christoph; Nguyen Viet Hung; Rao, Hui, On the open set condition for self-similar fractals, Proc. Am. Math. Soc. 134, No. 5, 1369-1374 (2006). ...
- 41.7k
9 votes
Convergence of the sequence $s_{n+1}=s_n^2-s_{n-1}^2$
Here is my approximation to the basin of attraction of $(0,0)$ for the map $F: (x,y) \to (y, y^2 - x^2)$ (shown in black).
- 54.7k
9 votes
Accepted
Does the airplane Julia set contain true circles?
The boundary of an attracting domain of $z^2+c$, with $c\neq 0$ cannot be a circle. This follows from the paper MR1031980 Zdunik, Anna Harmonic measure versus Hausdorff measures on repellers for ...
- 96.8k
8 votes
Accepted
$C^1$ function whose critical values are the middle thirds Cantor set
Let $X \subset [0,1]$ and $Y \subset [0,1]$ be two Cantor sets, meaning that they are closed, nowhere dense, and have no isolated points. Let $\psi : X \to Y$ be an order preserving continuous ...
- 161k
7 votes
Unexpected occurrences of the Sierpinski triangle
The Sierpinski triangle also emerges from a random geometrical process known as the chaos game. Consider the three vertices of any triangle, start from any one of them, and iterate the following ...
7 votes
Accepted
Singular distributions: Applications and Instances
In the so-called red and black, a player starts with a given fortune and wants to reach a given target. The reader may want to have a look at the exposition How to Gamble If You Must by Kyle Siegrist ...
- 86
7 votes
Accepted
Limit of homeomorphisms from square to square
Steve Ferry gave me an answer --- the answer is "no". In fact according to "A continuous decomposition of the plane into pseudo-arcs" by Wayne Lewis and John Walsh, there is a continuous subdivision ...
- 47.1k
7 votes
Who proved that the Mandelbrot set's Julia sets are locally connected?
If you mean are the Julia sets that correspond to parameter values of the Mandelbrot set connected, then I do believe that both Julia and Fatou proved this (in 1918 and 1916 respectively). are the ...
- 12k
7 votes
How can we not know the $s$-measure of the Sierpiński triangle?
The latest I could find is Móra, Péter, Estimate of the Hausdorff measure of the Sierpinski triangle, Fractals 17, No. 2, 137-148 (2009). ZBL1178.28007. where the best values are given as $$ 0.77 \...
- 41.7k
7 votes
Can the Mandelbrot set be designed through inequalities?
You should state your question more clearly. Yes, Mandelbrot set can be described by an inequality, namely $M=\{ z:u(z)\leq 0\}$ where $u(c)=v_c(c),$ and $$v_c(z)=\limsup_{n\to\infty}2^{-n}\log|(f^n_c)...
- 96.8k
7 votes
Accepted
Does finite Hausdorff dimension imply finite packing dimension?
A construction used (repeatedy) in the paper Edgar, G. A., Centered densities and fractal measures, New York J. Math. 13, 33-87 (2007). ZBL1112.28004. For more information, see that paper. We ...
- 41.7k
6 votes
Accepted
Box dimension of the graph of an increasing function
Pietro Majer's argument that you cited actually shows that the upper box dimension is $1,$ and hence the lower box, the upper and lower packing, and the Hausdorff dimensions are all equal to $1.$ Also,...
- 3,422
6 votes
Is the function Point -> Julia set "injective"?
You can deduce a positive answer to your question from Theorem 1 in the following paper: P. Atela, J. Hu, Commuting polynomials and polynomials with same Julia set, Internat. J. Bifur. Chaos Appl. Sci....
6 votes
Accepted
algebraic structure of fractals
It is a theorem of Douady and Hubbard that the hyperbolic points in the Mandelbrot set form a free noncommutative monoid in a natural way, and that this monoid has a natural action on the whole ...
- 58k
6 votes
Accepted
Does fractallity depend on the Riemannian metric?
The answer is no. Any two Riemannian metrics when restricted to a compact set are bi-Lipschitz equivalent and bi-LIpschitz homeomorphism preserves the Hausdorff dimension.
- 28.7k
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