Tagged Questions
1
vote
1answer
75 views
Heighway dragon and twindragon relation
The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1.
The ...
0
votes
2answers
53 views
Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?
Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer.
Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$
Is ...
1
vote
0answers
24 views
Do there exist periodic fractals $A_f$ of this type?
Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function ...
5
votes
1answer
199 views
Properties of the Mandelbrot set
Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology?
Considering the fact that the set is based on a quadratic function, are there any ...
1
vote
1answer
87 views
Classification of points in the Mandelbrot set
I am trying to understand the classification of points in the Mandelbrot set. There are an infinite number of baby Mandelbrots, each associated with a defined set of landing rays.
There are the pre ...
3
votes
1answer
73 views
Mandelbrot bulb's countable?
Are the Mandelbrot set's bulb's countably infinite?
My daughter asked me this question, after I pointed out that some Julia sets are a Cantor dust. For a point not in the Mandelbrot set, the ...
2
votes
0answers
71 views
Zoom out fractals?
It is well known that if we zoom in on the Mandelbrot set we get selfsimilarity.
So I wonder if $g$ is a fractal (in the complex plane) generated by a nonperiodic nonpolynomial entire function $f$
...
0
votes
0answers
78 views
About fractals , domains and iterations ( complex dynamics )
Let $f$ be an entire nonpolynomial function.
Consider $g:= f(f(f(...))).$
Define a hole as divergence surrounded completely by convergeance. ( a bit like the donut in topology )
Alot of questions ...
8
votes
2answers
184 views
Quadratic Julia sets and periodic cycles
Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting.
Infinity is always one ...
3
votes
1answer
151 views
Every basin of attraction contains a critical point?
Years and years ago, back when I first became interested in fractals [but didn't know much about anything], I vaguely remember coming across an interesting theorem. The gist of it was that "every ...
