Tagged Questions
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
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 ...
6
votes
1answer
184 views
The Mandelbrot Set Membership
To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions:
$$
\begin{cases}
z_{n+1} &= &z_n^2 + c &\text{ for }n\geq ...
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 ...
9
votes
1answer
195 views
Number of limit points of a continued exponential
Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
3
votes
1answer
130 views
coloring the inside point for Julia Fractal
I am trying to continuous coloring the inside point for a fractal image,such as $z \to z^2+C$. For those outside point, we can use the escape iteration to determine the color, just as the description ...
5
votes
2answers
341 views
Interesting but elementary properties of the Mandelbrot Set
I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know ...
0
votes
1answer
42 views
Why this Mandelbrot program add current point in the Mandelbrot
I read some posts about the Mandelbrot. I read that the Mandelbrot should be defined by $f(z)=z^2+C$. In my understanding, I think, the $C$ should be a constant, like $0.27$ or $2.1+4.5i$. However, in ...
3
votes
0answers
92 views
The Hausdorff dimension of the set of solutions of a system of coupled differential equations
I am interested in the relationship between non-linear differential equations and the Hausdorff, or fractal, dimension of the set of solutions.
For example, the Lorenz Attractor, which is a standard ...
14
votes
11answers
1k views
Mandelbrot-like sets for functions other than $f(z)=z^2+c$?
Are there any well-studied analogs to the Mandelbrot set using functions other than $f(z)= z^2+c$ in $\mathbb{C}$?
