All Questions
12 questions
12
votes
2
answers
750
views
Algorithm for computing external angles for the Mandelbrot set
Let $M$ be the Mandelbrot set: there exists a unique series
$$
\psi(z) := z + \sum_{m=0}^{+\infty} b_m z^{-m} = z - \frac{1}{2} + \frac{1}{8} z^{-1} - \frac{1}{4} z^{-2} + \cdots
$$
which defines a ...
- 32.4k
3
votes
0
answers
149
views
Is the Mandelbrot set weakly self-similar?
A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...
38
votes
1
answer
3k
views
Is the area of the Mandelbrot provably computable?
Recall the Mandelbrot set $M$ is the set of points $c$ in the complex plane such that the sequence $z_0 = 0, z_{n+1} = z_n^2 + c$ is bounded. It is well-known that $M$ is a compact set of positive ...
- 6,287
7
votes
3
answers
458
views
A question about Julia set for quadratic family
Let $P_{c}(z)=z^2+c$. It seems from the software that the map between the parameter $c$ and the Julia set $J(P_c)$ is an injective map. Is there some reference about it? Any comments and reference ...
- 1,706
44
votes
4
answers
8k
views
Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
I learned from Wikipedia that Gaston Julia died in 1978. Is it known if he ever got to see a computer-generated image of the set named after him?
- 457
27
votes
3
answers
948
views
A point set of power series with coefficients in {-1, 1}. Connected or not?
Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...
- 373
4
votes
3
answers
1k
views
Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision (...
- 445
6
votes
5
answers
2k
views
Precise location of the Mandelbrot Bulb Attachment to the main Cardioid
Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...
- 183
4
votes
2
answers
484
views
Algebraicity of the "outer" boundary of the Mandelbrot set
Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as
$$
t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu \...
- 3,897
9
votes
2
answers
1k
views
Is this a Julia set (and if so, for which function family is it the Julia set)?
Consider the function family given by $f_\lambda(z) = z - p_\lambda(z)/p_\lambda'(z)$ where $p_\lambda(z) = (z^2 - 1)(z - \lambda)$. Every attracting cycle and every rational neutral cycle of $f_\...
- 709
26
votes
6
answers
5k
views
Parametrization of the boundary of the Mandelbrot set
Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.
The ...
- 1,277
3
votes
1
answer
776
views
Attractive Basins and Loops in Julia Sets
I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with ...
- 41