Complex dynamics is the study of dynamical systems of functions over complex numbers.

learn more… | top users | synonyms

0
votes
0answers
30 views

optimal escape requirements for a point in the mandelbrot set

When determining if a point c is in the mandelbrot set one can use the fact that if 2 < |z| for a given z in the sequence generated by: zn+1 = zn2 + c then the sequence will diverge. Is there ...
1
vote
1answer
15 views

Intersection of the domains of the inverses of a one-parameter family

Context. Let $f_c:\mathbb{C}\rightarrow\mathbb{C}$ be a holomorphic function, depending holomorphically on the parameter $c\in\mathbb{C}$. Let $\alpha_0$ be a geometrically attracting fixed point of $...
7
votes
1answer
88 views

What is the shortest path to a “little Mandelbrot” from $i$?

As you all already know, the Mandelbrot set has little "copies" of itself strewn throughout the boundary region (some of them distorted somewhat), and these are all connected. The point $i$ (or $x = ...
1
vote
1answer
43 views

Topics in Geometry and Dynamical Systems

Dynamical Systems/Fractal Geometry and Differential Geometry/Topology are really interesting areas of study. My question is whether there is any direct connection between them? In other words, are ...
2
votes
1answer
84 views

Is every basin of attraction completely invariant?

I can't seem to find a definitive answer in the literature. I believe the answer is yes, but my focus has been on the rational maps on the Riemann sphere. At the very least I'm confident that if the ...
1
vote
1answer
69 views

How many completely invariant domains can there be for a rational function?

I am considering rational functions $R:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ of degree $d\geq 2$. A completely invariant set $U$ is a set for which it and its complement are ...
18
votes
0answers
162 views

Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some ...
3
votes
2answers
96 views

Different Coloring of Julia Sets

I have known about Julia Sets for a while now, and today I had an idea about the coloring of Julia and Mandelbrot Sets. What if someone were to color them not only by how quickly z diverges, but also ...
1
vote
1answer
89 views

Newton's Method and Julia sets and the Douady rabbit. What's the relation?

I have question about Newton's method and the Douady rabbit. I know that : The Douady rabbit is a Julia set with a specific choice of the complex number $c$ in the polynomial $z^2+c$ The rabbit is ...
2
votes
0answers
58 views

Convergence results for Durand-Kerner method

I've been using the mpmath library's (in Python) implementation of the Durand-Kerner method to find the roots of some "not small" polynomials (this is the polyroots function). Typically my ...
1
vote
1answer
24 views

Hybrid equivalence of Polynomial-like maps

I am reading Douady and Hubbards "On the dynamics of polynomial-like mappings". I am relatively new dynamics of complex maps, and I would appreciate some help with aspects of the following. ...
8
votes
1answer
134 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
5
votes
0answers
125 views

Where does the series of real Mandelbrot lobes end?

In the Mandelbrot Set, c=0 has a periodicity of 1 and is surrounded by a cartoid of non-periodic points that asymptotically approach periodicty 1. Extending left along the real axis are connected ...
2
votes
1answer
40 views

Discrete systems with complicated basin boundaries?

I am trying to come up with the strategy to write my Master's thesis in mathematics. At the moment it is as follows: Finding a (preferably) discrete dynamical system that possesses at least 3 ...
3
votes
0answers
85 views

Is it known whether the boundary of the Mandelbrot set is not continuous?

I might be missing something obvious here, but my understanding is that nobody currently knows whether the boundary of the Mandelbrot set is a Jordan curve because otherwise we would know that the ...
3
votes
0answers
350 views

An interesting way to visualize the Mandelbrot Set. Proofs? Simplifications? Extensions?

This is a multi-part Question. Please chime in with any interesting insights in addition to Answers. I have noticed some interesting properties of Mandelbrot series that lead to a different way to ...
1
vote
1answer
49 views

Filled Julia set

How would I find the filled Julia set for $f(z)=z^3$? I know it should be the filled unit circle, but I don't quite understand the math. This is what I have so far: Fixed points $z^3=z$ so $z=1,-1,0,...
0
votes
1answer
82 views

Why is the Mandelbrot Set to the power of 2?

I've been looking at the Mandelbrot set and playing around with the powers on it. In the sense that I know the equation is $z^2 +c$, and I've been changing up the $2$. As it happens gradual increases ...
1
vote
0answers
48 views

Is there a readable proof of the fact that the filled Julia set $K_c$ of a quadratic polynomial is a Cantor set if $f_c^n(0)$ goes to $\infty$?

I found some proofs online but I don't consider them acceptable. http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf there is a "proof" (page 22) that doesn't satisfy me. I don't see why the ...
0
votes
0answers
30 views

Can a (non-identity) rational and transcendental function commute?

As part of an attempt to find a uniqueness theorem for solutions to the functional equation $\phi \circ f =g \circ\phi$ in the neighborhood of a shared fixed point $z=0$ for holomorphic pairs $f,g$ ...
1
vote
1answer
70 views

The mandelbrot fractal and it's relations to algebras and groups

I have been fooling around with Mandelbrot fractal to and fro for many years. One of the latest years I learned some general algebra with some of the most basic groups, like cyclic groups, dihedral et....
0
votes
0answers
38 views

$\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$

Let $t,x$ be nonnegative reals. Let $* ^{[k]}$ denote k th iteration. Find real-analytic $f(x)$ such that $\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$ Holds. We require analytic iterations. ( $ f^{...
1
vote
1answer
55 views

Constructing the Koenigs function about a repelling fixed point

My question is rather simple and I hope someone has some sort of an answer. I am looking for a simple yes or no answer, and a reference if anyone has one. We have a holomorphic function $f$ defined ...
7
votes
0answers
87 views

is the Buddhabrot well-defined?

Define the Mandelbrot set $M = \{ c \in \mathbb{C} : P_c^n(0) \not\to \infty \text{ as } n \to \infty \}$ where $P_c(z) = z^2 + c$. Define the complement of the Mandelbrot set $\overline{M} = \mathbb{...
2
votes
1answer
85 views

Question About Filled Julia and Julia Sets

Question: Let $Q_{c}(z) = z^{2} +c $ which $ c \in \mathbb{C}$ and suppose that $z_{0} \in K _{c}$ for the filled Julia Set, $K_{c}$ of $Q_{c}$. Suppose further that $z_{1} = Q_{c}(z_{0})$ and it ...
4
votes
1answer
429 views

Prove that the Mandelbrot Set Is A Closed Set

The Problem: Suppose we define the Mandelbrot Set as the following For $c \in \mathbb{C}$ , $\mathbb{M}$ = $({c:|c| \leq 2}) \cap ({c: |c^2 + c| \leq 2}) \cap ({c: |(c^2+c)^2 +2| \leq 2}) \cap .....
1
vote
1answer
50 views

Example of polynomial in dynamics

I am looking for an example of a post-critically finite polynomial $P$ (i.e. all critical points have finite orbit), which has both the following: a critical point on its Julia set (such as the ...
2
votes
0answers
29 views

Asymptotic values of Meromorphic map

$f(z)=\frac{e^z}{z+1}$ I know that $0$ and $\infty$ are two asymptotic values of the above function. Question:Does there exist another asymptotic values other than $0$ and $\infty$ ?
8
votes
1answer
218 views

Mandelbrot set of $c \cdot \cos(z)$

I'm given a task to write a program, that determines if a given point $c \in \mathbb{C}$ is in the Mandelbrot set of the function $$f_c(z) = c \cdot \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : ...
4
votes
2answers
472 views

How to compute a negative “Multibrot” set?

The Mandelbrot set is defined as follows: given the function f(z, c) = z2 + c, a number z in the complex plane is in the Mandelbrot set if and only if the sequence ...
3
votes
1answer
80 views

Show that the Julia set of a function is a Cantor set

I'm asked to show that the Julia set of the function $f(z)=z^2-6$ is a Cantor set contained in $[-3,-\sqrt{3}]\cup [\sqrt{3},3]$. I have identified $3$ as a fixed point of $f$, and found that $-3,-\...
2
votes
2answers
70 views

The fixed points of a rational function of multiplicity 2 or more are in the Julia Set

Definition: The multiplicity of a fixed point $z_0$ of $f$ is the order of the zero the the function $f(z)-z$ at $z_0$. I have already shown the iff statement: Claim: The multiplicity of fixed $z_0$ ...
3
votes
1answer
89 views

Question about Eremenko's paper on iteration of entire functions

I have two questions about Eremenko's paper On the iteration of entire functions On the second page, it says "The family $\{f^n\}=\{\underbrace{f\circ f\circ\ldots\circ f}_{n}:n\in \mathbb{N}\}$ is ...
1
vote
0answers
38 views

Are the iterates of this function bounded?

I have the function $f(z) = \sqrt z + C.$ For the value of $C = i$ (complex number), would the iterates be bounded or not? Iterating from $z = 0: f(0) = i, f(i) = \sqrt i + i$ and it goes on, ...
1
vote
2answers
138 views

Proof of x-intersection of the Mandelbrot Set?

I'm trying to prove that the Mandelbrot set intersects the X-axis on the interval [-2,.25]. I understand and have proven that the Mandelbrot set lies in a radius of 2. Mostly, I'm wondering how to ...
4
votes
2answers
144 views

What is the algorithm hiding beneath the complexity in this paper?

So, I am a computer scientist (at least, I'm working to become one..) and I asked a question on here concerning some mathematics behind the Mandelbrot set. A reply I recieved pointed me to this paper. ...
3
votes
1answer
194 views

Finding external angles for Misiurewicz points in the Mandelbrot set

In the Mandelbrot set for the quadratic polynomial $z \to z^2 + c$, rational external angles with even denominator are pre-periodic and have corresponding external rays which land at Misiurewicz ...
3
votes
1answer
64 views

How to prove that $F(x) = \lim_{n \to \infty} F^{n}( f ' (0) \cdot F^{-n}(x)) $?

Let $F(x)$ be a real-analytic function near $0$ ,with $0$ as one of its fixpoints and $f ' (0) > 1$. $$F(x) = F \circ F \circ F^{-1} = \lim_{n \to \infty} F^{n} \circ F \circ F^{-n} = \lim_{n \to \...
1
vote
1answer
48 views

Entire $f,g$ such that $f(f(z)) = p(g(z))$

Let $p(z)$ be a given polynomial of degree $3$. How to find nonconstant entire functions $f,g$ such that $f(f(z)) = p(g(z))$ or prove they do not exist ? I considered the related equation $f(f(z)) ...
5
votes
1answer
227 views

Finding the location of an image of the Mandelbrot set

I've got an image of a segment of the Mandelbrot set that I generated with an iPhone app a long time ago (I use it as my background image). I now have no idea where in the set the image came from. ...
6
votes
4answers
395 views

prove conjecture; the limit of iterating is $\sqrt{z^2 - 2}$

$$\lim_{n \to \infty} f_n(x)=x-\frac{1}{nx}\;\;\; g(x)=f_n^{on}(x)$$ The conjecture is for values of $|x|>\sqrt{2}$: $g(x) = \sqrt{z^2 - 2}$ This question comes from another matstack question/...
4
votes
2answers
220 views

About fractional iterations and improper integrals

Let $g(x,0) = x$ and $g(x,t+1) = g(x,t) - \dfrac{1}{g(x,t)}$ for every real $t$. From the fact \begin{align} \int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right)dx&=\int_{0}^{\infty}f\left(x-\frac{...
7
votes
1answer
190 views

Convex hull of the Mandelbrot set

What is the convex hull of the Mandelbrot set? I know that the leftmost point is $c=-2$ and I thought the extreme vertical points were $c=\pm i$. Sheldon's answers says they're not. I think that the ...
4
votes
1answer
208 views

Mandelbrot and Julia Set

Consider a dynamical system $$z_{n+1}=\frac{\alpha+z_n}{1+z_{n-1}}$$ for $n=0,1,2,\dots$ In other words the system is $$z_{n+1}=f_{\alpha}(z_n,z_{n-1})$$ where $f_{\alpha}$ is defined from $B(z,r)\...
4
votes
2answers
183 views

Is it possible to prove that some point belongs to Mandelbrot set? Is this an example of Gödel's theorem?

Everybody knows about Mandelbrot set drawing computer programs. Program takes some point, builds sequence from it, and if found that sequence goes out of circle with 2 radius, then knows that this ...
1
vote
1answer
46 views

Show that $-2, -1, 0, i$ lies in the Mandelbrot set but that $1$ lies outside of it

The Question Let $c$ be a complex number. The complex numbers $z_n(c)$ are defined recursively by $z_1(c)=c$, $z_{n++1}(c)=(z_n(c))^2+c$ for $n\geq1$ The Mandelbrot set is defined by $M=\{c\in\...
9
votes
3answers
3k views

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
1
vote
1answer
66 views

software for studying Newton iterates of complex map $z \mapsto -a + 1/z + b/(1+z)$

I am looking for flexible software for studying complex dynamics (Julia sets, Newton iterations) with user-specified rational functions. Specifically, I wish to study the Newton iterates of complex ...
22
votes
3answers
858 views

Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the ...
14
votes
1answer
242 views

Integral over filled Julia sets

Defining the usual quadratic Julia set iteration $f_c(z)=z^2+c$ for complex $c$, and its $n$th iteration $f^n_c(z)=f_c(f_c(\cdots f_c(z)\cdots))$, you can define a function of 4 variables $$c_{x,y,z,w}...