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

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

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. ( $ ...
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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 ...
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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} = ...
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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 ...
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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 ...
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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 ...
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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 $\infty$ ?
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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) : ...
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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 ...
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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 ...
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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 $z_0$ ...
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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 ...
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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, ...
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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 ...
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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. ...
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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 ...
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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 ...
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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)) ...
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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. ...
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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 ...
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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} ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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Blow up in holomorphic dynamics

Someone could explain the concept of blow up used in holomorphic dynamics? Specifically in the context of iteration of holomorphics functions. This concept could be taken to some of the deformation ...
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More specific question on finite Blaschke products

I would like to make my previous question more precise. (1) If $B$ is a finite Blaschke product such that its Julia set $J_B$ is a Cantor subset of $S^1$ then is it true that $B$ is expanding on ...
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Julia sets of finite Blaschke products

Is there any book or paper where I can find a detailed discussion of the character of Julia set (Cantor set or the unit circle) depending on zeros and the constant unitary factor of a finite Blaschke ...
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Mandelbrot sets and radius of convergence

While watching this Numberphile video on Mandelbrot sets, it's more or less stated that the fractal will "blow up" if it's radius of convergence is greater than 2. What is the mathematical basis for ...
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Stable manifold for bidimensional nonlinear dynamic system with complex eigenvalues

Given a autonomous nonlinear dynamic system of the form $$f(x,y)=\begin{bmatrix} B_1 x + g_1(x,y) \\ B_2 y + g_2(x,y) \end{bmatrix}$$ with $B_i\in\mathbb{R}$ (bidimensional problem), with $g_i$ ...
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Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
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191 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
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Are mini-Mandelbrots known to be found in any fractals other than the Mandelbrot set itself?

This is a generalization of the question Are there mini-mandelbrots inside the julia set? @Hagen raises an issue I was afraid of, which is that even the mini-Mandelbrots in the Mandelbrot set are not ...
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Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
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Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...