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

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22
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3answers
871 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 ...
18
votes
0answers
180 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 ...
14
votes
1answer
247 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}...
12
votes
2answers
356 views

Is $\frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?

Let $z$ be a complex number. Is the alternating infinite series $ f(z) = \frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ an entire function ? Does it even converge ...
12
votes
9answers
2k views

half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and $x&...
10
votes
2answers
1k views

Properties of the Mandelbrot set, accessible without knowledge of topology?

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 ...
10
votes
3answers
585 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 ...
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?
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) : ...
8
votes
1answer
135 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 ...
7
votes
3answers
470 views

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 ...
7
votes
1answer
92 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 = ...
7
votes
2answers
589 views

Simplest way to determine if a number is a member of the Mandelbrot set?

I'm writing JavaScript code to plot the Mandelbrot set on an HTML5 Canvas element. (That's probably not relevant to the answer to this question). A core part of the problem is to write a simple ...
7
votes
1answer
192 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 ...
7
votes
0answers
90 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{...
6
votes
2answers
275 views

Help locating mini mandelbrots

I would like to be able to list the coordinates of all the first level minibrots. Here is a picture of the mandelbrot set generated by fraqtive: zooming in to the circled area we see a slightly ...
6
votes
4answers
396 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/...
6
votes
2answers
343 views

How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
6
votes
2answers
110 views

Concrete example of an entire function wanted

Let $X$ be the space of entire functions on $\mathbb{C}$ endowed with the topology of uniform convergence on compact sets. Let $a$ be a nonzero complex number. Let $T: X\to X$ be defined by $T(f)(z)=f(...
6
votes
1answer
77 views

Perfect circles in the Mandelbrot set?

It is known that the boundary of the period 2 hyperbolic component of the Mandelbrot set is a perfect circle of radius $\frac{1}{4}$ centered at $-1$. Moreover it is known that the boundaries of the ...
5
votes
2answers
224 views

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 ...
5
votes
1answer
230 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. ...
5
votes
3answers
348 views

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 ...
5
votes
2answers
432 views

Are there mini-mandelbrots inside the julia set?

I've seen a julia set zoom but it is not nearly as interesting as a mandelbrot zoom. I also have not seen corresponding julia sets for zooms in the mandelbrot deeper than the original image. I'm ...
5
votes
2answers
202 views

Computing a Julia set

In Ermentrout's book he computes the Julia set for $z \mapsto z^2 + c$ by starting with a point inside the unit circle and then randomly choosing iterates $z\mapsto \pm\sqrt{z-c}$. Since the Julia set ...
5
votes
2answers
263 views

radius of convergence of half iterate of sinh(z)?

The half iterate of sinh(z) has a formal power series, centered around z=0. Does the formal power series for the half iterate converge at the origin? This is equivalent to asking if the half iterate ...
5
votes
1answer
520 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 ...
5
votes
2answers
306 views

Are there an infinite number of minibrots on the real line?

This is at 25 zooms using Fractal Extreme. The red circle indicates that there are more minibrots inbetween the small one and the large one. The pattern of super big (bottom right), medium-sized (top-...
5
votes
1answer
187 views

two fixed points, same fractional iteration

Suppose a function f(z) has two fixed points, one repelling, and the other attracting. Call the repelling fixed point f(-1)=-1, and the attracting fixed point f(+1)=+1. I'm interested in functions ...
5
votes
1answer
267 views

Fixed Point of a complex dynamical spiral system

Last semester I finished my first class on complex variables and of course we had to show that $i^i$ was real. That got me wondering about quantities like $i^{i^i}$ and similar power towers. For my ...
5
votes
0answers
130 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 ...
4
votes
2answers
505 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 ...
4
votes
2answers
151 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. ...
4
votes
1answer
436 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 .....
4
votes
1answer
243 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 ...
4
votes
1answer
213 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
273 views

The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.

Problem Is there an irrational $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ such that the set $S= \{\,\{2^N\alpha\} :N\,\in\mathbb{N}\}$ is not dense in $[0,1]$. Here $\{x\}=x-\lfloor x\rfloor$ is the ...
4
votes
2answers
185 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 ...
4
votes
2answers
221 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{...
4
votes
2answers
86 views

Given $f(z)=z^2+c$. Prove that $|z|>|c|+1$ implies $|f(z)|>|z|$

Consider the quadratic function $f(z)=z^2+c$. If $|z|>|c|+1$, show that $|f(z)|>|z|$. Edit: This is not a homework problem. I found this in my textbook.
4
votes
1answer
135 views

Divisibility of an expression involving the Möbius function

Let $d$ and $n$ be integers, with $n\geq 2$, and define $$R(n,d) := \sum_{k\mid n} \mu( n/k)d^k,$$ where $\mu$ is the Möbius function. It can be shown (see below for an arithmetic proof) that $n\mid R(...
4
votes
0answers
76 views

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 ...
3
votes
2answers
526 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
223 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? ...
3
votes
2answers
97 views

What is the functional inverse (with respect to $h$ (!)) of $f^{\circ h}(x)={F(h) +x F(h-1) \over F(1+h) +x F(h) }$?

I've considered the fractional iteration of $f(x) = {1 \over 1+x} $ for which the general expression depending on the iteration-height parameter $h$ might be assumed by the formula $$ f^{\circ h}(x) =...
3
votes
2answers
379 views

What is the shape of parabolic critical orbit?

The parabolic critical orbits of discrete dynamical system form n-th arm stars : which shapes are conjugated with "regular" n-th arm stars Here are 2 images of parabolic critical orbits for 2 ...
3
votes
1answer
84 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,-\...
3
votes
2answers
97 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 ...
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 ...
3
votes
1answer
201 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 ...