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

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20
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3answers
492 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
119 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 ...
12
votes
2answers
326 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 ...
11
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 ...
9
votes
3answers
397 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 ...
7
votes
3answers
600 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?
7
votes
3answers
371 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
104 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 ...
6
votes
4answers
343 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 ...
6
votes
1answer
638 views

Properties of the Mandelbrot set

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 ...
6
votes
2answers
202 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 ...
6
votes
2answers
90 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 ...
6
votes
2answers
242 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 ...
5
votes
2answers
206 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
120 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
271 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
196 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 ...
5
votes
2answers
187 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
2answers
197 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 ...
5
votes
0answers
145 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 ...
4
votes
1answer
173 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
3answers
154 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 ...
4
votes
1answer
96 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 ...
4
votes
2answers
159 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 ...
4
votes
2answers
103 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
1answer
341 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 ...
4
votes
2answers
189 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} ...
4
votes
2answers
82 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
123 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 ...
4
votes
1answer
207 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 ...
3
votes
2answers
394 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
2answers
90 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
1answer
50 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 ...
3
votes
0answers
22 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
0answers
72 views

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 ...
3
votes
1answer
238 views

Zoom out fractals? (A question about selfsimilarity)

It is well known that if we zoom in on the Mandelbrot set we get selfsimilarity. So I wonder if $g$ is a fractal (in the complex plane) generated by a nonperiodic nonpolynomial entire function $f$ ...
2
votes
2answers
285 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 ...
2
votes
2answers
183 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 ...
2
votes
2answers
96 views

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 $g_i$ ...
2
votes
1answer
127 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? ...
2
votes
1answer
172 views

Heighway dragon and twindragon relation

The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1. The ...
2
votes
1answer
23 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 ...
2
votes
0answers
35 views

Blow up in holomorphic dynamics

Someone could explain the concept of blow up used in holomorphic dynamics? Especifaclly in the context of iteratation of holomorphics functions. This concept could be taken to some of the deformation ...
2
votes
0answers
23 views

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 ...
2
votes
2answers
143 views

M-set interior point probability on the real axis

For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are ...
2
votes
0answers
77 views

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$? I know this function has 2 other fixed points apart from $0$, so I'm not sure. Also ...
2
votes
0answers
60 views

Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function ...
1
vote
1answer
38 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)) ...
1
vote
1answer
162 views

What is the shape of external rays landing on fixed points in case of quadratic discrete dynamical system?

In case of parabolic discrete dynamical system based on the complex quadratic polynomial fc(z) = z^2 + c some external rays land on alfa fixed point. Hera are 34 external rays landing on fixed ...
1
vote
1answer
76 views

Simplify the nonlinear system of dynamic equations

I am working with a set of nonlinear dynamic equations that Mathematica has problems with solving. It is of the form $$ f_1(x_{t+1},y_{t+1},x_{t},y_t) = g_1(x_{t+1},y_{t+1},x_{t},y_t),\\ ...