# Tagged Questions

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

74 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 ...
15 views

47 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 ...
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 ...
85 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 ...
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 ...
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 ...
178 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 ...
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 ...
503 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 ...
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 ...
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 ...
25 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. ...
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 ...
586 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 ...
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 ...
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 ...
274 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 ...
49 views

50 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 ...
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 $f,g$ ...
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....
38 views

436 views

71 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 $z_0$ ...
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 ...
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 ...
48 views

### 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 ...
39 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, ...
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. ...
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 ...
Let $F(x)$ be a real-analytic function near $0$ ,with $0$ as one of its fixpoints and $f ' (0) > 1$. $$F(x) = F \circ F \circ F^{-1} = \lim_{n \to \infty} F^{n} \circ F \circ F^{-n} = \lim_{n \to \... 1answer 229 views ### Finding the location of an image of the Mandelbrot set I've got an image of a segment of the Mandelbrot set that I generated with an iPhone app a long time ago (I use it as my background image). I now have no idea where in the set the image came from. ... 1answer 49 views ### Entire f,g such that f(f(z)) = p(g(z)) Let p(z) be a given polynomial of degree 3. How to find nonconstant entire functions f,g such that f(f(z)) = p(g(z)) or prove they do not exist ? I considered the related equation f(f(z)) ... 4answers 396 views ### prove conjecture; the limit of iterating is \sqrt{z^2 - 2}$$\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/...
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{...