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

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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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Liouville's theorem (Hamiltonian) [closed]

can some one give me a link for a rigorous proof for Liouville's theorem (Hamiltonian) thanks
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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? Especifaclly in the context of iteratation 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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52 views

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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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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340 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 ...
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199 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 ...
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42 views

Complex polynomials: preimage of a large enough disc contains the disc compactly?

I'm studying a proof (by Hubbard) of an elementary result in complex dynamics: if a complex polynomial $p$ of degree at least 2 has all of its critical points in its filled Julia set $K_p$, then $K_p$ ...
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161 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 ...
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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.
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186 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 ...
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91 views

Stability of the optimal control steady state

I am struggling with a pretty much complicated optimal control problem, which I solve in Mathematica. The optimal controls are second order delayed, which makes it unclear to me how to analyse the ...
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104 views

Show basin of attraction has only one connected component

L.S., This is a homework question I find hard to answer, any help/hints would be greatly appreciated! I have to prove that every complex polynomial of degree 2 with an attracting fixed point has a ...
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2answers
128 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 ...
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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 ...
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156 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 ...
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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 ...
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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 ...
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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 ...
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246 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 ...
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184 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 ...
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118 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 ...
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Linear complex functions

$L_{\alpha}(z) = \alpha z$ where $z$ is a complex number and $\alpha$ is a constant and is also complex. How do I prove for $|\alpha|<1$ all orbits tend to a unique limit and how do I find this ...
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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) ...
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149 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 ...
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277 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 ...
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Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?

Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer. Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$ Is ...
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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 ...
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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),\\ ...
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About growth rate of the iterated exponential on the complex plane.

Let $n$ be a positive integer. Let $f(n,x) = exp(f(n-1,x))$ and $f(0,x)=x$. Let $Q(f(n,x)) =1$ if $Re(f(n,x))<2$. Let $S(n,x) = \Sigma_{1}^{n} Q(f(n,x))$. How to estimate $S(n,2+i)$ efficiently ? ...
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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 ...
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156 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 ...
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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 ...
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Is the alternating sum of exp(-exp(n z)) analytic?

Define $f(z) = \frac{1}{\exp(\exp(z))} - \frac{1}{\exp(\exp(2z))} + \frac{1}{\exp(\exp(3z))} - \frac{1}{\exp(\exp(4z))} + ...$ $f(z) = \sum_{n = 1}^{\infty} (-1)^{n-1} \exp(-\exp(n z))$. Is $f(z)$ ...
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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 ...
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543 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 ...
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369 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 ...
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151 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 ...
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123 views

Question about iterations : $g(c,z) = a$ always has a solution where $z$ is strictly real?

Let $c$ be any positive real number. Let $z$ be a complex number. Let $g(c,z)$ be some locally analytic function that is the $c$ th iteration of the entire function $f(z)$ with $g(0,z)=f(z)$. Let ...
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195 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 ...
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119 views

Boundary of $x^2+x$ Julia set

How do you calculate an infinite fixed point for $f=x^2+x$, so that $f^{o n}(x)$ never repeats, and doesn't go to infinity and doesn't go to zero? Can such a sequence of points densely cover the ...
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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 ...
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224 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 $f$ ...