Complex dynamics is the study of dynamical systems of functions over complex numbers.
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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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0answers
18 views
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 ...
3
votes
2answers
70 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
$$ f^{\circ h}(x) ...
1
vote
1answer
75 views
Heighway dragon and twindragon relation
The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1.
The ...
1
vote
2answers
130 views
What is the shape of parabolic critical orbit?
The parabolic critical orbits of discrete dynamical system form n-th arm stars :
which shapes are conjugated with "regular" n-th arm stars
Here are 2 images of parabolic critical orbits for 2 ...
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votes
2answers
53 views
Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?
Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer.
Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$
Is ...
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0answers
54 views
Is it possible to find the length of a limit cycle in advance?
Let $z$ be a complex number. Let $n$ be a positive integer. Let $f(z)$ be a given entire transcendental function.
Let $I$ be a given complex number. By definition $F_0=I,F_1=f(I),F_n=f(F_{n-1})$.
A ...
2
votes
2answers
78 views
The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.
Problem
Is there an irrational $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ such that the set $S= \{\,\{2^N\alpha\} :N\,\in\mathbb{N}\}$ is not dense in $[0,1]$.
Here $\{x\}=x-\lfloor x\rfloor$ is the ...
1
vote
1answer
54 views
Simplify the nonlinear system of dynamic equations
I am working with a set of nonlinear dynamic equations that Mathematica has problems with solving. It is of the form
$$
f_1(x_{t+1},y_{t+1},x_{t},y_t) = g_1(x_{t+1},y_{t+1},x_{t},y_t),\\
...
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0answers
47 views
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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0answers
39 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 ...
3
votes
1answer
59 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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vote
0answers
24 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 ...
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votes
2answers
131 views
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 $f(z)$ ...
10
votes
2answers
267 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 ...
5
votes
1answer
199 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 ...
1
vote
1answer
87 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
73 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 ...
0
votes
1answer
117 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 ...
2
votes
0answers
91 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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votes
1answer
79 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 ...
9
votes
9answers
838 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 ...
2
votes
0answers
71 views
Zoom out fractals?
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 $f$
...
0
votes
0answers
78 views
About fractals , domains and iterations ( complex dynamics )
Let $f$ be an entire nonpolynomial function.
Consider $g:= f(f(f(...))).$
Define a hole as divergence surrounded completely by convergeance. ( a bit like the donut in topology )
Alot of questions ...
8
votes
2answers
184 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 ...
3
votes
1answer
151 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 ...
