3
votes
0answers
53 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
1
vote
1answer
82 views

Why does an automorphism of the disk that is an involution not have fixed points in its boundary?

I'm reading Milnor's Dynamical Systems in One Complex Variable and I'm stuck with a detail in one of his proofs and would appreciate some help! Let $F$ be an automorphism of $\overline{\mathbb{D}}$ ...
3
votes
0answers
45 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 ...
0
votes
0answers
16 views

Looking for methods/results for explicitly bounding iterations of rational functions

This is a cross-post of http://mathoverflow.net/questions/155775/looking-for-methods-results-for-explicitly-bounding-iterations-of-rational-funct But I received no answer there to the actual ...
6
votes
3answers
324 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 ...
5
votes
2answers
181 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and maybe easy question, but I was not able to find an answer. If $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a linear change of coordinates it can be brought to the ...
1
vote
0answers
27 views

How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
3
votes
2answers
96 views

Derivative of $f(z)=az$ at infinity is $\frac{1}{a}$?

I was going through a dynamical system lecture note and there it is said that the derivative of $f(z)=az$ at infinity is $\frac{1}{a}$, where $z\in \mathbb{C}_\infty$ i.e., Riemann sphere. Normally I ...
4
votes
0answers
125 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 ...
6
votes
1answer
432 views

How to prove Mandelbrot set is simply connected?

In this lecture note of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is ...
2
votes
2answers
120 views

Conditions that Roots of a Polynomial be Less than Unity

Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
1
vote
1answer
173 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
0
votes
2answers
63 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 ...
0
votes
1answer
188 views

Is $\text{arc2sinh}(\dots(\exp(2\sinh(\dots z))$ an entire function?

Let $^{*n}$ denote the $n$-th iteration and $z$ be a complex number. Let $n$ be a positive integer. Let $2\sinh(z)$ be $\exp(z)-\exp(-z)$ and $\text{arc2sinh}$ its functional inverse. Is the limit for ...
4
votes
1answer
149 views

small circle inside embedding of complete graph in the plane

On the web, I found this beautiful drawing of the complete graph on 13 vertices: It is on the Geometry Daily tumblr page. A computer scientist drew a more interactive version up to about 40 ...
1
vote
0answers
85 views

A text or book in holomorphic foliations and vector fields over complex manifolds

For my master's degree dissertation, I am going to study some implications of the paper "SOME REMARKS ON INDICES OF HOLOMORPHIC VECTOR FIELDS" written by Marco Brunella. I just started it and I'm ...
0
votes
1answer
60 views

$Re[f(x + n i)] = 0$ and $f(z)$ is not periodic

Let $z$ be a complex number and $f(z)$ an entire function such that For $x$ real and $n$ any integer. $Re[f(x + n i)] = 0$ and $f(z)$ is not periodic. What are typical examples of such $f(z)$ ? Is ...
1
vote
0answers
63 views

Is there a “natural” embedding of the Henon map into a continuous flow?

I was wondering about this. Now, I'm a big fan of "continuous iteration", and I was curious about this problem. What I was wondering was whether or not there exists a continuous dynamical system which ...
0
votes
1answer
122 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 ...
4
votes
2answers
453 views

Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?

The simple plane pendulum $$\frac{d^2\theta}{dt^2} + \frac{g}{l}\sin{\theta} = 0$$ has the very perdy phase portrait Meanwhile, a domain coloring of $\sin(z)$ in the complex plane is Why are ...
4
votes
1answer
187 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 ...
4
votes
1answer
151 views

Constructing Riemann surfaces using the covering spaces

In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow: A polynomail-like map ...
1
vote
1answer
95 views

No fixed point problem for iterations

Let $f(z)$ be an entire function that is not a polynomial of degree 1 or degree 0 , where $z$ is a complex number. Let $f(z,1) = f(z)$ and let $f(z,n) = f(f(z,n-1))$. Let $g(f,1)$ be the amount of ...
9
votes
1answer
390 views

Number of limit points of a continued exponential

Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
4
votes
1answer
209 views

coloring the inside point for Julia Fractal

I am trying to continuous coloring the inside point for a fractal image,such as $z \to z^2+C$. For those outside point, we can use the escape iteration to determine the color, just as the description ...
8
votes
2answers
548 views

Interesting but elementary properties of the Mandelbrot Set

I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know ...
6
votes
1answer
156 views

Fixed, attracting points are Fatou points

Let $f$ be a holomorphic function on an open, connected set $\Omega\subset \mathbb{C}$ with $z_0\in \Omega$ a fixed point, and $\{f^n\}_{n\in \mathbb{N}}$ the sequence of iterates. I want to prove ...