# Tagged Questions

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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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### Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, $$z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots$$ where the parameters $\alpha, ~\beta$ ...
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### 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 $\overline{\mathbb{D}}$ ...
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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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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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 Routhâ€“Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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### $Re[f(x + n i)] = 0$ 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...