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

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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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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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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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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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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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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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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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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 derivation of Feigenbaum constant possible through Mandelbrot set?

this is Mandelbrot set: $z_{n+1}=z_n^2+C$ Is derivation of Feigenbaum constant possible through Mandelbrot set? $$\lim_{n\to\infty}\frac{z_{n+2}-z_{n+1}}{z_{n+1}-z_{n}}=\delta$$
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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 ...