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

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The fixed points of a rational function of multiplicity 2 or more are in the Julia Set

Definition: The multiplicity of a fixed point $z_0$ of $f$ is the order of the zero the the function $f(z)-z$ at $z_0$. I have already shown the iff statement: Claim: The multiplicity of fixed $z_0$ ...
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Show that $-2, -1, 0, i$ lies in the Mandelbrot set but that $1$ lies outside of it

The Question Let $c$ be a complex number. The complex numbers $z_n(c)$ are defined recursively by $z_1(c)=c$, $z_{n++1}(c)=(z_n(c))^2+c$ for $n\geq1$ The Mandelbrot set is defined by ...
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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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127 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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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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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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Blow up in holomorphic dynamics

Someone could explain the concept of blow up used in holomorphic dynamics? Specifically in the context of iteration 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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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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Are the iterates of this function bounded?

I have the function $f(z) = \sqrt z + C.$ For the value of $C = i$ (complex number), would the iterates be bounded or not? Iterating from $z = 0: f(0) = i, f(i) = \sqrt i + i$ and it goes on, ...
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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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25 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 ...
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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 ? ...