complex-dynamics

complex-dynamics

- Basics

complex-dynamics-00-misc

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## Fatou/Julia set

**TODO**

- Prop.
    - for $c = R e^{i\nu}$, we have
      $
        f_c(r e^{i \phi + i \frac{\nu}{2}})
          = (r^2 e^{i 2 \phi} + R) e^{i \nu}
          = f_c(r e^{i \phi + i \frac{\nu}{2} + \pi})
      $

- Prop.
    - $K_{f_c} : \t{connected} \iff 0 \in K_{f_c}$
        - $\implies$ : by symmetry around $0$

- Prop.
    - $z \in F_f \iff f(z) \in F_f$
    
- Prop. Koenigs linearization

  
**Preliminary**

- $f_c(z) \defeq z^2 + c$.


### Def. 1

Define ***Fatou set*** and ***Julia set*** of $f$ by:

- $
F_f \defeq
  \{\; z \mid
    \exists U : \t{neighborhood of \(z\) s.t.}
      \;\{\; f^{n} : U \to \CC \;\}_n : \t{normal} \;\}
$
- $J_f$

If $\infty \in F_f$, then we also define ***Filled Julia set*** of $f$ by:

- $K_f \defeq (\t{the connected component of \(F_f\) which includes \(\infty\)})$.