I am given ( a simplistic definition I think ) of Mandelbrot set:

M- set of complex numbers $c \in \mathbb{C}$ s.t. the sequence $(z_n)$ is bounded where $z_0=0 , z_{n+1}=z_n^2+c $

Need to show that:

(i)$-2\in M$

(ii) $1/4 \in M$.

Am I missing something here or is it kind of trivial?

(i) $ c=-2 $ gives $z_n=2 \space \forall n\geq 2$ , obviously bounded so $c \in M$ ?

(ii) $c=1/4$ we get that $z_n \longrightarrow \frac{1}{2} $ again bounded.

Am I doing it right or did I completely miss somehow the point of the exercise? It just seems too simple to be true. Sorry if this is a stupid question, I have never worked with Mandelbrot set before.

  • $\begingroup$ Looks fine to me. $\endgroup$ – lhf Dec 10 '15 at 0:45
  • $\begingroup$ Okay thank you, that is reassuring ! $\endgroup$ – Guestgreat Dec 10 '15 at 0:47
  • $\begingroup$ Actually, $M \cap \mathbb R = [-2,0.25]$, but this is harder to prove. $\endgroup$ – lhf Dec 10 '15 at 1:15

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