# Thread: Bounded set

1. ## Bounded set

Could someone explain the the conclusion of this proof that the Julia set is bounded to me?

Theorem: The Julia set $\displaystyle J$ of a function $\displaystyle f_c(z) = z^2 + c$ is compact for all $\displaystyle c \in \mathbb{C}$
Proof: First, we show that $\displaystyle J$ is bounded.
Choose $\displaystyle r = \textrm{max}(|c|,3)$ and let $\displaystyle |z| \geq r$.
Then we have,
$\displaystyle |z^2| = |f_c(z) - c| \leq |f_c(z)| + |c|$
And so we have,
$\displaystyle |f_c(z)| \geq |z^2| - |c| \geq 3|z| - |z| = 2|z|$
Hence, since $\displaystyle |z| \geq r$ then
$\displaystyle f_c^n(z) \geq 2^n |z| \rightarrow \infty \textrm{ as } n \rightarrow \infty$ (*)

And so $\displaystyle J(f_c) \subset B(0,r)$. (**)

I found this in a journal, I don't really understand how the last line (**) happens. Surley (*) shows the function diverges... How can we say it's bounded?

2. Originally Posted by Deadstar Could someone explain the the conclusion of this proof that the Julia set is bounded to me?

Theorem: The Julia set $\displaystyle J$ of a function $\displaystyle f_c(z) = z^2 + c$ is compact for all $\displaystyle c \in \mathbb{C}$
Proof: First, we show that $\displaystyle J$ is bounded.
Choose $\displaystyle r = \textrm{max}(|c|,3)$ and let $\displaystyle |z| \geq r$.
Then we have,
$\displaystyle |z^2| = |f_c(z) - c| \leq |f_c(z)| + |c|$
And so we have,
$\displaystyle |f_c(z)| \geq |z^2| - |c| \geq 3|z| - |z| = 2|z|$
Hence, since $\displaystyle |z| \geq r$ then
$\displaystyle f_c^n(z) \geq 2^n |z| \rightarrow \infty \textrm{ as } n \rightarrow \infty$ (*)

And so $\displaystyle J(f_c) \subset B(0,r)$. (**)

I found this in a journal, I don't really understand how the last line (**) happens. Surely (*) shows the function diverges... How can we say it's bounded?
What (*) shows is that if z is any point outside B(0,r) then the iterates of z under the map f_c go off to infinity, which means that z does not belong to the Julia set. Therefore the Julia set is contained in B(0,r).

3. Originally Posted by Opalg What (*) shows is that if z is any point outside B(0,r) then the iterates of z under the map f_c go off to infinity, which means that z does not belong to the Julia set. Therefore the Julia set is contained in B(0,r).
Thanks! I had a feeling that was it...

Could the '3' in the r = max... part be replaced by 'a' where a>2?

4. Originally Posted by Deadstar Could the '3' in the r = max... part be replaced by 'a' where a>2?
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