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Consider the complex map $z\rightarrow z^2+1$. Find any one point that is

a) not in the filled Julia set

b) in the filled Julia set

c) in the Julia set

I know that the Julia sets that are in the Mandelbrot set will be connected. While the ones that are outside are disconnected. But I don't know how to do this question please help ASAP!

Community
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Itsnhantransitive
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1 Answers1

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The filled-in Julia set is defined as the set of all points with bounded orbit. So for (a), look at $$ 0 \to 1 \to 2 \to 5 \to 26 \to ... $$ For (b), look at a root of $z^2+1=z.$ It is fixed under the map, so it has a bounded orbit.

Reiner Martin
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  • Thank you so much Mr.Martin. Can you explain the difference between b) and c)? – Itsnhantransitive Nov 21 '17 at 21:10
  • The Julia set is the boundary of the filled Julia set, which means that for any point from the Julia set and any $\epsilon>0$ there is a both a value from the filled Julia set and from outside the filled Julia set in the $\epsilon$-neighborhood of that point. Alas, I don't have a concrete example point for you at the moment. – Reiner Martin Nov 21 '17 at 21:12
  • is the border itself bounded then? – Itsnhantransitive Nov 21 '17 at 21:23
  • Yes, because the boundary of a bounded set is bounded, and it is not hard to show that the filled Julia set is contained in a circle of sufficient radius. – Reiner Martin Nov 21 '17 at 21:32