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Math Help - Bounded set

  1. #1
    Super Member Deadstar's Avatar
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    Bounded set

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

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

    And so 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?
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Deadstar View Post
    Could someone explain the the conclusion of this proof that the Julia set is bounded to me?

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

    And so 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).
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  3. #3
    Super Member Deadstar's Avatar
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    Quote Originally Posted by Opalg View Post
    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?
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  4. #4
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Deadstar View Post
    Could the '3' in the r = max... part be replaced by 'a' where a>2?
    Yes.
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