Julia set

Julia set is set of values inside of the range of holomorphism of some function, that fall out from the range of holomorphism of some integer iteration this function. Julia set is often defined with symbol $J$ or $\mathbb J$. The name of the function can be indicated either as the subscript or in the parenthesis immediately after this symbol.

Let $f$ be holomorphic function defined at some $C\in \mathbb C$.

Then

$\mathbb J(f) = \{ z \in C : \exists ~ n\in \mathbb N_+ ~,~ f^n(z) \bar \in C\}$

where the upper subscript after the name of the function indicates the number of iteration. On this case, the number of iteration is supposed to be integer.

The complementary set to Julia set is called Fateu set; symbol $\mathbb F$ is used to denote it:

$\mathbb F(f) = \{ z \in C : \forall ~ n\in \mathbb N_+ ~,~ f^n(z) \in C\}$

in such a way that

$\mathbb J(f) \cup \mathbb F(f)=C$