The Julia set of a complex polynomial of the form $z^2+c,~c\in\mathbb{C}$, is defined as the set of all $z\in\mathbb{C}$ whose neighbourhood behaves chaotically under iteration $($by chaotically, we mean that each value of $z$ could tend to a different value, or infinity, under iteration$)$, whereas the Fatou set of the same polynomial is the set of all $z\in\mathbb{C}$ whose neighbourhood behaves normally under iteration, where normally implies the opposite of chaotically.
Having studied these concepts rather arbitrarily, I was wondering why are they important to pure math specifically $($as in, I'm not particularly interested in real world applications$)$? Are there any important theorems that are worth knowing that depend on the Julia or Fatou set of a complex function?
Any responses are appreciated.