3Blue1Brown has 2 great videos on Newton's fractal. In the second video he describe a property that makes it inevitable. Does this property/theorem have a name?
For any rational map, if you were to assign colors to regions based on which limiting behavior points fall into ... then tiny circles that you draw either contain points with 1 limiting behavior or all of them, never in between.
This is Theorem 3.9 (Transitivity) in the following intro paper.
The idea of topological transitivity is that the forward iterations of an open set end up filling up the space. In our application, this says that $U$ contains points arbitrarily close to the different roots of the polynomial, and we know that for such points the iteration just pulls them towards the respective root.