Fatou Set

The Fatou set, in complex dynamics, is a region where the behavior of points under repeated application of a complex function is stable and predictable. If you imagine repeatedly applying a mathematical rule to points in this region, their outputs will behave in a smooth, consistent manner, often settling into patterns like cycles or staying close to a certain location. In contrast, points outside this set exhibit chaotic and unpredictable behavior. The Fatou set helps mathematicians understand the structure of complex functions by dividing their behavior into stable regions, aiding in the study of fractals and complex dynamical systems.