Julia Set

In complex dynamics, the Julia set of a function is the set of points at which the behavior of the function under repeated iteration is chaotic: an arbitrarily small change in the starting point can produce drastically different long-term outcomes. Its complement, the Fatou set, consists of points where nearby starting values behave similarly under iteration instead. Julia sets are named for Gaston Julia and Pierre Fatou, whose foundational papers on complex iteration appeared in 1918 and 1917 respectively. For the quadratic family f(z) = z^2 + c, the connectedness of the Julia set is determined by whether c lies in the Mandelbrot set, tying the two families of fractals together.