Douady–Hubbard Theorem

The Douady–Hubbard Theorem describes the behavior of certain complex functions called quadratic polynomials. It states that for these functions, the set of points where the function's behavior is stable (the Julia set) either forms a totally disconnected fractal or is a more connected, blob-like shape depending on the parameter. Essentially, it explains how the complex dynamics of these functions split into two distinct types: chaotic, dust-like sets or solid, connected structures. This result helps mathematicians understand the intricate boundary between stable and chaotic behavior in complex systems.