Douady–Hubbard mateability conjecture for quadratic polynomials

Let $f_{c}(z)=z^2+c$ be a quadratic polynomial, and let $c_1,c_2$ be points in the parameter plane. Two parameters lie in conjugate limbs of the Mandelbrot set when their corresponding limbs are paired by the conjugation symmetry relevant to quadratic mating.

Douady–Hubbard's conjecture. The points $c_1$ and $c_2$ do not lie in conjugate limbs of the Mandelbrot set if and only if

$$f_{c_1}(z)=z^2+c_1$$

and

$$f_{c_2}(z)=z^2+c_2$$

are conformally mateable.

The paper presents this as a foundational conjecture about quadratic polynomial matings and then recalls results settling important cases, including post-critically finite and hyperbolic cases, while non-hyperbolic cases remain only partly understood.