Fatou's density of hyperbolicity conjecture for rational maps

Let $f$ be a rational map of the Riemann sphere, and fix its degree. A rational map is hyperbolic when every critical point is attracted to an attracting periodic cycle. Fatou's density of hyperbolicity conjecture. Any rational function is approximated by hyperbolic rational functions of the same degree. This is one of the central conjectures in rational dynamics and is related to the conjecture that rational maps carry no invariant line fields on their Julia sets.