The density of hyperbolicity conjecture

Let ${\cal F}$ be one of the following families: unicritical polynomials of degree $d$, rational maps of degree $d$, the exponential family, or a family of entire maps with finitely many singular values as defined in Epstein and Lyubich. A map is hyperbolic when all its singular values belong to attracting basins. Density of hyperbolicity conjecture. Hyperbolic maps are dense in ${\cal F}$. This conjecture generalizes the density of hyperbolic quadratic polynomials implied by the Mandelbrot local connectivity conjecture. It is known for several real polynomial families and some classes of transcendental entire maps, but remains open in the stated generality.