No invariant line fields conjecture for rational maps

Let $R$ be a rational map of the Riemann sphere, and let its Julia set be the set on which the associated dynamics is chaotic. An invariant line field for $R$ is a Beltrami form $\mu=\mu(z)\frac{{\mathrm d}\overline z}{{\mathrm d}z}$ with $|\mu(z)|=1$ on a positive-measure subset of the Julia set, vanishing elsewhere, and satisfying $R^*\mu=\mu$ almost everywhere. No invariant line fields conjecture. A rational map $R$ carries no invariant line field on its Julia set, except when $R$ is double covered by an integral torus endomorphism. The conjecture remains open, even for quadratic polynomials of the form $z^2+c$, where it is equivalent to density of hyperbolicity.