Complex analytic dynamics on the Riemann sphere

Holomorphic, non-invertible dynamical systems of the Riemann sphere are surprisingly intricate and beautiful. Often the indecomposable, completely invariant sets are fractals (a la Mandelbrot M1) because, in fact, they are quasi-self-similar (see Sullivan S3 and (8.5)). Sometimes they are nowhere differentiable Jordan curves whose Hausdorff dimension is greater than one (Sullivan S4 and Ruelle R). Yet these sets are determined by a single analytic function zn+1 = R(zn).