Natural Invariant Measures, Divergence Points and Dimension in One-dimensional Holomorphic Dynamics

In this paper we discuss dimension-theoretical properties of rational maps on the Riemann sphere. In particular, we study existence and uniqueness of generalized physical measures for several classes of maps including hyperbolic, parabolic, non-recurrent and Topological Collet-Eckmann maps. These measures have the property that their typical points have maximal Hausdorff dimension. On the other hand, we prove that the set of divergence points (the set of points which are non-typical for any invariant measure) also has maximal Hausdorff dimension. Finally, we prove that if (fa)a is a holomorphic family of stable rational maps, then the dimension d(fa) is a continuous and plurisubharmonic function of the parameter a. In particular, d(f) varies continuously and plurisubharmonically on an open and dense subset of Ratd, the space of all rational maps with degree d ≥ 2.