Frontiers in complex dynamics

Abstract

Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The rational maps of a given degree form a finite-dimensional manifold, so exploration of this {\em parameter space} is especially tractable. Finally, some of the conjectures once proposed for {\em smooth} dynamical systems (and now known to be false) seem to have a definite chance of holding in the arena of rational maps. In this article we survey a small constellation of such conjectures centering around the density of {\em hyperbolic} rational maps --- those which are dynamically the best behaved. We discuss some of the evidence and logic underlying these conjectures, and sketch recent progress towards their resolution.