The Structure of Expanding Rational Maps as Holomorphic Dynamical Systems

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The structure of expanding, or hyperbolic, rational maps as holomorphic dynamical systems on the Riemann sphere will be investigated. The approach is motivated by the analogy between the theories of rational maps and Kleinian groups developed by McMullen and Sullivan. To an expanding, (equivalently, geometrically finite without cusps) torsion-free Kleinian group one may associate a topological object-the quotient compact three-manifold obtained by quotienting hyperbolic space together with the domain of discontinuity on the Riemann sphere at infinity by the action of the group. To an expanding rational map one may also associate a topological object-the map viewed as a branched covering of the sphere to itself, up to a homotopy-theoretic form of equivalence. A characterization of those branched coverings which are equivalent to an expanding rational map is known. However, in contrast to the case of Kleinian groups, little is known in general about how combinatorial properties of these branched coverings are reflected in the geometry and dynamics of the corresponding rational maps and in their parameter spaces. Motivated by the role played by the theory of incompressible surfaces in the study of hyperbolic 3-manifolds, the combinatorics of branched coverings will studied. This analysis will be used to develop general combination and decomposition theories for ratimnal maps and general characterizations of adjacencies and compactness properties of hyperbolic components in parameter spaces of rational maps. A dynamical system is a mathematical model now widely used in many disciplines to describe biological, physical, and social phenomena. The main theme in the subject is that even a very simple model can yield extremely complicated behavior. Therefore one often simplifies the model first to make the problem tractable. Next, one needs tools to attack these problems. Finally, one looks for a reduction of the problem to a combinatorial one, i.e., one which, in principle, a computer can sol ve. This project concerns reducing the dynamics of one-dimensional holomorphic functions (which are the simplest dynamical systems to which the powerful tools of complex analysis apply) to a combinatorial problem. This problem, when formulated correctly, looks remarkably similar to one in a different field whose solution is already known. The main goal of this project is to take these known results and, reasoning by analogy, formulate and prove new results in holomorphic dynamics.