In many instances, a dynamical system $f: X o X$ having good metric expanding properties can be shown to be determined by purely combinatorial data. That is, there is a computable model $f^{comb}: X^{comb} o X^{comb}$ and a homeomorphism $h: X^{comb} o X$ conjugating $f^{comb}$ to $f$. A recent new construction shows that for a very wide class of examples, one may take $X^{comb}$ to be the Boundary at infinity of a certain associated infinite Gromov hyperbolic one-complex and $f^{comb}$ to be a simplicial covering map from this complex to itself. The construction is analogous to describing the action of a Kleinian group on its limit set via its action on the boundary of its Cayley graph. Techniques from geometric group theory imply the existence of a natural class of metrics on $X^{comb}$ such that the system becomes ``uniformly quasiregular'', i.e. iterates of $f^{comb}$ distort the roundness of balls by at most a constant factor independent of scale and of iterate. Succintly: this process uniformizes $f: X o X$ by providing a metric on $X$ for which $f$ is nearly conformal. The PI proposes to apply this new point of view to the study of rational maps of the Riemann sphere to itself. The PI will develop a unified framework with which to describe the dynamics of rational maps, Kleinian groups, and their natural generalizations. The PI will thereby (i)reinterpret Thurston's characterization of rational functions from this point of view; (ii) bring new methods into the study of dynamical systems; (iii) further the development of the ``dictionary'' between rational maps and Kleinian groups, focusing on those aspects related to Cannon's conjecture and the regularity of the new metric; (iv) suggest new directions for the study of dynamical systems by formulating and developing a self-contained theory of uniformly quasiregular expanding dynamical systems, and (v) connect several areas of mathematics of current active interest, namely conformal dynamics, geometric group theory, geometric topology, and analysis on metric spaces.
