Constructing an Expanding Metric for Dynamical Systems in One Complex Variable

We describe a rigorous computer algorithm for attempting to construct an explicit, discretized metric for which a polynomial map f : C → C is expansive on a neighborhood of the Julia set, J . We show construction of such a metric proves the map is hyperbolic. We also examine the question of whether the algorithm can be improved, and the related question of how to build a metric close to euclidean. Finally, we give several examples generated with our implementation of this algorithm.