Rigorous numerical models for the dynamics of complex Henon mappings on their chain recurrent sets

Abstract

We describe a rigorous and efficient computer algorithm for building a model of the dynamics of a polynomial diffeomorphism of C^2 on its chain recurrent set, and for sorting points into approximate chain transitive components. Further, we give explicit estimates which quantify how well this algorithm approximates the chain recurrent set and distinguishes the chain transitive components. We also discuss our implementation for the family of Henon mappings, f_{a,c}(x,y) = (x^2 + c - ay, x), into a computer program called Hypatia, and give several examples of running Hypatia on Henon mappings.