A numerical method for constructing the hyperbolic structure of complex Henon mappings

Abstract

For complex parameters a,c, we consider the Henon mapping H_{a,c}: C^2 -> C^2 given by (x,y) -> (x^2 +c -ay, x), and its Julia set, J. In this paper, we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J, which is preserved by DH, and a continuous norm in which DH (and DH^{-1}) uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is hyperbolic on J. We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.