A numerical method for constructing the hyperbolic structure of complex Henon mappings
For complex parameters a, c, we consider the Henon mapping H, : 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.
hyperbolicity henon mapping rigorous numerics complex dynamics