A Numerical Method for Constructing the Hyperbolic Structure of Complex Henon Mappings

Abstract

Abstract. For complex parameters a, c, we consider the Hénon mapping Ha,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.

Uniformly Hyperbolic Attractor of the Smale–Williams Type for a Poincaré Map in the Kuznetsov System

“…Recently, Hruska [Hr1,Hr2] proposed a method for computer assisted verification of uniform hyperbolicity. She successfully applied the method to the complex Hénon map.…”

Expansivity and Cone-fields in Metric Spaces

“…Recently, Sheldon Newhouse [10] obtained new conditions for dominated and hyperbolic splittings on compact invariant sets with the use of cone-fields. It is also worth mentioning that the notion of cone-field can be very useful in the study of hyperbolicity [1,3,4,10].…”

Recent development in rigorous computational methods in dynamical systems

“…We remark that Hruska [19] also constructed a rigorous numerical method for proving hyperbolicity of the complex Hénon map. The main difference between our method and Hruska's method is that our method does not prove hyperbolicity directly.…”