Abstract
For complex parameters a,c, we consider the Henon mapping \(H_{a,c}: {\Bbb C}^2 \rightarrow {\Bbb C}^2,\) given by \((x,y) \mapsto (x^2 +c -ay, x),\) 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 \(DH \ (\rm{and }\ DH^{-1})\) 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.
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Hruska, S. A Numerical Method for Constructing the Hyperbolic Structure of Complex Henon Mappings. Found Comput Math 6, 427–455 (2006). https://doi.org/10.1007/s10208-006-0141-2
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DOI: https://doi.org/10.1007/s10208-006-0141-2