Hyperbolic polynomial diffeomorphisms of C2. I: A non-planar map

Abstract In this paper we develop a general framework for verifying hyperbolicity of holomorphic dynamical systems in C 2 . Our framework in particular enables us to construct the first example of a hyperbolic Henon map of C 2 which is non-planar, i.e. which is not topologically conjugate on its Julia set to a small perturbation of any expanding polynomial in one variable. The key ideas in its proof are: the Poincare box, which is a building block to apply our criterion for hyperbolicity, an operation called fusion, to merge two polynomials in one variable to obtain essentially two-dimensional dynamics, and rigorous computation by using interval arithmetic. Some conjectures and problems are also presented.

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