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

In this paper we develop a general framework for verifying hyperbolicity of holomorphic dynamical systems in C². Our framework in particular enables us to construct the first example of a hyperbolic Hénon map of C² 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 Poincaré 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.