The Fig. 1 was drawn by Shigehiro Ushiki using his software called HenonExplorer. This complicated object is the Julia set of a complex Hénon map fc , b(x, y) = (x2+ c- by, x) defined on C2 together with its stable and unstable manifolds, hence it is a fractal set in the real 4-dimensional space! The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of C2 including complex Hénon maps with an emphasis on the combinatorial and topological aspects of their Julia sets.
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