Dynamics of Polynomial Diffeomorphisms of C²: Combinatorial and Topological Aspects

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 f_{c , 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.