Dynamics of Polynomial Diffeomorphisms of $$\mathbb {C}^2$$ C 2 : 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 Henon map $$f_{c, b}(x, y)=(x^2+c-by, x)$$ defined on $$\mathbb {C}^2$$ 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 $$\mathbb {C}^2$$ including complex Henon maps with an emphasis on the combinatorial and topological aspects of their Julia sets.