In some sense, the filtration property is the easiest property of the Hénon map. It is the most
fundamental of the tools we have. When n > 2, the group of polynomial automorphisms of Cn is not at
all understood. (The paper of Friedland-Milnor made essential use of the Theorem of Jung on the structure of the polynomial
automorphisms of C2.) However, there is the class of regular automorphisms which can be approached
following the general approach used for the Hénon family.
Shafikov and Wolf have shown that some of the
theory can be carried over to higher dimension.