A compactification of Henon mappings in C^2 as dynamical systems

Abstract

In \cite {HO1}, it was shown that there is a topology on $\C^2\sqcup S^3$ homeomorphic to a 4-ball such that the H\'enon mapping extends continuously. That paper used a delicate analysis of some asymptotic expansions, for instance, to understand the structure of forward images of lines near infinity. The computations were quite difficult, and it is not clear how to generalize them to other rational maps. In this paper we will present an alternative approach, involving blow-ups rather than asymptotics. We apply it here only to H\'enon mappings and their compositions, but the method should work quite generally, and help to understand the dynamics of rational maps $f:\Proj^2\ratto\Proj^2$ with points of indeterminacy. The application to compositions of H\'enon maps proves a result suggested by Milnor, involving embeddings of solenoids in $S^3$ which are topologically different from those obtained from H\'enon mappings.