Polynomial diffeomorphisms of \({\mathbb C}^2\): Currents, equilibrium measure and hyperbolicity
\textit{Sh. Friedland} and \textit{J. Milnor} [Ergodic Theory Dyn. Syst. 9, No.1, 67-99 (1989; Zbl 0651.58027)] have classified polynomial diffeomorphisms of \({\mathbb{C}}^ 2\) up to conjugation. One of these classes, noted by \({\mathcal G}\), consists of the finite compositions of generalized Hénon mappings, which have the form \(g(x,y)=(y,p(y)-ax)\) for a monic polynomial p of degree at least 2 and \(a\neq 0.\) The authors study the dynamics of maps \(g\in {\mathcal G}\), using the methods of plurisubharmonic Green functions and positive currents. An analogue of Brolin's theorem is proved. The notion of hyperbolicity for diffeomorphisms in \({\mathcal G}\) is introduced. If f is a hyperbolic diffeomorphism, then its periodic points are dense in the nonwandering set. It is shown that the set of hyperbolic polynomial diffeomorphisms is open in \({\mathcal G}\).
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