Polynomial diffeomorphisms of \({\mathbb C}^2\): Currents, equilibrium measure and hyperbolicity. (English) Zbl 0721.58037

Bedford, Eric; Smillie, John

Invent. Math. 103, No. 1, 69-99 (1991).

Sh. Friedland and 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 \({\mathcal G}\).

Reviewer: I.Mihai (Bucureşti)

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
31C10	Pluriharmonic and plurisubharmonic functions
37D25	Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
58C25	Differentiable maps on manifolds
32C30	Integration on analytic sets and spaces, currents
32U05	Plurisubharmonic functions and generalizations
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables

Keywords:

polynomial diffeomorphisms; generalized Hénon mappings; plurisubharmonic Green functions; currents; hyperbolic diffeomorphism; periodic points

Citations:

Zbl 0665.58031; Zbl 0651.58027

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