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

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 \({\mathcal G}\).

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
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References:

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