Polynomial diffeomorphisms of \(\mathbb{C}^ 2\). III: Ergodicity, exponents and entropy of the equilibrium measure. (English) Zbl 0765.58013

Bedford, Eric; Smillie, John

Math. Ann. 294, No. 3, 395-420 (1992).

In a series of papers, the authors deal with polynomial diffeomorphisms of \(\mathbb{C}^ 2\) with nontrivial dynamics.
Let \(f\) be such a diffeomorphism and \(K=K^ +\cap K^ -\), where \(K^ \pm\subset\mathbb{C}^ 2\) denotes the set of points of \(\mathbb{C}^ 2\) bounded in forward/backward time. In the first part [Invent. Math. 103, No. 1, 69-99 (1991; Zbl 0721.58037)] the stable/unstable currents \(\mu^ \pm\) associated to \(K^ \pm\) are studied. The equilibrium measure \(\mu=\mu^ +\wedge\mu^ -\) of \(K\) has finite total mass and is invariant under \(f\).
In the paper under review, the dynamics of \(f\) with respect to \(\mu\) are considered. The mapping \(f\) is mixing on \(\mu\) and ergodic, and applications are given. The measure theoretic entropy of \(\mu\) is determined and some consequences on its Hausdorff dimension are derived. It is proved that the larger Lyapunov exponent is pluri(sub)harmonic. In the last section the parameter space is extended.
[For part II see the authors, J. Am. Math. Soc. 4, No. 4, 657-679 (1991; Zbl 0744.58068)].

Reviewer: I.Mihai (Bucureşti)

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37D25	Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables

Keywords:

ergodicity; equilibrium measure; polynomial diffeomorphisms; Lyapunov exponent

Citations:

Zbl 0721.58037; Zbl 0744.58068

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