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Polynomial diffeomorphisms of \({\mathbb{C}}^2\). V: Critical points and Lyapunov exponents. (English) Zbl 1004.37027
Over the last 10 years, E. Bedford and J. Smillie (in part with M. Lyubich) have published a steady stream of works on Polynomial Diffeomorphisms of \(\mathbb{C}^2\): For Parts I–IV, see E. Bedford and J. Smillie, Currents, equilibrium measure and hyperbolicity. Invent. Math. 103, No. 1, 69-99 (1991; Zbl 0721.58037); Stable manifolds and recurrence. J. Am. Math. Soc. 4, No. 4, 657-679 (1991; Zbl 0744.58068); Ergodicity, exponents and entropy of the equilibrium measure. Math. Ann. 294, No. 3, 395-420 (1992; Zbl 0765.58013); The Measure of maximal entropy and laminar currents. Invent. Math. 112, No. 1, 77-125 (1993; Zbl 0792.58034).
In this paper, Part V, the authors give an integral formula for the Lyapunov exponents of a dynamically non-trivial polynomial automorphism, associated with the unique invariant measure of maximal entropy.
For Parts VI–VIII, see the following review (Zbl 1004.37034).
Main Theorem: For any \(t>0\), the Lyapunov exponent of a polynomial automorphism \(f\) with degree \(d\), is estimated as follows: \[ \Lambda_\mu(f)= \log d+\int_{t\leq G^+<td} G^+d\mu_c^-, \] where \(\mu_c^-\) is a critical measure \(G^+\), Green function associated to \(f\).
This is analogous to the Brolin-Manning formula for a polynomial map \(p\) of \(\mathbb{C}\) with degree \(d\): Manning-Przytycki Result: The Lyapunov exponent is as follows: \[ \lambda_\nu (p)=\log d+\sum_iG(c_i), \] where \(\nu\) is Brolin measure, \(G\) Green function of the Julia set of \(p\), and \(\{c_i\}\) the set of critical points.
Further information on this theory will be available on the World Wide Web at:
http://www.math.sunysb.edu/dynamics/preprints/preprint.html.

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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