Polynomial Diffeomorphisms of C 2 : v. Critical Points and Lyapunov Exponents

§0. Introduction This paper deals with the dynamics of polynomial diffeomorphisms f : C → C. To exclude trivial cases we make the standing assumption that the dynamical degree d = d(f) is greater than one (see Section 1 for a definition). It is often quite useful in dynamics to focus attention on invariant objects. A natural invariant set to consider is K = Kf , the set of points with bounded orbits. Pluripotential theory allows us to associate to this set the harmonic measure, μ = μf of Kf . For polynomial diffeomorphisms this measure is finite and invariant, and we normalize it to have total mass one. In previous papers we have shown that this measure has considerable dynamical significance. We have shown that μ is ergodic [BS3] and that the support of μ is the closure of the set of periodic saddle orbits [BLS1]. Further, μ is the unique measure of maximal entropy [BLS1], and μ describes the distribution of periodic points [BLS2]. To any measure we can associate Lyapunov exponents. The rate of expansion and contraction of tangent vectors at a point p by f is measured by a pair of Lyapunov exponents, λ(p) and λ(p). In the presence of an ergodic invariant measure such as μ these exponents are constant almost everywhere, and we denote them by λ(μ) and λ(μ). By [BS3] the (complex) Lyapunov exponents of μ satisfy λ(μ) < 0 < λ(μ). This condition is known as (nonuniform) hyperbolicity of the measure μ. Nonuniform hyperbolicity implies that at μ almost every point p there is a spitting of the tangent space into complex one dimensional subspaces E(p) and E(p) so that for v ∈ E(p) we have ||Dfn(v)|| ∼ exp(nλ+)||v|| and for v ∈ E(p) we have ||Dfn(v)|| ∼ exp(nλ−)||v||. In this paper we will prove an integral formula for the Lyapunov exponents. In many ways our formula is analogous to the Brolin-Manning formula for Lyapunov exponents with respect to harmonic measure for polynomial maps of C, which we now describe. Let g be a polynomial map of C. We let Kg denote the set of points with bounded orbits. We denote by μ = μg the harmonic measure of Kg. There is a single Lyapunov exponent λ(μ) which gives the average rate of expansion along the orbit μ almost everywhere. The Green function of K is given by the following formula: