Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity

Let L(f) = ∫log∥Df∥dμ_f denote the Lyapunov exponent of a rational map, f : P¹ → P¹. In this paper, we show that for any holomorphic family of rational maps {f_λ : λ ∈ X} of degree d > 1, T(f) = dd^cL(f_λ defines a natural, positive (1,1)-current onX supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula for the Lyapunov exponent: L(f) = ∑ G_F(c_j) - log d + (2d - 2) log (capK_F). Here F : C² → C² is a homogeneous polynomial lift of f; | dot DF(z)| = ∏ |z ∧ C_j|; G_F is the escape rate function of F; and cap K_F is the homogeneous capacity of the filled Julia set of F. We show, in particular, that the capacity of K_F is given explicitly by the formula capK_F = |Res(F)|^-1/d(d-1), where Res(F) is the resultant of the polynomial coordinate functions of F. We introduce the homogeneous capacity of compact, circled and pseudoconvex sets K ⊂ C² and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such K ⊂ C² correspond to metrics of non-negative curvature on P¹, and we obtain a variational characterization of curvature.