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

Abstract

Let L(f)=∫log∥Df∥dμ f denote the Lyapunov exponent of a rational map, f:P 1→P 1 . In this paper, we show that for any holomorphic family of rational maps {f λ :λX} of degree d>1, T(f)=dd c L(f λ ) defines a natural, positive (1,1)-current on X supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula for the Lyapunov exponent: Here F:C 2 →C 2 is a homogeneous polynomial lift of f; ; G F is the escape rate function of F; and capK 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 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⊂C 2 and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such K⊂C 2 correspond to metrics of non-negative curvature on P 1, and we obtain a variational characterization of curvature.