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.