zbMATH — the first resource for mathematics

Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. (English) Zbl 1032.37029
Author’s abstract: Let \(L(f) = \int \log \|Df \|d\mu_f\) denote the Lyapunov exponent of a rational map, \(f : {\mathbb P}^1 \to {\mathbb P}^1\). We show that for any holomorphic family of rational maps \(\{ f_\lambda : \lambda \in X\}\) of degree \(d >1\), \(T(f) = d d^c L(f_\lambda)\) 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: \[ L(f) = \sum G_F (c_j)-\log d + (2d-2)\log ( \text{cap} K_F). \] Here \(F : {\mathbb C}^2 \to {\mathbb C}^2\) is a homogeneous polynomial lift of \(f\); \(|\det DF (z)|=\Pi |z \wedge c_j |\); \(G_F\) is the escape rate function of \(F\); and \(\text{cap} K_F\) 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: \[ \text{cap} K_F = |\text{Res} (F) |^{-1/d(d-1)}, \] where \(\text{Res} (F)\) is the resultant of the polynomial coordinate functions of \(F\). We introduce the homogeneous capacity of compact, circled and pseudoconvex sets \(K \subset {\mathbb C}^2\) and show that the Levi measure (determined by the geometry of \(K\)) is the unique equilibrium measure. Such \(K \subset {\mathbb C}^2\) corresponds to metrics of nonnegative curvature on \({\mathbb P}^1\), and we obtain a variational characterization of curvature.

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32U20 Capacity theory and generalizations
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
Full Text: DOI