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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.

##### MSC:
 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
##### Keywords:
holomorphic family of rational maps
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