## 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 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 : C^{2} → C^{2} 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 ⊂ 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.

Original language | English (US) |
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Pages (from-to) | 43-73 |

Number of pages | 31 |

Journal | Mathematische Annalen |

Volume | 326 |

Issue number | 1 |

DOIs | |

State | Published - May 2003 |

## ASJC Scopus subject areas

- Mathematics(all)