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

Laura DeMarco*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

55 Scopus citations

Abstract

Let L(f) = ∫log∥Df∥dμf denote the Lyapunov exponent of a rational map, f : P1 → P1. In this paper, we show that for any holomorphic family of rational maps {fλ : λ ∈ X} of degree d > 1, T(f) = ddcL(fλ 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) = ∑ GF(cj) - log d + (2d - 2) log (capKF). Here F : C2 → C2 is a homogeneous polynomial lift of f; | dot DF(z)| = ∏ |z ∧ Cj|; GF is the escape rate function of F; and cap KF is the homogeneous capacity of the filled Julia set of F. We show, in particular, that the capacity of KF is given explicitly by the formula capKF = |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 ⊂ C2 and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such K ⊂ C2 correspond to metrics of non-negative curvature on P1, and we obtain a variational characterization of curvature.

Original languageEnglish (US)
Pages (from-to)43-73
Number of pages31
JournalMathematische Annalen
Volume326
Issue number1
DOIs
StatePublished - May 2003

ASJC Scopus subject areas

  • Mathematics(all)

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