Mathematische Annalen

, Volume 326, Issue 1, pp 43-73

First online:

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

  • Laura DeMarcoAffiliated withDepartment of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 01238, USA (e-mail:

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


 Let L(f)=∫log∥Dfdμ f denote the Lyapunov exponent of a rational map, f:P 1P 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 KC 2 and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such KC 2 correspond to metrics of non-negative curvature on P 1, and we obtain a variational characterization of curvature.