Mathematische Annalen

, Volume 326, Issue 1, pp 43–73 | Cite as

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

  • Laura DeMarco


 Let L(f)=∫log∥Dfdμ f denote the Lyapunov exponent of a rational map, f:P1P 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 KC2 and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such KC2 correspond to metrics of non-negative curvature on P1, and we obtain a variational characterization of curvature.


Lyapunov Exponent Coordinate Function Unique Equilibrium Equilibrium Measure Escape Rate 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Laura DeMarco
    • 1
  1. 1.Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 01238, USA (e-mail:

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