Dynamics of rational maps: Lyapunov exponents,bifurcations, and capacity |
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Authors: | Laura DeMarco |
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Institution: | (1) Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 01238, USA (e-mail: demarco@math.harvard.edu), US |
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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
:λX} 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 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.
Received: 28 November 2001 / Revised version: 2 April 2002 /
Published online: 10 February 2003 |
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Keywords: | |
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