Iteration at the boundary of the space of rational maps

Details Iteration at the boundary of the space of rational maps Iteration at the boundary of the space of rational maps

2004-03-03

arxiv.org/abs/math/0403078v2

Mathematics

Dynamical Systems

25 pages

Scientific paper

Let $Rat_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of degree $d\geq 2$, and $\mu_f$ the measure of maximal entropy for $f\in Rat_d$. The map of measures $f\mapsto\mu_f$ is known to be continuous on $Rat_d$, and it is shown here to extend continuously to the boundary of $Rat_d$ in $\bar{Rat}_d \simeq {\bf P}^{2d+1}$, except along a locus $I(d)$ of codimension $d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map $f\mapsto f^n$ for every $n\geq 2$. The limiting measures are given explicitly, away from $I(d)$. The degenerations of rational maps are also described in terms of metrics of non-negative curvature on the Riemann sphere: the limits are polyhedral.

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