Iteration at the boundary of the space of rational maps
Abstract
Let $Rat_d$ denote the space of holomorphic selfmaps 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 nonnegative curvature on the Riemann sphere: the limits are polyhedral.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403078
 Bibcode:
 2004math......3078D
 Keywords:

 Dynamical Systems;
 Complex Variables
 EPrint:
 25 pages