Iteration at the boundary of the space of rational maps

Abstract

Let ${\displaystyle {\mathrm{Rat}}_d}$ denote the space of holomorphic self-maps of $P^1$ of degree $d\ge 2$, and let ${\mu }_f$ be the measure of maximal entropy for $f\in {\mathrm{Rat}}_d$. The map of measures $f\to {\mu }_f$ is known to be continuous on ${\mathrm{Rat}}_d$, and it is shown here to extend continuously to the boundary of ${\mathrm{Rat}}_d$ in ${\stackrel{̲}{\mathrm{Rat}}}_d\simeq P{H}^0(P^1\times P^1,O(d,1))\simeq P^{2d+1}$, except along a locus $I(d)$ of codimension $d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map $f\to f^n$ for every $n\ge 2$. 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