Iteration at the boundary of the space of rational maps

Abstract

Let Ratd denote the space of holomorphic self-maps of P1 of degree d≥2, and let μf be the measure of maximal entropy for f∈Ratd. The map of measures f→μf is known to be continuous on Ratd, and it is shown here to extend continuously to the boundary of Ratd in Rat̲d≃PH0(P1×P1,O(d,1))≃P2d+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→fn for every n≥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