Let Ratd denote the space of holomorphic self-maps of P 1 of degree d ≥ 2, and let μf be the measure of maximal entropy for ∈ 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 Rat d ≃ PH0(P1 × P1, Φ(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 → f n 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.
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