TY - JOUR
T1 - Iteration at the boundary of the space of rational maps
AU - Demarco, Laura
PY - 2005/10/1
Y1 - 2005/10/1
N2 - 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.
AB - 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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U2 - 10.1215/S0012-7094-05-13015-0
DO - 10.1215/S0012-7094-05-13015-0
M3 - Article
AN - SCOPUS:28444452257
SN - 0012-7094
VL - 130
SP - 169
EP - 197
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 1
ER -