### Abstract

Let Rat_{d} denote the space of holomorphic self-maps of P ^{1} of degree d ≥ 2, and let μ_{f} be the measure of maximal entropy for ∈ Rat_{d}. The map of measures f → μ_{f} is known to be continuous on Rat_{d}, and it is shown here to extend continuously to the boundary of Rat_{d} in Rat _{d} ≃ PH^{0}(P^{1} × P^{1}, Φ(d, 1)) ≃ 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 → 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.

Original language | English (US) |
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Pages (from-to) | 169-197 |

Number of pages | 29 |

Journal | Duke Mathematical Journal |

Volume | 130 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2005 |

### ASJC Scopus subject areas

- Mathematics(all)