Iteration at the boundary of the space of rational maps

الأصل · EN

Let Ratd denote the space of holomorphic self-maps of P¹ of degree d≥ 2, and μf the measure of maximal entropy for f∈ Ratd. The map of measures fμf is known to be continuous on Ratd, and it is shown here to extend continuously to the boundary of Ratd in Ratd ≃ 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 non-negative curvature on the Riemann sphere: the limits are polyhedral.