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Abstract. Let Rat d denote the space of holomorphic self-maps of P 1 of degree d ≥ 2, and µ f the measure of maximal entropy for f ∈ 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 ≃ 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 non-negative curvature on the Riemann sphere: the limits are polyhedral.For each integer d ≥ 1, let Rat d denote the space of holomorphic maps f : P 1 → P 1 of degree d with the topology of uniform convergence. Fixing a coordinate system on the projective line, each such map can be expressed as a ratio of homogeneous polynomials f (z : w) = (P (z, w) : Q(z, w)), where P and Q have no common factors and are both of degree d. Parameterizing the space Rat d by the coefficients of P and Q, we havewhere V (Res) is the hypersurface of polynomial pairs (P, Q) for which the resultant vanishes. In particular, Rat d is smooth and affine. In this paper, we aim to describe the possible limiting behavior of a sequence of rational maps which diverges in Rat d , for each d ≥ 2, in terms of the measures of maximal entropy and corresponding conformal metrics on the Riemann sphere. This is the first step in describing a natural compactification of this space, or a boundary of the moduli space Rat d / PSL 2 C, which is well-behaved under iteration. A compactification of the moduli space has been studied by Milnor [Mi] and Epstein [Ep] in degree 2 and Silverman [Si] in all degrees, but iteration does not extend continuously to this boundary, as first seen in [Ep]. See [De2] for more details.We can associate to every point in Rat d ≃ P 2d+1 a self-map of the Riemann sphere of degree ≤ d, together with a finite set of marked points. Namely, each f ∈ Rat d determines the coefficients for a pair of homogeneous polynomials, defining a map on P 1 away from finitely many holes,

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In this paper we study branched coverings of metrized, simplicial trees F : T → T which arise from polynomial maps f : C → C with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space PT D compactifying the moduli space of polynomials of degree D; that F records the asymptotic behavior of the multipliers of f ; and that any meromorphic family of polynomials over ∆ * can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial enumeration of the trees that arise, and show that PT 3 is itself a tree. ResuméDans ce travail, nous étudions des revêtements ramifiés d'arbres métriques simpliciaux F : T → T qui sont obtenus à partir d'applications polynômiales f : C → C possédant un ensemble de Julia non connexe. Nous montrons que la collection de tous ces arbres, à un facteur d'échelle près, forme un espace contractile PT D qui compactifier l'espace des modules des polynômes de degré D. Nous montrons aussi que F enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille méromorphe de polynômes définis sur ∆ * peut être complétée par un unique arbre comme sa fibre centrale. Dans le cas cubique, nous donnons une énumération combinatoire des arbres ainsi obtenus et montrons que PT 3 est lui-même un arbre.