Iteration at the boundary of the space of rational maps

Abstract

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,

Degenerations of Complex Dynamical Systems

“…Our work is very much inspired by these results. The Berkovich space viewpoint allows us to recover the results in [4], and it provides a conceptual explanation for the form of the limiting measures. In a sequel to this article, we describe a countable-state Markov process that allows one to compute the residual measure explicitly [6].…”

“…In [4], the first author obtained a version of Theorem A under an additional hypothesis. The observations of Lemma 2.1 and (the more refined result in) Lemma 2.5 concerning Möbius rescalings allow us to obtain the complete statement.…”

“…In [4], the first author showed that for 'most' degenerating sequences f k → Φ ∈ ∂Rat d , a limit of the maximal measures µ f k will exist, depending only on the limit point Φ, and it can be expressed as a countably infinite sum of atoms. (The measures µ f k themselves are atomless.)…”

“…(Compare [4,Lemma 4.1].) It remains to handle the case where z 0 ∈ η = ∂ D. By construction, the boundary γ k of U k is mapped with degree m(x) over η; and by viewing z 0 as the point ∞, we see that there must be s(x) preimages of z 0 converging to x as k → ∞.…”

A Compactification of the Space of Expanding Maps on the Circle

“…Along the way we show every f ∈ Exp d (S 1 ) has a natural measure of maximum entropy µ f , which varies continuously with f . An analogous result for rational maps and algebraic correspondences appears in [D1,Thm. 0.1].…”