Laura DeMarco

Preperiodic points and unlikely intersections

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of one-dimensional complex dynamical systems. We show that for any fixed a, b ∈ C, and any integer d ≥ 2, the set of c ∈ C for which both a and b are preperiodic for z + c is infinite if and only if a = b. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions φ,ψ ∈ C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and in particular the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that φ and ψ are defined over Q. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with non-archimedean Berkovich spaces playing an essential role.

The moduli space of quadratic rational maps

Let be the space of quadratic rational maps , modulo the action by conjugation of the group of Mobius transformations. In this paper a compactification of is defined, as a modification of Milnors , by choosing representatives of a conjugacy class such that the measure of maximal entropy of has conformal barycenter at the origin in and taking the closure in the space of probability measures. It is shown that is the smallest compactification of such that all iterate maps extend continuously to , where is the natural compactification of coming from geometric invariant theory.

Iteration at the boundary of the space of rational maps

Let Ratd denote the space of holomorphic self-maps of P of degree d ≥ 2, and let μf be the measure of maximal entropy for f ∈ 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 Ratd P H(P1 × P1,O(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. 0. Introduction For each integer d ≥ 1, let Ratd denote the space of holomorphic maps f : P1 → P1 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. Parametrizing the space Ratd by the coefficients of P and Q, we have Ratd P2d+1\V (Res), where V (Res) is the hypersurface of polynomial pairs (P,Q) for which the resultant vanishes. In particular, Ratd is smooth and affine. In this article, we aim to describe the possible limiting behavior of a sequence of rational maps which diverges in Ratd 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 dynamically natural compactification of this space or a boundary of the moduli space Ratd/PSL2 C, where the group of Mobius transformations acts by conjugation on Ratd . A compactification of the moduli space has been studied by Milnor [Mi] and Epstein [E] in degree 2 and Silverman [S] in all DUKE MATHEMATICAL JOURNAL Vol. 130, No. 1, c

Axiom A polynomial skew products of 2 and their postcritical sets

A polynomial skew product of C is a map of the form f(z, w) = (p(z), q(z, w)), where p and q are polynomials, such that f extends holomorphically to an endomorphism of P of degree ≥ 2. For polynomial maps of C, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson ([Jon99]) established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present an analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.

The conformal geometry of billiards

This article provides an introduction to some recent results in billiard dynamics. We present results that follow from a study of compact Riemann surfaces (equipped with a holomorphic 1-form) and an SL2R action on the moduli spaces of these surfaces. We concentrate on the progress toward classification of “optimal” billiard tables, those with the simplest trajectory structure.

Transfinite diameter and the resultant

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E ⊂ℂ N by a regular polynomial map F, expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F.

Degenerations of complex dynamical systems II: analytic and algebraic stability: With an appendix by Jan Kiwi

We study pairs

On the postcritical set of a rational map

(f, \Gamma )