We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on P 1 (Q). We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over C. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over C, and a uniform Bogomolov bound for the family over Q.Date: December 4, 2019. 1 Heights, measures, and energiesThis section develops the background and notation needed for the proofs that follow. Throughout, K is a number field and M K its set of places.

We prove the Dynamical Bogomolov Conjecture for endomorphisms Φ : P 1 ×P 1 −→ P 1 ×P 1 , where Φ(x, y) := (f (x), g(y)) for any rational functions f and g defined overQ. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we can prove the Dynamical Manin-Mumford Conjecture for endomorhisms Φ = (f, g) of P 1 × P 1 , where f and g are rational functions defined over an arbitrary field of characteristic 0.2010 Mathematics Subject Classification. Primary: 37P05. Secondary: 37P30. Key words and phrases. Dynamical Manin-Mumford Conjecture, equidistribution of points of small height, symmetries of the Julia set of a rational function.

Abstract. We study critical orbits and bifurcations within the moduli space M 2 of quadratic rational maps, f : P 1 → P 1 . We focus on the family of curves, Per 1 (λ) ⊂ M 2 for λ ∈ C, defined by the condition that each f ∈ Per 1 (λ) has a fixed point of multiplier λ. We prove that the curve Per 1 (λ) contains infinitely many postcritically-finite maps if and only if λ = 0, addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of f define distinct bifurcation measures along Per 1 (λ).

In the moduli space MP d of degree d polynomials, special subvarieties are those cut out by critical orbit relations, and then special points are the post-critically finite polynomials. It was conjectured that in MP d , subvarieties containing a Zariskidense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case d = 3.2010 Mathematics Subject Classification. Primary 37F50; Secondary 37F05.

We give a dynamical proof of a result of Masser and Zannier [MZ2, MZ3]: for any a = b ∈ Q \ {0, 1}, there are only finitely many parameters t ∈ C for which points P a = (a, a(a − 1)(a − t)) andOur method also gives the finiteness of parameters t where both P a and P b have small Néron-Tate height. A key ingredient in the proof is the equidistribution theorem of [BR1, FRL, CL]. For this, we prove two statements about the degree-4 Lattès family f t on P 1 : (1) for each c ∈ C(t), the bifurcation measure µ c for the pair (f t , c) has continuous potential across the singular parameters t = 0, 1, ∞; and (2) for distinct points a, b ∈ C \ {0, 1}, the bifurcation measures µ a and µ b cannot coincide. Combining our methods with the result of [MZ3], we extend their conclusion to points t of small height also for a, b ∈ C(t).1.1. Dynamical and potential-theoretic statements. The Lattès map f t is postcritically finite for all t, meaning that each of its critical points has finite forward orbit, with postcritical set equal to {0, 1, t, ∞}. We consider marked points c that are rational functions of t (or constant ∞), viewed as holomorphic maps from the parameter space C \ {0, 1} to the dynamical space P 1 . We begin with an observation that the marked points in the postcritical set are the only rational points that are preperiodic for all t.Remark. In terms of the Legendre family of elliptic curves E t , Proposition 1.4 states that P c = (c, c(c − 1)(c − t)) is torsion for all t ∈ C \ {0, 1} if and only if c = 0, 1, t, ∞. This could also be deduced from more traditional methods in the study of elliptic curves. Proposition 3.1 provides a more precise statement, computing the degrees of t → f n t (c(t)) for all n ≥ 1 and all c ∈ C(t).