Torsion points and the lattès family

We give a dynamical proof of a result of Masser and Zannier: 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)) and P_b = (b, √ b(b−1)(b−t)) are both torsion on the Legendre elliptic curve E_t = {y² = x(x−1)(x−t)}. Our 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 an arithmetic equidistribution theorem on ℙ¹. For this, we prove two statements about the degree-4 Lattès family f_t on ℙ¹: (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 Masser and Zannier, we extend their conclusion to points t of small height also for a,b ∈ ℂ(t).