Torsion points and the Lattès family

We give a dynamical proof of a result of Masser and Zannier: for any a not equal b is an element of (Q) over bar\{0,1}, there are only finitely many parameters t is an element of C for which points P-a = (a, root a(a-1)(a-t)) and P-b = (b, root b(b-1)(b-t)) are both torsion on the Legendre elliptic curve E-t = {y(2) = x(x-1)(x-t)}. Our method also gives the finiteness of parameters t where both P-a and P-b have small Neron-Tate height. A key ingredient in the proof is an arithmetic equidistribution theorem on P-1. For this, we prove two statements about the degree-4 Lattes family f(t) on P-1: (1) for each c is an element of C(t), the bifurcation measure mu(c) for the pair (f(t), c) has continuous potential across the singular parameters t = 0,1, infinity; and (2) for distinct points a, b is an element of C \ {0,1}, the bifurcation measures mu(a) and mu(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 is an element of C(t).