Bounded Height in Families of Dynamical Systems

Let a, b ∈ $\bar{\mathbb{Q}}$ be such that exactly one of a and b is an algebraic integer, and let f$_{t}$(z) := z$^{2}$ + t be a family of polynomials parameterized by t ∈ $\bar{\mathbb{Q}}$. We prove that the set of all t ∈ $\bar{\mathbb{Q}}$ for which there exist m, n ≥ 0 such that f$^{m}_{t}$(a) = f$^{n}_{t}$(b) has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics.L.D. was partially supported by National Science Foundation grants DMS-1517080 and DMS-1600718. D.G. was partially supported by a Discovery grant from the National Sciences and Engineering Research Council of Canada. H.K. was partially supported by National Science Foundation grant DMS-1303770. K.N. was partially supported by a fellowship from the Pacific Institute for the Mathematical Sciences and T.T. was partially supported by National Science Foundation grant DMS-1200749