Bounded Height in Families of Dynamical Systems

Abstract

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.