Bounded Height in Families of Dynamical Systems

Abstract

Abstract Let $a,b\in\overline{\mathbb{Q}}$ 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 $t\in\overline{\mathbb{Q}}$. We prove that the set of all $t\in\overline{\mathbb{Q}}$ for which there exist $m,n\geq 0$ such that $f_t^m(a)=f_t^n(b)$ has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics.