  In these lecture notes, we present a connection between the complex dynamics
of a family of rational functions $f_t: \mathbb{P}^1\to \mathbb{P}^1$,
parameterized by $t$ in a Riemann surface $X$, and the arithmetic dynamics of
$f_t$ on rational points $\mathbb{P}^1(k)$ where $k = \mathbb{C}(X)$ or
$\bar{\mathbb{Q}}(X)$. An explicit relation between stability and canonical
height is explained, with a proof that contains a piece of the Mordell-Weil
theorem for elliptic curves over function fields. Our main goal is to pose some
questions and conjectures about these families, guided by the principle of
"unlikely intersections" from arithmetic geometry, as in [Zannier 2012]. We
also include a proof that the hyperbolic postcritically-finite maps are Zariski
dense in the moduli space of rational maps of any given degree $d>1$. These
notes are based on four lectures at KAWA 2015, in Pisa, Italy, designed for an
audience specializing in complex analysis, expanding upon the main results of
[Baker-DeMarco 2013, DeMarco 2016, DeMarco-Wang-Ye 2016].
