I will give a brief introduction to canonical heights (Weil and Moriwaki), using the Tate telescoping limit. I will use this along with a variant of the Klein-Tits ping-pong lemma to prove a result about the structure of semigroups generated by two rational functions that do not share the same set of preperiodic points.

The talk should be mostly fairly elementary, modulo some technical facts about Weil and Moriwaki heights (which will not be proved).