Counting points of bounded height in monoid orbits

Abstract

Given a set of endomorphisms on ${\mathbb{P}}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.

Main result strengthened to generic sets of rational functions. An appendix by Umberto Zannier is included