Counting points by height in semigroup orbits

TL;DR

This paper refines estimates for counting points of bounded height in semigroup orbits of dynamical systems, providing exact asymptotics for generic cases on the projective line using advanced Tauberian theorems.

Contribution

It introduces a novel application of the Wiener-Ikehara Tauberian theorem to count functions in semigroups of bounded degree, improving existing bounds.

Findings

Exact asymptotics for semigroup actions on the projective line

Enhanced estimates for points of bounded height in dynamical systems

Application of Tauberian theorems to dynamical counting problems

Abstract

We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient is the Wiener-Ikehara Tauberian theorem, which we use to count functions in semigroups of bounded degree.