Orbit counting in polarized dynamical systems

TL;DR

This paper extends orbit counting methods from finitely generated to certain infinitely generated polarized semigroups acting on projective varieties, with applications to orbits generated by infinite sets of unicritical polynomials.

Contribution

It introduces new orbit counting techniques for infinitely generated polarized semigroups on projective varieties, expanding prior finite-generation results.

Findings

Extended orbit counts to infinitely generated semigroups

Applied results to unicritical polynomial semigroup orbits

Provided new bounds and asymptotic formulas

Abstract

We extend recent orbit counts for finitely generated semigroups acting on $\mathbb{P}^N$ to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.