Towards Common Zeros of Iterated Morphisms

TL;DR

This paper investigates the distribution of common zeros of iterated morphisms, providing new results on their non-density in specific dynamical systems like Hénon maps and polynomial skew products.

Contribution

It extends previous work by proving non-density of common zeros for certain classes of iterated morphisms over algebraic numbers.

Findings

Confirmed non-density of common zeros for Hénon maps on 

Established non-density for endomorphisms on ()^n

Proved a Tits' alternative for semigroups generated by polynomial skew products

Abstract

Recently, the authors have proved the finiteness of common zeros of two iterated rational maps under some compositional independence assumptions. In this article, we advance towards a question of Hsia and Tucker on a Zariski non-density of common zeros of iterated morphisms on a variety. More precisely, we provide an affirmative answer in the case of H\'{e}non type maps on $\mathbb{A}^2$, endomorphisms on $(\mathbb{P}^1)^n$, and polynomial skew products on $\mathbb{A}^2$ defined over $\overline{\mathbb{Q}}$. As a by-product, we prove a Tits' alternative analogy for semigroups generated by two regular polynomial skew products.