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arXiv · 1510.07684

The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

Junyi Xie
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Abstract

In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$ satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose orbit under $f$ is Zariski dense in $\mathbb{A}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.

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Junyi Xie. 2015-10-26. The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane. https://doi.org/10.1112/s0010437x17007187

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