Density of orbits of endomorphisms of commutative linear algebraic groups

Dragos Ghioca; Fei Hu

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arXiv · 1803.03928

Density of orbits of endomorphisms of commutative linear algebraic groups

Dragos Ghioca·Fei Hu

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Abstract

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon G\longrightarrow G$ is a dominant endomorphism, we prove that one of the following holds: either there exists a non-constant rational function $f\in \mathbb{k}(G)$ preserved by $\Phi$ (i.e., $f\circ \Phi = f$), or there exists a point $x\in G(\mathbb{k})$ whose $\Phi$-orbit is Zariski dense in $G$.

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Dragos Ghioca, Fei Hu. 2018-03-11. Density of orbits of endomorphisms of commutative linear algebraic groups. https://arxiv.org/abs/1803.03928

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