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$\def\CC{\mathbb{C}}$Let $K = \CC(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$; in the cases I care about, $G \cong \mathbb{Z}$. Then the field of $G$-invariants, $K^G$, is an extension of $\CC$ of some transendence degree $d$. I would like to know under what circumstances I can say that the Zariski closure of the $G$-orbit of a generic $n$-tuple has dimension $n-d$.

I want to be a little sloppy about what I mean by generic, but I don't think there is anything deep in that issue. For "generic" $x \in \CC^n$, all the rational maps $g$ are defined at $x$, so we can consider $\{ g(x) \}_{g \in G} \subset \CC^n$ and take its Zariski closure. Every function in $K^G$ is constant on this Zariski closure, so the Zariski closure has dimension $\leq n-d$.

Question Can I conclude that the dimension is "generically" $n-d$?

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Yes for $G=\mathbb Z$, see Theorem 4.1 of this paper by Amerik-Campana.

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  • $\begingroup$ Thanks! Following up on your answer, I have found a paper which does the general case; see my answer. $\endgroup$ – David E Speyer Mar 1 at 5:24
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Following citation links from the paper Jorge directed me to, the case of general $G$ is done in

Bell, Jason; Ghioca, Dragos; Reichstein, Zinovy, On a dynamical version of a theorem of Rosenlicht, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 1, 187-204 (2017). ZBL1401.14071. (arXiv link)

Disclaimer -- I have only read the MR review.

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