# Dimension of orbit versus invariant functions

$$\def\CC{\mathbb{C}}$$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" $$n-d$$?

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

• Thanks! Following up on your answer, I have found a paper which does the general case; see my answer. – David E Speyer Mar 1 at 5:24

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.