Dimension of orbit versus invariant functions

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


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.

Tags:

Ag.Algebraic Geometry

Ds.Dynamical Systems

Invariant Theory

Birational Geometry

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