Let $ R$ be a integral domain and $ \phi$ be a an automorphism of $ R$ . For a given element $ x \in R$ , we consider a sequence $ (\phi^n(x))_{n=0}^{\infty}$ .

I wonder if there is any related theory to determine when $ \phi^n(x)$ is irreducible for all $ n \in \mathbb Z_{\geq 0}$ . It depends on $ x$ and $ \phi$ of course.

More precisely, I want to know about a specific case, not so general: $ R=k[x_1,\ldots,x_n]$ , where $ k$ is a field, and $ \phi=D_R$ is a derivation on $ R$ (which means, $ \phi$ satisfies $ \phi(xy)=x\phi(y)+y\phi(x)$ , the Leibniz rule).

Any suggestions and comments are welcome.