Let $R$ be a integral domain and $\phi$ be 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.

Any suggestions and comments are welcome.

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I want to know about the derivation, so I edit the question.

Let $R$ be an algebra not just a ring. In fact I wonder only when $R$ is a polynomial ring, which is an $k$-algebra, and $\phi$ is not a ring homomorphism, but just a derivation. (Which is an $k$-module homomorphism).

Or more generally, $R$:integral domain, and $\phi$ is a derivation (satisfying Leibniz rule and group homomorphism).