Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \subset k\{x_1,..,x_n\}$.

The algebraic power series are in some sense "controlled" by the ring polynomials. (e.g. because they satisfy polynomial equations). Is there some way "to control" the analytic power series by the algebraic ones?

e.g. given a morphism of two "large" rings, $R\stackrel{\phi}{\to}S$, each containing subrings of algebraic/analytic power series, suppose I know how $\phi$ acts on polynomials (and thus on the algebraic power series). Is there any way to determine how $\phi$ acts on the analytic power series? (Here $\phi$ is just an algebraic morphism, without any assumption of continuity in the classical topology, coming from the topology on $k$.)

In the extreme case: suppose an automorphism $\phi\circlearrowright R$ is identity on the subring of algebraic power series. Is $\phi$ identity on the analytic power series? At least, does $\phi$ send analytic to analytic?

(Here $R$ can be non-Noetherian, e.g. $C^\infty(\Bbb{R}^1,0)$.)