The first lemma in Lubin-Tate theory says the following:

Let $K$ be a local field, $A$ its ring of integers, and $f\in A[[T]]$ be such that $f(0) = 0$, $f'(0)$ is a uniformizer, and $f$ induces Frobenius over the residue field. Then there exists a unique formal group law $F_f(X,Y)\in A[[X,Y]]$ that makes $f$ into a formal $A$-endomorphism.

If you go over the details of the lemma, you can (I think) generalize it as follows:

If $R$ is any ring, $f\in R[[T]]$ such that $f(0) = 0$ and $f'(0)\in R^\times$ (

Edit: $u=f'(0)$ then $u^n - u\in R^\times$ for all $n$), then there exists a unique formal group law $F_f(X,Y)\in R[[X,Y]]$ that makes $f$ into a formal $R$-endomorphism.

The business about uniformizers and Frobenius in the Lubin-Tate lemma is just to ensure that everything converges on the maximal ideal of the ring of integers in the separable closure of $K$, so that you get an actual group.

So this is pretty cool---it says that you can take something purely analytic, $f$, and magically give it an algebraic structure. Specifically, the roots of the iterates $f^{(n)} = f\circ\cdots\circ f$ become a torsion $A$-module.

If the *existence* of $F_f$ generalizes like I think it does, a natural question is where does $F_f$ *converge*? I want to be able to answer the question for specific $f$, a simple example would be the following: if $R=\mathbb{C}$ and $f(z) = uz + z^2$, then what can you say about the convergence of $F_f$?

**Edit:** Okay, $\mathbb{C}$ was a bad choice, but suppose $R$ is a ring complete with respect to some $\mathfrak{a}$-adic topology. Would there be a reason not to study this case? Maybe the question I should be asking is, for what other $R$ and $f$ do people study these formal groups $F_f$?