What are the entire functions that commute with the exponential function? I am looking for entire functions $f(z)$ that satisfy $f(\exp(z))=\exp(f(z))$ on $\mathbb{C}$. I see that this is satisfied by $e^z$, $z$, and constant functions where the constant is a solution of $e^c=c$. Are there any other entire functions that commute with the exponential?
 A: if $f(z)$ is any function that satisfies your relation, so does
$$ g(z) = f(e^z), $$
so you can make more.
The most complete treatment of your question that i know is BAKER 1958 but it certainly does not give a complete answer to your question. Not sure it is published anywhere, but Baker is the right author to begin with. He did move from entire functions to problems around fixed points; some of this material was completed much later, and independently by Ecalle. The best book for the vocabulary necessary is Milnor, Dynamics in One Complex Variable. The big question is this: when your function has a fixpoint, is the derivative at that point $0$ (bad) or of magnitude $1$ (worse)? If not, you may be able to continue with entire holomorphic functions. 
Should point out that commutativity is how Noel Baker phrased his study of, for example, fractional iteration of holomorphic functions. See, for example, Iterative Functional Equations by Kuczma, Choczewski, and Ger. Interesting, Hellmuth Kneser was Baker's adviser. H. Kneser constructed a real analytic function on the real line with $h(h(x)) = e^x;$ this cannot be extended to the entire complex plane.  
