Let $f,g:\mathbb{C} \to \mathbb{C}$ be holomorphic and have the property $f^{-1}(S)=g^{-1}(S)$ where S is the unit circle centered at 0. What can be said about $f$ and $g$.



Holomorphic functions with equal inverse images of unit circle
An exhaustive study of this question is contained in the paper


MR0825840
Stephenson, Kenneth
Analytic functions sharing level curves and tracts. 
Ann. of Math. (2) 123 (1986), no. 1, 107–144 (freely available online).


