Koenigs (Linearization) Theorem

If the magnitude (absolute value) of the multiplier of a holomorphic map f(t) at the point t=0 is not strictly equal to 0 or 1( that is, not super-attractive or indifferent) then a local holomorphic change of coordinates w = φ(z), called the Koenig's function, unique up to a scalar multiplication by nonzero constant, exists, having a fixed-point at the origin φ(0) = 0 such that Schröder's equation f(φ(x))=λφ(x) is true for some neighborhood of the origin.

See:

• John Milnor. Dynamics in One Complex Variable. Annals of Mathematics Studies 160. Princeton University Press, 2ndedition, 2006
• J. H. Shapiro. Composition operators and schroder's functional equation. Contemporary Mathematics, (213):213-228, 1998.
• D.S. Alexander, F. Iavernaro, and A. Rosa. Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906-1942. History of mathematics. American Mathematical Society, 2012.