Abel equation

The Abel equation, named after Niels Henrik Abel, is a type of functional equation which can be written in the form
or, equivalently, 
and controls the iteration of   .
Equivalence.
These equations are equivalent. Assuming that is an invertible function, the second equation can be written as
Taking , the equation can be written as
For a function assumed to be known, the task is to solve the functional equation for the function , possibly satisfying additional requirements, such as .
The change of variables , for a real parameter , brings Abel's equation into the celebrated Schröder's equation, .
The further change into Böttcher's equation, .
The Abel equation is a special case of (and easily generalizes to) the translation equation,
e.g., for formula_6, 
The Abel function further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).
History.
Initially, the equation in the more general form
was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.
In the case of a linear transfer function, the solution is expressible compactly. 
Special cases.
The equation of tetration is a special case of Abel's equation, with .
In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,
and so on,
Solutions.
Fatou coordinates describe local dynamics of discrete dynamical system near a parabolic fixed point.