Group iteration for Abel’s functional equation

Studied is the Abel functional equation α ( f ( x ) ) = α ( x ) + 1 and its generalization α ( f ( x ) ) = g ( α ( x ) ) . Given an increasing function f , possibly having fixed points in its domain ( a , b ) , a group-theoretic iterative explicit construction is given for infinitely many solutions α which are infinite at fixed points of f and otherwise monotonic. The group-theoretic structure is suitable for studying solution properties of Abel functional equations. The methods apply in particular to Abel functional equations for which the domain ( a , b ) is a finite interval, a half-line or the real line with f possibly having many fixed points.