Group iteration for Abel’s functional equation

Studied is the Abel functional equation α(f(x))=α(x)+1α(f(x))=α(x)+1 and its generalization α(f(x))=g(α(x))α(f(x))=g(α(x)). Given an increasing function ff, possibly having fixed points in its domain (a,b)(a,b), a group-theoretic iterative explicit construction is given for infinitely many solutions αα which are infinite at fixed points of ff 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)(a,b) is a finite interval, a half-line or the real line with ff possibly having many fixed points.