  We study linearization models for continuous one-parameter semigroups of
parabolic type. In particular, we introduce new limit schemes to obtain
solutions of Abel's functional equation and to study asymptotic behavior of
such semigroups. The crucial point is that these solutions are univalent
functions convex in one direction. In a parallel direction, we find analytic
conditions which determine certain geometric properties of those functions,
such as the location of their images in either a half-plane or a strip, and
their containing either a half-plane or a strip. In the context of semigroup
theory these geometric questions may be interpreted as follows: is a given
one-parameter continuous semigroup either an outer or an inner conjugate of a
group of automorphisms? In other words, the problem is finding a fractional
linear model of the semigroup which is defined by a group of automorphisms of
the open unit disk. Our results enable us to establish some new important
analytic and geometric characteristics of the asymptotic behavior of
one-parameter continuous semigroups of holomorphic mappings, as well as to
study the problem of existence of a backward flow invariant domain and its
geometry.
