Local geometry of trajectories of parabolic type semigroups

It is well known that the geometric nature of semigroup trajectories essentially depends on the semigroup type. 
In this work, we concentrate on parabolic type semigroups of holomorphic self-mappings of the open unit disk and of the right
half-plane, and study the structure of semigroup trajectories near the Denjoy--Wolff point.  In particular, we find the limit
order of contact and the limit curvature of trajectories and their `closeness’,  determine whether these trajectories have
asymptotes. For these purposes, we suggest that two terms in the asymptotic power expansion of semigroup generators are known.
Our methods are based on the asymptotic expansion of a semigroup  that we find on the first step. Inter alia, this enable us
to establish a new rigidity property for semigroups of parabolic type.

The talk is based on a joint work with F. Jacobzon.