17. Slopes of Orbits at the Denjoy-Wolff Point

In this chapter we consider the slope of orbits of non-elliptic semigroups when converging to the Denjoy-Wolff point. In other words, we study the possible angles of approach of the trajectories of a semigroup toward its Denjoy-Wolff point. We show that the angle of approach of the orbits of a hyperbolic semigroup is a harmonic function whose level sets are exactly the maximal invariant curves of the semigroup and whose range is \((-\pi /2,\pi /2)\). While, the orbits of a parabolic semigroup of positive hyperbolic step always converge tangentially to the Denjoy-Wolff point. The situation becomes more complicated for parabolic semigroups of zero hyperbolic step. In such a case all orbits of the same semigroup have the same slopes, but an orbit can behave in all possible ways: it can converge tangentially, non-tangentially, have some subsequence which converges tangentially and some non-tangentially. Moreover, using harmonic measure theory, for any closed interval I of \([-\pi /2,\pi /2]\) we construct an example of a parabolic semigroup of zero hyperbolic step whose slope is I. We will show however how to detect the type of convergence from the image of the Koenigs function of a semigroup: for instance, the convergence is non-tangential if and only if the image of the Koenigs function is “quasi-symmetric” with respect to vertical axes. These types of results are based on the construction of suitable quasi-geodesics in starlike domains at infinity and on suitable estimates of the hyperbolic distance.