Models and Koenigs Functions

Abstract

In this chapter we construct models for semigroups in $\mathbb {D}$. Starting from a given semigroup, the basic idea is to define an abstract space (the space of orbits of a semigroup or abstract basin of attraction) which inherits a complex structure of simply connected Riemann surface, in such a way that the semigroup is conjugated to a continuous group of automorphisms of such a Riemann surface. Moreover, our construction is universal, which implies that all possible (semi-)conjugations of the semigroup factorize through the model. Also, the model respects basic properties of the semigroup, in particular, the divergence rate, which is a measure in the hyperbolic distance of the rate of convergence to the Denjoy-Wolff point. After having defined and discussed models (and the more general concept of semi-models), we concentrate in studying the canonical models, where it makes its appearance the second main subject of our study: the Koenigs function of a semigroup, which intertwines the semigroup to a very simply group of automorphisms of either $\mathbb C$, or of a half-plane or of a strip, depending on the properties of the starting semigroup. Those Koenigs functions are univalent functions which are either spirallike or starlike at infinity, and we spend some time in understanding properties of those maps. The upshot is to translate dynamical properties of a semigroup into geometrical properties of the image of the associated Koenigs function. We will also study in details models for semigroups of linear fractional maps and non-canonical semi-models with base space $\mathbb {D}$ and $\mathbb C$ and we see how conjugation can be naturally reads through models. We end up the chapter by considering topological models for semigroups, showing that, from a topological point of view, there are only three possible models: the model of hyperbolic rotations, the model of elliptic semigroups (which are not groups) with spectral value 1 and the model of hyperbolic semigroups with spectral value $\pi$.