Dynamics of finite Blaschke products

Abstract

[en] The aim of this project is to characterise the dynamics of finite Blaschke products, which are precisely the proper maps of the unit disk. It is proven that, inside the unit disk, all points converge to a unique point, the Wolff-Denjoy point. We build a classification of finite Blaschke products according to the position of the Wolff-Denjoy point and the dynamics around it.

Finally, we study the restriction of finite Blaschke products to the unit circle and calculate explicitly a conjugacy to $z^d$. We end this work by showing a brief example of generalised Blaschke products, a nuanced variation of the previous family that presents rich dynamical phenomena, such as the emergence of Herman rings.