Ritt's theory on the unit disk

Abstract

The aim of this paper is to revisit Ritt's theory from a topological perspective by extensively using the concept of fundamental groups. This enables us to regard the theory as an example which illustrates many ideas of a letter of Grothendieck and to put Ritt's theory into a more general analytic setting. In particular, Ritt's theory on the unit disk will be carefully developed and a special class of finite Blaschke products will be introduced as the counterpart of Chebyshev polynomials in Ritt's theory. These finite Blaschke products will be shown to be closely related to the elliptic rational functions, which are of great importance in the filter design theory.