Dynkin operators, renormalization and the geometric β function

In this paper, I show a close connection between renormalization and a generalization of the Dynkin operator in terms of logarithmic derivations. The geometric $\beta$ function, which describes the dependence of a Quantum Field Theory on an energy scale defines is defined by a complete vector field on a Lie group $G$ defined by a QFT. It also defines a generalized Dynkin operator.