Mobius Inversion Formulas For Flows of Arithmetic Semi Groups

Abstract

We define a convolution-like operator which transforms functions on a space X via func tions on an arithmetical semigroup S, when there is an action or flows of S on X. This operator includes the well known classical Mobius transforms and associated inversion formulas as special cases. It is defined in a sufficiently general context so as to empha size the universal and fuctorial aspects of arithmetical Mobius inversion. We give general analytic conditions guaranteenig the existance of the transform and the validity of the corresponding inversion formulas, in terms of operators on certain function spaces. A number of examples are studied that illustrate the advantages of the convolutional point of view for obtaining new inversion formulas.