Holomorphic Flows, Cocycles, and Coboundaries

摘要

Cocycles appear in many areas of analysis (harmonic analysis, representation theory, operator theory, ergodic theory, etc.) and, indeed, they are present whenever a group or a semigroup acts as a transformation group on some space. In a sense, cocycles are generalizations of the exponential function and provide a measure of "normality" of the underlying group action. We are primarily concerned with the semigroup action provided by a holomorphic flow on a domain in the complex plane. The properties of semigroups of holomorphic flows may be studied by replacing these semigroups by any member of a large class of isospectral operators generated from the above semigroups by certain types of cocycles called coboundaries. This motivation has led us to investigate when cocycles are coboundaries, and in doing so, we are led to a complete description of all holomorphic flows on C. Our approach and techniques are quite direct and independent of operator-theoretic considerations.