Smooth Koopman eigenfunctions

Abstract

Any dynamical system, whether it is generated by a differential equation or a transformation map on a manifold, induces a dynamics on functional-spaces. The choice of functional-space may vary, but the induced dynamics is always linear, and codified by the Koopman operator. The eigenfunctions of the Koopman operator are of extreme importance in the study of the dynamics. They provide a clear distinction between the mixing and non-mixing components of the dynamics, and also reveal embedded toral rotations. The usual choice of functional-space is $L^2$, a class of square integrable functions. A fundamental problem with eigenfunctions in $L^2$ is that they are often extremely discontinuous, particularly if the system is chaotic. There are some prototypical systems called skew-product dynamics in which $L^2$ Koopman eigenfunctions are also smooth. The article shows that under general assumptions on an ergodic system, these prototypical examples are the only possibility. Moreover, the smooth eigenfunctions can be used to create a change of variables which explicitly characterizes the weakly mixing component too.