Lie group valued Koopman eigenfunctions

TL;DR

This paper generalizes Koopman eigenfunctions to Lie group-valued functions, introducing a geometric framework and a Lie group exterior derivative to analyze their dynamical properties.

Contribution

It develops a new geometric approach for Lie group-valued Koopman eigenfunctions and introduces a Lie group exterior derivative to study their properties.

Findings

Lie group valued eigenfunctions generalize classical eigenfunctions

The geometric approach reveals properties like behavior under time-rescaling

Introduction of a Lie group exterior derivative for analysis

Abstract

Every continuous-time flow on a topological space has associated to it a Koopman operator, which operates by time-shifts on various spaces of functions, such as $C^r$, $L^2$, or functions of bounded variation. An eigenfunction of the vector field (and thus for the Koopman operator) can be viewed as an $S^1$-valued function, which also plays the role of a semiconjugacy to a rigid rotation on $S^1$. This notion of Koopman eigenfunctions will be generalized to Lie-group valued eigenfunctions, and we will discuss the dynamical aspects of these functions. One of the tools that will be developed to aid the discussion, is a concept of exterior derivative for Lie group valued functions, which generalizes the notion of the differential $df$ of a real valued function $f$. The extended notion of Koopman eigenfunctions utilizes a geometric property of usual eigenfunctions. We show that the generalization in a geometric sense can be used to reveal fundamental properties of usual Koopman eigenfunctions, such as their behavior under time-rescaling, and as submersions.