Generic Properties of Koopman Eigenfunctions for Stable Fixed Points and Periodic Orbits

TL;DR

This paper explores the properties and classification of Koopman eigenfunctions for stable fixed points and periodic orbits, extending results to smooth systems and improving understanding of their applications in dynamical systems analysis.

Contribution

It generalizes the classification of Koopman eigenfunctions to broad classes of smooth systems with attracting hyperbolic points or orbits, building on previous analytic results.

Findings

Koopman eigenfunctions can be completely classified for almost all smooth flows with attracting hyperbolic points or orbits.

Every smooth eigenfunction is uniquely determined by its eigenvalue up to scalar multiplication.

The work extends classical results from analytic to smooth dynamical systems.

Abstract

Our recent work established existence and uniqueness results for $\mathcal{C}^{k,\alpha}_{\text{loc}}$ globally defined linearizing semiconjugacies for $\mathcal{C}^1$ flows having a globally attracting hyperbolic fixed point or periodic orbit (Kvalheim and Revzen, 2019). Applications include (i) improvements, such as uniqueness statements, for the Sternberg linearization and Floquet normal form theorems; (ii) results concerning the existence, uniqueness, classification, and convergence of various quantities appearing in the "applied Koopmanism" literature, such as principal eigenfunctions, isostables, and Laplace averages. In this work we give an exposition of some of these results, with an emphasis on the Koopmanism applications, and consider their broadness of applicability. In particular we show that, for "almost all" $\mathcal{C}^\infty$ flows having a globally attracting hyperbolic fixed point or periodic orbit, the $\mathcal{C}^\infty$ Koopman eigenfunctions can be completely classified, generalizing a result known for analytic systems. For such systems, every $\mathcal{C}^\infty$ eigenfunction is uniquely determined by its eigenvalue modulo scalar multiplication.