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

arXiv: https://arxiv.org/abs/2010.04008 · HTML: https://mlsanity.com/paper/2010.04008/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2020.

## Built on by (most cited first, 2 of 2)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [CKNet: A Convolutional Neural Network Based on Koopman Operator for Modeling Latent Dynamics from Pixels](https://mlsanity.com/paper/2102.10205/) (2021)

## Abstract (arXiv)

Our recent work established existence and uniqueness results for Ck,αloc globally defined linearizing semiconjugacies for C1 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" C^∞ flows having a globally attracting hyperbolic fixed point or periodic orbit, the C^∞ Koopman eigenfunctions can be completely classified, generalizing a result known for analytic systems. For such systems, every C^∞ eigenfunction is uniquely determined by its eigenvalue modulo scalar multiplication.