Pennsylvania, Philadelphia, PA 19104'
- 'Department of Electrical Engineering and Computer Science, Ecology and Evolutionary Biology Department, Robotics Institute, University of Michigan, Ann Arbor, MI 48109'
author:
- 'Matthew D. Kvalheim'
- Shai Revzen
bibliography:
- 'ref.bib'
title: Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits
---

Introduction
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In this paper, we consider $C^1$ dynamical systems $\Phi\colon Q\times {\mathbb{T}}\to Q$ having a globally attracting hyperbolic fixed point or periodic orbit. Here $Q$ is a smooth manifold and either ${\mathbb{T}}= {\mathbb{Z}}$ or ${\mathbb{T}}= {\mathbb{R}}$; when ${\mathbb{T}}= {\mathbb{R}}$, a common example is that of $t\mapsto \Phi^t(x_0)$ being the solution to the initial value problem $$\frac{d}{dt}x(t) = f(x(t)),\qquad x(0) = x_0$$ determined by a complete $C^1$ vector field $f$ on $Q$. Our main contributions are existence and uniqueness results regarding globally defined ${C_{\textnormal{loc}}^{k,\alpha}}$ linearizing semiconjugacies $\psi\colon Q\to {\mathbb{C}}^m$ which, by definition, make the diagram $$\label{eq:cd-gen-efunc}
    \begin{tikzcd}
    &Q \arrow{r}{\Phi^t}\arrow{d}{\psi}&Q\arrow{d}{\psi}\\
    &{\mathbb{C}}^m\arrow{r}{e^{tA}}&{\mathbb{C}}^m
    \end{tikzcd}$$ commute for some $A\in {\mathbb{C}}^{m\times m}$ and all $t\in {\mathbb{T}}$. By ${C_{\textnormal{loc}}^{k,\alpha}}$ with $k\in {\mathbb{N}}_{\geq 1}$ and $0\leq \alpha \leq 1$ we mean that $\psi\in C^k(Q,{\mathbb{C}}^m)$ and that all $k$-th partial derivatives of $\psi$ are locally $\alpha$-Hölder continuous in local coordinates.

Linearizing semiconjugacies are also known 