Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits

Matthew D. Kvalheim, Shai Revzen

Published 2019-11-27Version 1

We consider $C^1$ dynamical systems having a globally attracting hyperbolic fixed point or periodic orbit and prove existence and uniqueness results for $C^{k,\alpha}_{\text{loc}}$ globally defined linearizing semiconjugacies, of which Koopman eigenfunctions are a special case. Our main results both generalize and sharpen Sternberg's $C^k$ linearization theorem for hyperbolic sinks, and in particular our corollaries include uniqueness statements for Sternberg linearizations and Floquet normal forms. Additional corollaries include existence and uniqueness results for $C^{k,\alpha}_{\text{loc}}$ Koopman eigenfunctions, including a complete classification of $C^\infty$ eigenfunctions assuming a $C^\infty$ dynamical system with semisimple and nonresonant linearization. We give an intrinsic definition of "principal Koopman eigenfunctions" which generalizes the definition of Mohr and Mezi\'{c} for linear systems, and which includes the notions of "isostables" and "isostable coordinates" appearing in work by Ermentrout, Mauroy, Mezi\'{c}, Moehlis, Wilson, and others. Our main results yield existence and uniqueness theorems for the principal eigenfunctions and isostable coordinates and also show, e.g., that the (a priori non-unique) "pullback algebra" defined in \cite{mohr2016koopman} is unique under certain conditions. We also discuss the limit used to define the "faster" isostable coordinates in \cite{wilson2018greater,monga2019phase} in light of our main results.