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

Abstract

Our recent work established existence and uniqueness results for Cᵏ (actually Cᵏᵃ ₗₒ꜀ ) linearizing semiconjugacies for C flows defined on the entire basin of an 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, and (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 consider the broadness of applicability of these results with an emphasis on the Koopmanism applications. In particular we show that, for the flows of “typical” c∞ vector fields having an attracting hyperbolic fixed point or periodic orbit with a fixed basin of attraction, the c∞ Koopman eigenfunctions can be completely classified, generalizing a result known for analytic eigenfunctions of analytic systems.