Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?

TL;DR

This paper explores the connection between local Lyapunov exponents and Koopman eigenfunctions, explicitly constructing polynomial vector fields where the finite-time Lyapunov exponent field is a Koopman eigenfunction, bridging geometric and operator-theoretic approaches.

Contribution

It provides the first explicit construction of vector fields where the Lyapunov exponent field is a Koopman eigenfunction, linking geometric and operator-theoretic analyses of dynamical systems.

Findings

Existence of polynomial vector fields with Lyapunov exponent as Koopman eigenfunction

Explicit construction of such vector fields

Potential structural properties linking geometric and transfer operator theories

Abstract

This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis and the global, operator theoretic, Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov exponent field is a Koopman eigenfunction. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure, thus making a link between the geometric theory and the transfer operator theory.