EDMD for expanding circle maps and their complex perturbations

TL;DR

This paper demonstrates an effective EDMD-based algorithm for computing the spectral data of Koopman operators associated with analytic expanding circle maps and their complex perturbations, with proven exponential convergence.

Contribution

It introduces a novel combination of collocation and Galerkin methods for spectral approximation of Koopman operators on expanding circle maps, with explicit convergence conditions.

Findings

Spectral data can be computed efficiently using the proposed method.

Convergence is exponential when the collocation order exceeds a multiple of the Galerkin order.

Results extend to general expansive maps on annuli containing the unit circle.

Abstract

We show that spectral data of the Koopman operator arising from an analytic expanding circle map $\tau$ can be effectively calculated using an EDMD-type algorithm combining a collocation method of order m with a Galerkin method of order n. The main result is that if $m \geq \delta n$, where $\delta$ is an explicitly given positive number quantifying by how much $\tau$ expands concentric annuli containing the unit circle, then the method converges and approximates the spectrum of the Koopman operator, taken to be acting on a space of analytic hyperfunctions, exponentially fast in n. Additionally, these results extend to more general expansive maps on suitable annuli containing the unit circle.