EDMD for expanding circle maps and their complex perturbations

Abstract

We show that spectral data of the Koopman operator arising from an analytic expanding circle map $τ$ 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 $m \geq δn$, where $δ$ is an explicitly given positive number quantifying by how much $τ$ 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.