# Dynamic mode decomposition for analytic maps

## TL;DR

This paper demonstrates that EDMD modes for analytic maps correspond to spectra of Perron-Frobenius and Koopman operators on Hardy-Hilbert spaces, clarifying spectral interpretation in complex dynamical systems.

## Contribution

It establishes a theoretical link between EDMD modes and spectral properties of operators on Hardy-Hilbert spaces for analytic maps.

## Key findings

- EDMD modes align with Perron-Frobenius and Koopman spectra
- Spectral interpretation of EDMD clarified for analytic maps
- Numerical simulations support theoretical results

## Abstract

Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps.