On the approximation of Koopman spectra for measure preserving transformations

TL;DR

This paper introduces a finite-dimensional approximation method for Koopman operators associated with measure-preserving automorphisms, enabling spectral analysis through permutation operators and efficient numerical algorithms.

Contribution

It proposes a novel spectral approximation technique using permutation-based discretizations and provides an efficient computational framework for volume-preserving maps.

Findings

Spectral measures and projectors converge weakly to their infinite-dimensional counterparts.

The method effectively computes spectra for automorphisms on the torus, including the Chirikov standard map.

Discretizations can be computed with reduced complexity leveraging FFT algorithms.

Abstract

For the class of continuous, measure-preserving automorphisms on compact metric spaces, a procedure is proposed for constructing a sequence of finite-dimensional approximations to the associated Koopman operator on a Hilbert space. These finite-dimensional approximations are obtained from the so-called "periodic approximation" of the underlying automorphism and take the form of permutation operators. Results are established on how these discretizations approximate the Koopman operator spectrally. Specificaly, it is shown that both the spectral measure and the spectral projectors of these permutation operators converge weakly to their infinite dimensional counterparts. Based on this result, a numerical method is derived for computing the spectra of volume-preserving maps on the unit $m$-torus. The discretized Koopman operator can be constructed from solving a bipartite matching problem with $\mathcal{O}(\tilde{n}^{3m/2})$ time-complexity, where $\tilde{n}$ denotes the gridsize on each dimension. By exploiting the permutation structure of the discretized Koopman operator, it is further shown that the projections and density functions are computable in $\mathcal{O}(m \tilde{n}^{m} \log \tilde{n})$ operations using the FFT algorithm. Our method is illustrated on several classical examples of automorphisms on the torus that contain either a discrete, continuous, or a mixed spectra. In addition, the spectral properties of the Chirikov standard map are examined using our method.