Ergodic and synthetic Koopman analyses of cat maps onto classical 2-tori

TL;DR

This paper analyzes classical cat maps on 2-tori using Koopman theory, deriving analytical formulas for Koopman modes and spectra across different dynamical regimes, including ergodic and chaotic cases.

Contribution

It introduces analytical formulas for Koopman modes and spectra for various types of cat maps, enhancing understanding of their ergodic and chaotic behaviors.

Findings

Derived explicit Koopman mode formulas for all cat map types.

Analyzed the spectrum of the Koopman operator in cyclic, quasi-cyclic, critical, and chaotic regimes.

Studied the synthetic spectrum related to ergodic decomposition.

Abstract

We study classical continuous automorphisms of the torus (cat maps) from the viewpoint of the Koopman theory. We find analytical formulae for Koopman modes defined coherently on the whole of the torus, and their decompositions associated with the partition of the torus into ergodic components. The spectrum of the Koopman operator is studied in four cases of cat maps: cyclic, quasi-cyclic, critical (transition from quasi-cyclic to chaotic behaviour) and chaotic. The synthetic spectrum associated with the ergodic decomposition is also studied.