Koopman analysis of CAT maps onto classical and quantum 2-tori

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. We generalise these results to quantum CAT maps defined onto a noncommutative torus (and on its dual space). Finally, we study usual quantum chaos indicators onto quantum CAT maps from the viewpoint of the Koopman picture. The analogy with the classical case suggests that couples of these indicators are in fact necessary to certify a quantum chaotic behaviour.