标题： CAT映射在经典和量子2环上的Koopman分析 显示英文标题

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

摘要： 我们从Koopman理论的角度研究环面（CAT映射）的经典连续自同构。 我们在整个环面上定义了相干的Koopman模式，并研究了与环面分解为遍历成分相关的分解。 研究了四种CAT映射情况下Koopman算子的谱：循环、准循环、临界（从准循环到混沌行为的过渡）和混沌。 我们将这些结果推广到定义在非交换环面（及其对偶空间）上的量子CAT映射。 最后，我们从Koopman图景的角度研究了通常的量子混沌指标在量子CAT映射上的表现。 与经典情况的类比表明，这些指标的组合实际上对于确认量子混沌行为是必要的。 显示英文摘要

摘要： 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.